Algebraic Curves: Towards Moduli Spaces 9783030029432, 3030029433

Table of contents :
Front Matter ....Pages i-xiv
Preliminaries (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 1-11
Algebraic Curves (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 13-31
Complex Structure and the Topology of Curves (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 33-49
Curves in Projective Spaces (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 51-57
Plücker Formulas (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 59-69
Mappings of Curves (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 71-90
Differential 1-Forms on Curves (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 91-102
Line Bundles, Linear Systems, and Divisors (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 103-111
Riemann–Roch Formula and Its Applications (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 113-123
Proof of the Riemann–Roch Formula (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 125-130
Weierstrass Points (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 131-138
Abel’s Theorem (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 139-156
Examples of Moduli Spaces (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 157-162
Approaches to Constructing Moduli Spaces (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 163-175
Moduli Spaces of Rational Curves with Marked Points (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 177-192
Stable Curves (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 193-200
A Backward Look from the Viewpoint of Characteristic Classes (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 201-212
Moduli Spaces of Stable Maps (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 213-220
Exam Problems (Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov)....Pages 221-226
Back Matter ....Pages 227-231

Citation preview

Moscow Lectures 2

Maxim E. Kazaryan · Sergei K. Lando Victor V. Prasolov

Algebraic Curves Towards Moduli Spaces

Moscow Lectures Volume 2

Series Editors Lev D. Beklemishev, Moscow, Russia Vladimir I. Bogachev, Moscow, Russia Boris Feigin, Moscow, Russia Valery Gritsenko, Moscow, Russia Yuly S. Ilyashenko, Moscow, Russia Dmitry B. Kaledin, Moscow, Russia Askold Khovanskii, Moscow, Russia Igor M. Krichever, Moscow, Russia Andrei D. Mironov, Moscow, Russia Victor A. Vassiliev, Moscow, Russia Managing Editor Alexey L. Gorodentsev, Moscow, Russia

More information about this series at http://www.springer.com/series/15875

Maxim E. Kazaryan • Sergei K. Lando • Victor V. Prasolov

Algebraic Curves Towards Moduli Spaces

123

Maxim E. Kazaryan Steklov Mathematical Institute of RAS National Research University Higher School of Economics Skolkovo Institute of Science and Technology Moscow, Russia

Sergei K. Lando National Research University Higher School of Economics Skolkovo Institute of Science and Technology Moscow, Russia

Victor V. Prasolov Independent University of Moscow Moscow, Russia

Translated from the Russian by Natalia Tsilevich.: Originally published as Алгебраические Кривые по направлению к пространствам модулей by MCCME, 2018.

ISSN 2522-0314 ISSN 2522-0322 (electronic) Moscow Lectures ISBN 978-3-030-02942-5 ISBN 978-3-030-02943-2 (eBook) https://doi.org/10.1007/978-3-030-02943-2 Library of Congress Control Number: 2018964412 Mathematics Subject Classification (2010): 14H10, 14H37, 14H55, 14H70 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: https://www.istockphoto.com/de/foto/panorama-der-stadt-moskau-gm49008001475024685, with kind permission This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface to the Book Series Moscow Lectures

You hold a volume in a textbook series of Springer Nature dedicated to the Moscow mathematical tradition. Moscow mathematics has very strong and distinctive features. There are several reasons for this, all of which go back to good and bad aspects of Soviet organization of science. In the twentieth century, there was a veritable galaxy of great mathematicians in Russia, while it so happened that there were only few mathematical centers in which these experts clustered. A major one of these, and perhaps the most influential, was Moscow. There are three major reasons for the spectacular success of Soviet mathematics: 1. Significant support from the government and the high prestige of science as a profession. Both factors were related to the process of rapid industrialization in the USSR. 2. Doing research in mathematics or physics was one of very few intellectual activities that had no mandatory ideological content. Many would-be computer scientists, historians, philosophers, or economists (and even artists or musicians) became mathematicians or physicists. 3. The Iron Curtain prevented international mobility. These are specific factors that shaped the structure of Soviet science. Certainly, factors (2) and (3) are more on the negative side and cannot really be called favorable but they essentially came together in combination with the totalitarian system. Nowadays, it would be impossible to find a scientist who would want all of the three factors to be back in their totality. On the other hand, these factors left some positive and long lasting results. An unprecedented concentration of many bright scientists in few places led eventually to the development of a unique “Soviet school”. Of course, mathematical schools in a similar sense were formed in other countries too. An example is the French mathematical school, which has consistently produced first-rate results over a long period of time and where an extensive degree of collaboration takes place. On the other hand, the British mathematical community gave rise to many prominent successes but failed to form a “school” due to a lack of collaborations. Indeed, a v

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school as such is not only a large group of closely collaborating individuals but also a group knit tightly together through student-advisor relationships. In the USA, which is currently the world leader in terms of the level and volume of mathematical research, the level of mobility is very high, and for this reason there are no US mathematical schools in the Soviet or French sense of the term. One can talk not only about the Soviet school of mathematics but also, more specifically, of the Moscow, Leningrad, Kiev, Novosibirsk, Kharkov, and other schools. In all these places, there were constellations of distinguished scientists with large numbers of students, conducting regular seminars. These distinguished scientists were often not merely advisors and leaders, but often they effectively became spiritual leaders in a very general sense. A characteristic feature of the Moscow mathematical school is that it stresses the necessity for mathematicians to learn mathematics as broadly as they can, rather than focusing on a narrow field in order to get important results as soon as possible. The Moscow mathematical school is particularly strong in the areas of algebra/algebraic geometry, analysis, geometry and topology, probability, mathematical physics and dynamical systems. The scenarios in which these areas were able to develop in Moscow have passed into history. However, it is possible to maintain and develop the Moscow mathematical tradition in new formats, taking into account modern realities such as globalization and mobility of science. There are three recently created centers—the Independent University of Moscow, the Faculty of Mathematics at the National Research University Higher School of Economics (HSE) and the Center for Advanced Studies at Skolkovo Institute of Science and Technology (SkolTech)—whose mission is to strengthen the Moscow mathematical tradition in new ways. HSE and SkolTech are universities offering officially licensed full-time educational programs. Mathematical curricula at these universities follow not only the Russian and Moscow tradition but also new global developments in mathematics. Mathematical programs at the HSE are influenced by those of the Independent University of Moscow (IUM). The IUM is not a formal university; it is rather a place where mathematics students of different universities can attend special topics courses as well as courses elaborating the core curriculum. The IUM was the main initiator of the HSE Faculty of Mathematics. Nowadays, there is a close collaboration between the two institutions. While attempting to further elevate traditionally strong aspects of Moscow mathematics, we do not reproduce the former conditions. Instead of isolation and academic inbreeding, we foster global sharing of ideas and international cooperation. An important part of our mission is to make the Moscow tradition of mathematics at a university level a part of global culture and knowledge. The “Moscow Lectures” series serves this goal. Our authors are mathematicians of different generations. All follow the Moscow mathematical tradition, and all teach or have taught university courses in Moscow. The authors may have taught mathematics at HSE, SkolTech, IUM, the Science and Education Center of the Steklov Institute, as well as traditional schools like MechMath in MGU or MIPT. Teaching and writing styles may be very different. However, all lecture notes are

Preface to the Book Series Moscow Lectures

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supposed to convey a live dialog between the instructor and the students. Not only personalities of the lecturers are imprinted in these notes, but also those of students. We hope that expositions published within the “Moscow lectures” series will provide clear understanding of mathematical subjects, useful intuition, and a feeling of life in the Moscow mathematical school. Moscow, Russia

Igor M. Krichever Vladlen A. Timorin Michael A. Tsfasman Victor A. Vassiliev

Introduction

The foundations of the theory of Riemann surfaces were laid in the second half of the nineteenth century. It became the convergence point of the then advanced developments in analysis, algebra, and yet-to-be-created topology. During the whole twentieth century, the theory of Riemann surfaces, merged with the theory of complex algebraic curves, would repeatedly move into the foreground of mathematics. It explained many difficulties arising when integrating various functions and made it possible to develop efficient integration methods, clarified Galois theory and led to a new understanding of arithmetics and number theory, became a testing ground for the theory of complex varieties and functional classes. In the last decades of the twentieth century, Riemann surfaces and their moduli spaces proved to be much needed as one of the most efficient tools in the study of integrable systems and related models of mathematical physics. It is these aspects of the theory of Riemann surfaces that are the main subject of the book. The principal feature of modern applications is that they use not individual curves, but families of curves depending analytically on a parameter. For this reason, the book pays much attention to those properties of individual curves that play a crucial role when passing to families of curves, such as, for example, the existence of nontrivial symmetries. Curves with nontrivial symmetries form natural subvarieties in moduli spaces of curves and, in a sense, determine orbifold characteristics of these spaces. Most monographs and textbooks on algebraic curves either work with individual curves and are fairly elementary (like, for example, [6, 16], or [5]), or reckon on using advanced algebro-geometric techniques and thus require a solid background (like [11] or [2]). The purpose of this book is to fill the existing gap while remaining in the framework of more or less elementary methods, and provide senior students with a tool for dealing with moduli spaces. We’d like to hope that physicists, for whom moduli spaces of curves are a tool rather than an object of study, will also find here answers to some natural questions without going into technicalities. The main properties of moduli spaces of curves and maps interesting for applications are their topological characteristics. These characteristics can be expressed primarily in terms of vector bundles over moduli spaces and their characteristic ix

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classes. When presenting this material, we tried not to overstep the bounds of the theory of Chern classes of vector bundles and demonstrate only calculations that can be performed manually. There is no doubt that the topology of moduli spaces of curves of higher genus is extremely complicated, and one can hardly expect to obtain its complete description, at least in the near future. However, information on these spaces needed for applications (“tautological cohomology ring,” defined one way or another) can prove to be much more available. In this respect, moduli spaces of rational curves with marked points can serve as a good testing ground for developing an intuition for the structure of more complicated spaces, so they are covered in a separate chapter. In this chapter we discuss Kapranov’s model, which represents stable rational curves as Veronese curves in appropriate projective spaces, as well as Keel’s and Kontsevich and Manin’s theorems describing the cohomology of moduli spaces of rational curves. The introduction to Graphs on Surfaces and Their Applications by S. K. Lando and A. K. Zvonkin (Springer-Verlag, Berlin–Heidelberg, 2004) says: It is clear that every chapter of our book could give rise to an entire book. Probably, one day such a series of books will be written.

At least one of the authors of the present book regards it as an element of this future series, namely, as the background for Chaps. 4 and 5 in Graphs on Surfaces. In turn, the latter book contains a fair amount of information on modern applications of algebraic curves and their moduli spaces. The present book is based on a 1-year course taught for a number of years at the Independent University of Moscow, and then at the Department of Mathematics of the Higher School of Economics. The course is intended for third and fourth year undergraduates and graduate students with no previous knowledge of algebraic curves. A regular lecture course requires a large number of exercises, in particular intended for the evaluation of student learning. For the reader’s convenience, the exercises scattered throughout the book are also collected in the concluding chapter. This chapter also contains sample exam problems and questions. While working on the book, M. E. Kazaryan and S. K. Lando were supported by the RSF grant 16-11-10316 Characteristic classes and representation theory.

Contents

1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Complex Projective Spaces . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 The Topology of Projective Spaces. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Two-Dimensional Surfaces . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Gluing Two-Dimensional Surfaces Out of Polygons . . . . . . . . . . . . . . . 1.5 Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Ramified Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.7 Riemann–Hurwitz Formula .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 2 3 3 4 6 8 9

2

Algebraic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Plane Algebraic Curves . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Bézout’s Theorem and Its Applications . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Rational Parametrization .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

13 13 19 26

3

Complex Structure and the Topology of Curves . . . .. . . . . . . . . . . . . . . . . . . . 3.1 A Complex Structure on a Curve .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 The Genus of a Smooth Plane Curve .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Hessian and Inflection Points . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Hyperelliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Lifting a Complex Structure . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6 Quotient Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7 Meromorphic Functions.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

33 33 34 41 42 44 45 46

4

Curves in Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Definition and Examples .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Embeddings and Immersions of Curves. . . . . . . . .. . . . . . . . . . . . . . . . . . . .

51 51 54

5

Plücker Formulas .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Projective Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Plücker Formulas for Nonsingular Curves . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Plücker Formulas for Singular Curves . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Newton Polygons .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

59 59 64 65 68

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6

Contents

Mappings of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Automorphisms of the Riemann Sphere . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Mappings of Elliptic Curves .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Moduli of Elliptic Curves .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Lattices and Cubic Curves . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 The j -Invariant Revisited . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Automorphisms of Elliptic Curves and Poncelet’s Closure Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.7 Automorphisms of Curves of Higher Genus: Hurwitz’s Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

71 71 72 74 77 81 85 87

7

Differential 1-Forms on Curves . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 91 7.1 Tangent and Cotangent Bundles .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 91 7.2 How to Define Vector Fields and Differential Forms .. . . . . . . . . . . . . . 94 7.3 The Dimension of the Space of Holomorphic 1-Forms on a Plane Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 96 7.4 Integrating 1-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 99 7.5 The Dimension of the Space of Holomorphic 1-Forms on a Curve with Singularities .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 101 7.6 Residues and Integrals of Meromorphic 1-Forms . . . . . . . . . . . . . . . . . . 101

8

Line Bundles, Linear Systems, and Divisors. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 The Divisor of a Meromorphic Section of a Line Bundle .. . . . . . . . . 8.2 The Degree of a Divisor and the Degree of a Bundle.. . . . . . . . . . . . . . 8.3 The Tautological Line Bundle Over the Projective Line . . . . . . . . . . . 8.4 Recovering a Line Bundle from a Class of Divisors.. . . . . . . . . . . . . . . 8.5 Mappings from Curves to Projective Spaces Associated with Line Bundles .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.6 Linear Systems and Mappings Between Curves .. . . . . . . . . . . . . . . . . . .

103 103 105 106 107

Riemann–Roch Formula and Its Applications . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Mittag-Leffler’s Problem .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 The Rational Curve.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Elliptic Curves .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Hyperelliptic Curves and Curves of Genus 2 . . .. . . . . . . . . . . . . . . . . . . . 9.5 Riemann’s Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.6 Curves of Genus 3, 4, and 5 . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

113 113 116 117 119 120 121

9

108 109

10 Proof of the Riemann–Roch Formula . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 125 10.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 125 10.2 Divisors on the Canonical Curve . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 129 11 Weierstrass Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 131 11.1 Definition of Weierstrass Points . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 131 11.2 Weierstrass Points on Curves of Genus 3 and Inflection Points of Plane Quartics.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 133

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11.3 Weights of Weierstrass Points . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 133 11.4 Weierstrass Points and the Finiteness of the Automorphism Group .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 137 12 Abel’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.1 Jacobian .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2 Proof of the Necessity Part. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.3 Proof of the Sufficiency Part: Beginning .. . . . . . .. . . . . . . . . . . . . . . . . . . . 12.4 Abelian Differentials of the First, Second, and Third Kind . . . . . . . . 12.5 Riemann’s Bilinear Relations . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.6 Completing the Proof of the Sufficiency Part . . .. . . . . . . . . . . . . . . . . . . . 12.7 Proof of Jacobi’s Inversion Theorem .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.8 Theta Divisor and Theta Functions .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

139 139 142 143 144 145 148 150 155

13 Examples of Moduli Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.1 First Examples .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.2 The Space M1;1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.3 The Universal Curve Over M1;1 . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.4 The Cohomology of the Space M1;1 . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

157 157 159 160 161

14 Approaches to Constructing Moduli Spaces . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.1 Requirements on Moduli Spaces . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.2 A Naive Attempt to Construct a Moduli Space .. . . . . . . . . . . . . . . . . . . . 14.3 Hilbert Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.4 Hilbert Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.5 Pluricanonical Embeddings .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.6 The Quotient by the Action of the Group of Projective Transformations and Stability . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.7 Moduli Spaces of Curves with Marked Points . .. . . . . . . . . . . . . . . . . . . .

163 163 165 167 170 170 171 174

15 Moduli Spaces of Rational Curves with Marked Points.. . . . . . . . . . . . . . . 15.1 Kapranov’s Construction .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.2 Compactification of Moduli Spaces . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.3 Poincaré Polynomials of Moduli Spaces .. . . . . . .. . . . . . . . . . . . . . . . . . . . 15.4 Compactified Spaces: Keel’s Description of Cohomology .. . . . . . . . 15.5 Compactified Spaces: Kontsevich and Manin’s Description of Cohomology .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

177 178 180 182 186

16 Stable Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.1 Definition and Examples of Stable Curves .. . . . .. . . . . . . . . . . . . . . . . . . . 16.2 The Genus of a Nodal Curve . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.3 Degenerations of Smooth Curves.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.4 Compactification of Moduli Spaces by Stable Curves and Pluricanonical Embeddings .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

193 193 195 197

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17 A Backward Look from the Viewpoint of Characteristic Classes . . . . . 201 17.1 The First Chern Class of a Line Bundle .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 201 17.2 Chern Classes of Vector Bundles .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 202

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Contents

17.3 17.4 17.5 17.6

Other Approaches to Defining Chern Classes . .. . . . . . . . . . . . . . . . . . . . The Genus of a Smooth Plane Curve .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The Genus of a Complete Intersection . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Plücker Formulas .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

206 209 210 211

18 Moduli Spaces of Stable Maps . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.1 Rational Curves in the Plane .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.2 Moduli Spaces of Stable Maps . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.3 Quantum Cohomology and the Associativity Equation . . . . . . . . . . . .

213 213 216 218

19 Exam Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 19.1 First Term .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 19.2 Exam Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 19.3 Second Term .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 19.4 Exam Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

221 221 223 224 226

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 227 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 229

Chapter 1

Preliminaries

The main focus of this course will be on complex algebraic curves, or, which is the same, Riemann surfaces. From the topological point of view, a Riemann surface is a two-dimensional oriented surface; its topological properties are uniquely determined by a nonnegative integer, the genus. At the same time, individual characteristics of algebraic curves are complicated, and two different curves, even of the same genus, usually bear little resemblance to each other. However, if we look at curves not one by one, but in families, it turns out that such families have a relatively simple structure and many remarkable properties, which find various applications in mathematics and theoretical physics. It is the transition from individual curves to families of curves that is the topic of the course. In our everyday understanding, curves are primarily real curves in a real plane. But it turns out that for algebraic curves such an understanding has numerous drawbacks. First and foremost, in algebra it is more convenient to deal with algebraically closed fields. For instance, over C every polynomial of degree n in one variable has exactly n roots counted with multiplicity, while over R this is not the case: a polynomial of degree n can have less than n roots. For curves, this difference manifests itself in the fact that two complex plane curves of degrees n and m have mn intersection points, while for real curves this is not true in general: the number of intersection points can be less than mn. Besides, curves in the affine plane can intersect at points at infinity (like, for example, parallel lines), which also hampers the study of such curves. Thus, it is natural to consider curves in complex projective spaces, and now we proceed to the corresponding definition.

© Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_1

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1.1 Complex Projective Spaces A point of the n-dimensional complex projective space CPn is a complex line in the (n + 1)-dimensional complex space Cn+1 passing through the origin. Such a line is uniquely determined by any nonzero point (z0 , z1 , . . . , zn ) belonging to it. In turn, this point is uniquely determined up to multiplication by a nonzero complex number λ: the point (λz0 , λz1 , . . . , λzn ) determines the same line as (z0 , z1 , . . . , zn ). In other words, CPn is the quotient of the (n + 1)-dimensional complex space punctured at the origin by the action of the multiplicative group of nonzero complex numbers, CPn ∼ = (Cn+1 \ {0})/C∗ . A point of CPn will be written as (z0 : z1 : . . . : zn ), with not all zi zero, meaning the equivalence class of (n + 1)-tuples of coordinates with respect to the equivalence relation described above. The (n + 1)-tuple (z0 : z1 : . . . : zn ) is called the homogeneous coordinates of the point under consideration. Successively setting zi = 1 for i = 0, 1, . . . , n, we obtain n + 1 charts in CPn . Each chart covers the whole space except a hyperplane zi = 0 and coincides with the n-dimensional complex space Cn . All coordinates except the ith one form the affine coordinates in the corresponding chart. For n = 1, we have the one-dimensional projective space CP1 . Along with the complex line C, this is one of the first examples of complex curves. It is also called the rational curve, or the Riemann sphere. Setting z0 = 1 or z1 = 1 in CP1 , we obtain two affine charts in CP1 , each coinciding with the complex line C. The first of them has the coordinate z1 , the second one, the coordinate z0 . Each of these charts does not contain exactly one point of CP1 , which is called the point at infinity for this affine chart; it is denoted by z1 = ∞ in the first chart and z0 = ∞ in the second chart, see Fig. 1.1.

Fig. 1.1 An affine chart and the corresponding point at infinity on the projective line

1.3 Two-Dimensional Surfaces

3

The choice of affine coordinates on the projective plane (and, more generally, on a projective space of arbitrary dimension n) is not unique. One can pass from one coordinate system to another one using a projective transformation, that is, a linear transformation Z = AZ  regarded up to a multiplicative constant (here Z, Z  are column vectors and A is an (n + 1) × (n + 1) nondegenerate matrix, A : Cn+1 → Cn+1 ). A projective transformation does not change the geometric properties of subsets in a projective space, in particular, of curves. Thus we will often choose the most convenient coordinates, in which the subsets we are interested in are given by the simplest equations.

1.2 The Topology of Projective Spaces The complex projective line is called the Riemann sphere for a good reason: from the topological point of view, it is a two-dimensional sphere. The topology of projective spaces of higher dimension is also simple. These spaces are compact, and their homology can be described inductively: the (n+1)-dimensional complex projective space is the union of the affine chart Cn+1 and the n-dimensional projective space. Hence CPn+1 is the result of gluing a 2(n+1)-dimensional cell to a 2n-dimensional complex. This property allows one to compute the homology of projective spaces with complex coefficients: H2i (CPn ) = C

for i = 0, 1, . . . , n,

and all the other homology groups are zero. The complex cohomology ring of the projective space CPn is generated by an element h of degree 2. It is the ring H ∗ (CPn ) = C[h]/(hn+1 = 0) of polynomials in one variable truncated at degree n + 1. The element hi in this ring is the Poincaré dual of the projective subspace CPn−i representing the generator in H2(n−i) (CPn ), i = 0, . . . , n. In particular, the element h itself is dual to the hyperplane CPn−1 .

1.3 Two-Dimensional Surfaces A two-dimensional surface is a compact two-dimensional manifold (possibly with boundary). If the boundary is empty, then the surface is said to be closed. For every point of a two-dimensional surface there are two possible directions of rotation around this point. If such a direction is chosen at each point, and these directions are compatible for nearby points, then one says that the surface is endowed with an orientation. A surface that can be endowed with an orientation

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1 Preliminaries

Fig. 1.2 Spheres with 0, 1, 2 handles

is said to be orientable. One can easily see that the plane R2 is orientable, and that deleting several points does not affect the orientability. Therefore, the sphere S 2 is orientable. We will be interested in orientable surfaces only, because, from the topological point of view, a smooth complex curve is a closed orientable surface. The connected sum of two surfaces M and N is the surface M#N obtained by removing small open disks D1 and D2 from M and N and gluing the surfaces M \ D1 and N \ D2 together by a homeomorphism h : ∂D1 → ∂D2 . What remains after removing a small open disk D from the torus S 1 × S 1 is called a handle. The connected sum of g tori is called a sphere with g handles (a sphere with 0 handles is just the ordinary sphere S 2 ). Every closed orientable two-dimensional surface is homeomorphic to a sphere with g handles (for some g), and spheres with different numbers of handles are not homeomorphic. Figure 1.2 shows spheres with g handles for g = 0, 1, 2. Let us also mention that every closed nonorientable surface is homeomorphic to the connected sum of several copies of the real projective plane RP2 , and these connected sums with different numbers of copies are not homeomorphic. For a proof of this classification theorem for two-dimensional surfaces, see, e.g., [19].

1.4 Gluing Two-Dimensional Surfaces Out of Polygons Every closed two-dimensional surface can be triangulated, i.e., cut into triangles in such a way that any two triangles either have no common points, or have one common vertex, or have one common side (but are not allowed to share only a part of a side). Given a triangulation of a surface M, let V be the number of vertices, E be the number of edges (sides), and F be the number of faces in this triangulation. The number V − E + F , called the Euler characteristic of M, does not depend on the choice of a triangulation. It is denoted by χ(M). The Euler characteristic of a sphere with g handles is 2 − 2g, and the Euler characteristic of the connected sum of g real projective planes is 2 − g. More generally, every closed two-dimensional surface can be glued out of polygons. Each side of each polygon is glued to exactly one side of the same or

1.4 Gluing Two-Dimensional Surfaces Out of Polygons

5

a

b Fig. 1.3 Gluing surfaces out of polygons

Fig. 1.4 The standard gluing of an orientable surface of genus g = 2 out of a (4g = 8)-gon

another polygon in such a way that vertices are glued to vertices. The sides of the polygons, pairwise glued together, form a graph on the resulting surface, whose vertices are the vertices of the polygons (also glued together). The interiors of the polygons are called the faces of the embedded graph. The Euler formula, which expresses the Euler characteristic of the surface obtained in this way in terms of the number of vertices, edges, and faces of the graph, remains valid also for such a gluing. To obtain an orientable (and oriented) surface, one should orient each of the polygons to be glued and take care of the orientations when gluing. Since for every pair of sides there is only one way to glue them together so that the orientations agree, a gluing is uniquely determined by a partition of all sides of all polygons into disjoint pairs, see Fig. 1.3. It is often convenient to realize a surface of a given genus by gluing together the sides of a single polygon. The minimum number of sides in such a polygon for a surface of genus g is 4g. For g > 1, an orientable surface of genus g can be glued out of a 4g-gon in several ways. By the standard gluing one usually means the gluing according to the scheme ababcdcd . . . (the first side is glued to the third one, the 2nd to the 4th, the 5th to the 7th, etc.), see Fig. 1.4.

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The trail the polygon leaves under the standard gluing has one vertex, 2g edges, and one face. Hence the Euler characteristic of this surface is V − E + F = 1 − 2g + 1 = 2 − 2g, i.e., its genus is indeed g. Exercise 1.1 Using the Euler formula, show that a surface of genus g cannot be glued out of a polygon with less than 4g sides. Exercise 1.2 Using the Euler formula, find the genus of the orientable surface glued out of a single polygon according to the following scheme: a) abcabc; b) abcdabcd; c) abcdabdc. Exercise 1.3 Using the Euler formula, find the genus of the surface obtained by gluing together the polygons from Fig. 1.3 according to the scheme indicated in the figure.

1.5 Coverings A continuous mapping p : M → N between two-dimensional surfaces is called a covering if every point y of N has a neighborhood U = U (y) ⊂ N whose ppreimage is a disjoint union of several copies of U and the restriction of p to each of the copies is a homeomorphism. A typical example of a covering is the mapping z → zn from the punctured unit disk |z| < 1, z = 0, in C to itself. In what follows, we always assume that the covered surface N is connected. In this case, either every point of N has infinitely many preimages, or the set of preimages of every point is finite and any two points have the same number of preimages. This common number of preimages is called the degree, or the number of sheets, of the covering. If the covered surface N is orientable and one of the two orientations of N is chosen, then the covering surface M is also orientable, and it can be endowed with the orientation induced from N. Exercise 1.4 Give an example of a covering p : M → N with M orientable and N nonorientable. Theorem 1.1 Let p : M → N be an n-sheeted covering with M and N connected compact two-dimensional surfaces. Then χ(M) = nχ(N). Proof Consider a sufficiently fine triangulation of N. Then the preimage of every its triangle consists of n pairwise disjoint triangles, and all these triangles together form a triangulation of M. The latter contains exactly n triangles corresponding to each triangle of the original triangulation of N, exactly n edges corresponding to each edge, and exactly n vertices corresponding to each vertex. Therefore, χ(M) = nχ(N).

1.5 Coverings

7

Exercise 1.5 Show that a sphere with g handles covers a sphere with h handles if and only if h − 1 divides g − 1 (here g, h ≥ 2). Coverings are closely related to fundamental groups of surfaces. Let y0 ∈ N be an arbitrary point; we will regard it as the base point of the fundamental group π1 (N, y0 ). Consider an arbitrary covering p : M → N of N by a connected surface M. Let γ : [0, 1] → N be an arbitrary continuous loop with γ (0) = γ (1) = y0 , i.e., starting and ending at y0 . Take an arbitrary preimage of y0 . Then γ admits a unique continuous lifting γ˜ : [0, 1] → M such that γ˜ (0) coincides with the chosen preimage of y0 and p ◦ γ˜ = γ . The endpoint γ˜ (1) of γ˜ is also a preimage of y0 , but it may not coincide with the original one. This preimage does not depend on the choice of γ in the given homotopy class of closed paths. Thus, with every element of the fundamental group π1 (N, y0 ) we have associated a permutation of the (finite or infinite) set of preimages p−1 (y0 ): every preimage x0 is mapped to the endpoint of the lifted path starting at x0 . This correspondence determines an action of the fundamental group on the set of preimages. It is called the monodromy of the covering. Example 1.1 Consider a path γ in the punctured unit disk D \ {0} that starts and ends at the same point y0 and makes one turn around the puncture in the positive direction. Under the mapping f : z → zn from the punctured unit disk to itself, the point y0 has n preimages located at the vertices of a regular n-gon centered at 0. The monodromy along the path γ is a cyclic permutation of the vertices of this n-gon, the result of rotating it by 2π/n around the center. Exercise 1.6 Find the monodromy of the same mapping along a path that makes two turns around the puncture in the negative direction. The elements of π1 (N, y0 ) giving rise to permutations of the fiber p−1 (y0 ) that fix a point x0 ∈ p−1 (y0 ) form a subgroup in π1 (N, y0 ). It is isomorphic to the fundamental group π1 (M, x0 ) of the covering surface. One can prove that this construction establishes a one-to-one correspondence between the subgroups of the fundamental group of a surface and the equivalence classes of its coverings. Here two coverings p1 : M1 → N and p2 : M2 → N are equivalent if there is a homeomorphism h : M1 → M2 completing the commutative triangle, i.e., such that p2 ◦ h = p1 . In particular, the fundamental group of the punctured disk is isomorphic to the infinite cyclic group Z. Its subgroups are the groups of the form nZ ⊂ Z, consisting of the multiples of a given positive integer n, as well as the zero subgroup. Coverings corresponding to finite index subgroups are described above. The zero subgroup gives rise to a covering which can be described as follows. The covering surface is the horizontal strip {z ∈ C : 0 < Im z < 1} (which is, of course, homeomorphic to the disk), and the mapping is the exponential mapping z → e2πiz . In this case, the covering surface is simply connected: its fundamental group is trivial.

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1.6 Ramified Coverings Let p : M → N be a finite-sheeted covering of a surface N obtained by puncturing  at several points. We construct a surface M  ⊃ M, obtained a connected surface N →N  extending p by adding several points to M, and a continuous mapping pˆ : M  \ N has a punctured neighborhood (a punctured disk) as follows. Each point from N such that its preimage in M is a disjoint union of punctured disks and the restriction of p to each of these preimages is equivalent to the mapping z → zni for some ni . Add the central point to each of these disks and extend the mapping p by continuity . to this point. The extended mapping is called a ramified covering of the surface N Note that a ramified covering is a covering only in the case where the values ni are equal to 1 for all punctures. In the simplest and extremely important situation, we deal with finite-sheeted coverings of a punctured sphere. Let N be the sphere S 2 punctured at a finite set of points T = {t1 , . . . , tc }. Take a point t0 ∈ S 2 \ T . Denote by d the number |f −1 (t0 )| of preimages of t0 , i.e., the degree of the covering f . Connect t0 with each of the points ti by a non-self-intersecting segment of a smooth curve that does not pass through tj for j = i. With such a segment we can associate the path γi that follows this segment from t0 to ti , then makes a counterclockwise turn around the point ti , and, finally, returns back to t0 along the same segment. Do this for each of the punctures ti making sure that • the segments leading to the punctures pairwise intersect only at the base point t0 ; • the cyclic order in which the segments leave the point t0 coincides with the order in which the punctures are labeled. As a result, we obtain a star graph on S 2 with center at t0 and rays from t0 to ti (see Fig. 1.5). Exercise 1.7 What is the fundamental group of the sphere S 2 punctured at c points? Thus we have constructed a collection of permutations σ1 , . . . , σc on the preimage f −1 (t0 ) ∈ Y satisfying the following properties: • the subgroup of the group Sd of permutations of the preimages of t0 generated by σ1 , . . . , σc acts transitively on the fiber f −1 (t0 ), i.e., for every pair of points of f −1 (t0 ) there is a permutation in this subgroup sending the first point to the second one; • the product σc ◦ . . . ◦ σ2 ◦ σ1 is the identity permutation.

Fig. 1.5 Star graphs

1.7 Riemann–Hurwitz Formula

9

Exercise 1.8 Prove these properties. Exercise 1.9 Show that for every collection of points T = {t1 , . . . , tc } on the sphere, every star on these points, and every collection of permutations σ1 , . . . , σc satisfying the above two properties there exists a covering Y → S 2 \ T for which the monodromy along the path γi coincides with σi . This covering is unique, i.e., for any two such coverings f1 : Y1 → S 2 , f2 : Y2 → S 2 there exists a homeomorphism h : Y1 → Y2 such that f2 = h ◦ f1 . Exercise 1.10 Verify that the cycle type (i.e., the number of cycles and their lengths) of the permutation σi does not depend on the choice of a base point t0 and a path connecting it with ti . Exercise 1.11 Assume that the base surface is not a sphere, but a punctured surface of higher genus. How should one modify the notion of a star and a collection of permutations to ensure that the covering surface still can be uniquely recovered?

1.7 Riemann–Hurwitz Formula For a covering, the Euler characteristic of the covering surface is given by a simple formula in terms of the Euler characteristic of the covered surface and the degree of the covering. For ramified coverings, the formula turns out to be more complicated, it involves some additional characteristics of the covering. Theorem 1.2 (Riemann–Hurwitz) Let p : M → N be an n-sheeted ramified covering with k ramification points having m1 , . . . , mk preimages, respectively. Then χ(M) = n(χ(N) − k) + m1 + . . . + mk . Proof Divide the surface N into two closed sets: N = NA ∪ NB , where NA is the union of the closures of small circular neighborhoods of the ramification points and NB is the closure of the complement N \ NA to NA . Then χ(N) = χ(NA ) + χ(NB ) − χ(NA ∩ NB ). But the set NA ∩ NB consists of several circles, hence χ(NA ∩ NB ) = 0. Therefore, χ(N) = AN + BN ,

where AN = χ(NA ), BN = χ(NB ).

In a similar way, divide the surface M into the closed sets MA = p−1 (NA ) and MB = p−1 (NB ). Then χ(M) = AM + BM ,

where AM = χ(MA ), BM = χ(MB ).

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The restriction of p to MB is a covering, hence BM = nBN , and then χ(M) − AM = n(χ(N) − AN ). The set MA consists of m1 + . . .+ mk disjoint disks. Clearly, the Euler characteristic of a disk is 1. Hence AM = m1 + . . . + mk and AN = k. Here is another proof of the Riemann–Hurwitz formula. Proof Consider a sufficiently fine triangulation of the surface N whose vertices include all ramification points (the triangulation may also have other vertices). Here the words “sufficiently fine” mean that the preimage of the interior of each triangle of the triangulation consists of interiors of n triangles, where n is the degree of the covering, and the same holds for the preimages of interiors of edges. Over the interior of each triangle of the triangulation, as well as over the interior of each its edge, p is an unramified covering. Hence the p-preimages of the edges and the triangles of the triangulation of N form the induced triangulation of M. Let vN , eN , and fN be the number of vertices, edges, and faces of the triangulation of N, and vM , eM , and fM be the number of vertices, edges, and faces of the induced  triangulation of M. Then eM = neN , fM = nfN , and vM = n(vN − k) + mi , whence  χ(M) = vM − eM + fM = nχ(M) − nk + mi , as required. In this argument, a triangulation can be replaced by an arbitrary sufficiently small partition of the surface N into polygons. The Riemann–Hurwitz formula can be rewritten in another form, which is often more convenient for applications. It uses information not on the ramification over the image surface N, but on the ramification at the preimage surface M. If in a small neighborhood of a point x0 ∈ M, in some local complex coordinates, the mapping p looks as z → zd , then this point is said to have ramification index d (for almost all points, the ramification index is equal to 1). Let d1 , . . . , dm1 be the ramification indices over the first ramification point, dm1 +1 , . . . , dm1 +m2 be the ramification indices over the second ramification point, etc. Then d1 + . . . + dm1 = dm1 +1 + . . . + dm1 +m2 = . . . = n. Let x1 , . . . , xl be all the preimages of all the ramification points. Then l  (di − 1) = (n − m1 ) + (n − m2 ) + . . . = kn − (m1 + m2 + . . . + mk ). i=1

1.7 Riemann–Hurwitz Formula

Hence the Riemann–Hurwitz formula can be rewritten in the form  χ(M) = nχ(N) − (di − 1);

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(1.1)

here we may assume that the sum ranges over all points of M with ramification index different from 1. Formula (1.1) immediately implies the assertions stated in Exercises 1.12–1.14. Exercise 1.12 Show that every ramified covering of a torus by a torus is actually unramified, i.e., the ramification index of every point is equal to 1. Exercise 1.13 Let p : M → N be a ramified covering of orientable surfaces. Show that χ(M) ≤ χ(N). Exercise 1.14 Let p : M → N be a ramified covering of orientable surfaces. Show that if χ(M) = χ(N) < 0, then p is an isomorphism.

Chapter 2

Algebraic Curves

Algebraic curves are curves given by polynomial equations in projective spaces. On the other hand, algebraic curves are one-dimensional complex manifolds, and to define them, there is no need to embed them anywhere. We will consider various ways to define curves and discuss how one can decide whether they result in the same curve.

2.1 Plane Algebraic Curves We start from the most illustrative and familiar object, curves in the plane, and then proceed to curves in complex projective spaces. A curve in the plane is given by one polynomial equation. If all its coefficients are real, then the complex curve is real. The real points (x : y : z) lying on a real curve form the real part of this curve. There are no good methods for visualizing complex curves, so for illustration we will use real parts of plane real curves that belong to a chosen affine chart. The simplest algebraic curve is a line. It is given by a linear homogeneous equation ax + by + cz = 0. For any pair of distinct points in the projective plane there is exactly one line through them, and any two distinct lines intersect in exactly one point. Figure 2.1 shows also a curve of degree 2 (a conic) and three curves of degree 3 (cubics). A plane algebraic curve is a curve in CP2 given by a homogeneous polynomial  equation aij k x i y j zk = 0, where i, j, k are nonnegative integers and not all i+j +k=n

of the coefficients aij k are zero. The number n is called the degree of the curve. To write the equation of this curve in one of the affine charts x = 1, y = 1, or z = 1, one should simply substitute the corresponding value into the equation, turning it into a nonhomogeneous equation in the remaining two variables.

© Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_2

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Fig. 2.1 Various plane algebraic curves

Fig. 2.2 A reducible curve of degree 4

A plane algebraic curve F (x, y, z) = 0 is said to be irreducible if the homogeneous polynomial F cannot be written as the product of homogeneous polynomials F1 and F2 of positive degree. Otherwise, the curve is said to be reducible. As a set of points, a reducible curve is the union of the curves F1 = 0 and F2 = 0. The simplest reducible curve of degree n is given by the equation l1 · · · ln = 0, where l1 , . . . , ln are pairwise distinct linear functions. As a set of points, this curve is the union of the lines l1 = 0, . . . , ln = 0, see Fig. 2.2. In many cases, the simplest reducible curve l1 · · · ln = 0 helps one to understand the situation with an arbitrary curve of degree n. For example, curves l1 · · · lm = 0 and l1 ···ln = 0 that have no common lines and satisfy the condition that every point of the plane belongs to at most two lines among l1 , . . . , lm , l1 , . . . , ln have exactly mn pairwise distinct intersection points. Below we will show that arbitrary curves of degrees m and n have either mn common points (counted with multiplicity), or infinitely many common points. Along with irreducibility, another important property of curves is smoothness. A smooth curve in the complex projective plane is a subset C ⊂ CP2 given by an equation F (x, y, z) = 0, where F is a nondegenerate homogeneous polynomial in three variables. Here the nondegeneracy condition means that the curve F (x, y, z) = 0 contains no singular points of F , i.e., points at which the

2.1 Plane Algebraic Curves

15

differential dF =

∂F ∂F ∂F dx + dy + dz ∂x ∂y ∂z

vanishes. A smooth curve is irreducible. Exercise 2.1 Show that a reducible curve cannot be smooth. Exercise 2.2 Give an example of an irreducible nonsmooth curve. It is sometimes easier to check that a curve C is nondegenerate not in homogeneous coordinates, but in some chart. Then the required condition takes the following form. Let A ∈ C be a point of the curve. Consider an arbitrary chart that contains A, say z = 1. Then the condition “for the polynomial f (x, y) = F (x, y, 1), the differential df =

∂f ∂f dx + dy ∂x ∂y

does not vanish at A” is equivalent to the previous one, since an arbitrary homogeneous polynomial F (x, y, z) of degree n satisfies the Euler identity x Thus, if

∂F ∂x

= 0 and

∂F ∂y

∂F ∂F ∂F +y +z = nF. ∂x ∂y ∂z

= 0 at some point of the curve given by the equation

F = 0, then, for z = 0, we also have

∂F ∂z

= 0 at this point.

Exercise 2.3 Prove the Euler identity. Remark 2.1 It is convenient to check the nondegeneracy condition acting backwards. First one should find the singular points of the polynomial F (x, y, z) in CP2 . Usually, there are finitely many of them. Then one should check whether any of these points lies on the curve C. Example 2.1 Consider the plane curve given by the equation F (x, y, z) = x 2 + y 2 + z2 = 0. ∂F ∂F The degeneracy condition for F means that ∂F ∂x = ∂y = ∂z = 0, i.e., x = y = z = 0. Since no point of the projective plane has such homogeneous coordinates, F has no singular points at all.

Example 2.2 Consider the plane curve given by the equation F (x, y, z) = x 2 + y 2 = 0.

16

2 Algebraic Curves

Fig. 2.3 An ordinary double point of a curve

The degeneracy condition for F means that

∂F ∂x

=

∂F ∂y

=

∂F ∂z

= 0, i.e., x = y = 0.

It singles out the point (0 : 0 : 1) in CP . This point lies on the curve x 2 + y 2 = 0, which is thus singular. In fact, it is the pair of projective lines given by the equations x + iy = 0 and x − iy = 0. These lines intersect in one point, the singular point of the curve. 2

Exercise 2.4 Let C be a nondegenerate conic in the plane, i.e., a nondegenerate curve given by an equation of degree 2. Show that in an appropriate coordinate system it has the form x 2 + y 2 + z2 = 0. Let (0, 0) be a singular point of a curve given by an equation f (x, y) = 0 in an affine chart. Then the constant and linear terms of the polynomial f are zero. The singular point is called an ordinary double point if the quadratic part of f is nondegenerate. In this case, the quadratic part of f can be written as the product of two distinct linear functions, and the curve f = 0 in the vicinity of (0, 0) looks as a pair of intersecting lines (see Fig. 2.3). Now we define a tangent to a curve in CP2 . Let A be a point of a curve F = 0; thus F (A) = 0. Take an arbitrary point P and consider the line P A. In the affine chart centered at A, its points have coordinates tP = (tp1 , tp2 ), where t ∈ C and (p1 , p2 ) are the coordinates of P in this affine chart. Hence the intersection points of the line P A and the curve F (X) = 0 correspond to the roots of the polynomial equation F (tP ) = F (A) + t

 ∂F (A)pi + . . . = 0. ∂xi

(2.1)

Since F (A) = 0, it follows that t = 0 isa root of (2.1). In the case where the ∂F multiplicity of this root is at least two, i.e., ∂xi (A)pi = 0, one says that P A is a tangent to the curve F = 0 at the point A. Every line passing through a singular point of a curve is tangent to this curve. However, in the case of an ordinary double point A, there are two lines passing through it that can naturally be called tangent to the branches of the curve at this point. Namely, these are the lines for which the product of the equations is the quadratic part of the equation of the curve at A. Exercise 2.5 Show that in Cartesian coordinates (x, y), the tangent to a curve f (x, y) = 0 at a smooth point (x0 , y0 ) is given by the equation ∂f ∂f (x0 , y0 ) · (x − x0 ) + (x0 , y0 ) · (y − y0 ) = 0. ∂x ∂y

2.1 Plane Algebraic Curves

17

Exercise 2.6 Let A = (x0 : y0 : z0 ) be a point of a curve F = 0 in CP2 . Show that the equations x

∂F ∂F ∂F (A) + y (A) + z (A) = 0 ∂x ∂y ∂z

and (x − x0 )

∂F ∂F ∂F (A) + (y − y0 ) (A) + (z − z0 ) (A) = 0 ∂x ∂y ∂z

are equivalent. If A is a singular point of a curve F = 0, then F (A + tP ) = a0 (P ) + a1 (P )t + a2 (P )t 2 + . . . , where a0 (P ) = a1 (P ) = 0. In the case where a0 (P ) = . . . = ak−1 (P ) = 0 for all points P and ak (P ) = 0 for some point P , the number k is called the multiplicity of A. For instance, for a curve l1 · · · lk = 0 where l1 , . . . , lk are linear functions vanishing at a point A, the multiplicity of A is k. Exercise 2.7 a) Find the multiplicity of the point (0, 0) on the curve y 2 = x 2 (x − 1). b) Find the multiplicity of the point (0, 0) on the curve y 2 = x 3 . Exercise 2.8 Let A be a singular point of multiplicity k on a curve F = 0. Show that the restriction of the polynomial F to any line passing through A has a root at A of multiplicity at least k; this multiplicity is greater than k only for finitely many lines. Exercise 2.9 Find the singular points of the following curves in CP2 : a) y 2 z = x 3 ; b) y 2 zn−2 =

n 

(x − ai z), where n ≥ 4 and a1 , . . . , an ∈ C are pairwise distinct.

i=1

Let us discuss the following question: how many points in CP2 are required to uniquely determine a curve of degree n passing through them? First, we simply compute how many parameters are required to determine a curve of degree n. Such  a curve is given by an equation of the form aij k x i y j zk = 0. The number i+j +k=n

of coefficients aij k is equal to the number of different representations of n as an ordered sum of three nonnegative integers. To obtain such a representation, one should insert two separating bars into a sequence i.e., choose 2 elements  of n stars, (n+2)(n+1) = out of n+2. Thus the number of coefficients is n+2 . But proportional 2 2 equations determine the same curve, hence a curve of degree n is determined by

18

2 Algebraic Curves

  n(n+3) d = n+2 parameters. So, it is most likely that through d = n(n + 2 2 −1= 3)/2 points there are finitely many curves of degree n. For instance, if n = 1 we have d = 2, and indeed there is exactly one line through any two distinct points. Exercise 2.10 Show that there is one curve of degree 2 through 2 · 5/2 = 5 points in general position in the plane. To make these informal arguments rigorous, we need the so-called Veronese embedding vn : CP2 → CPd , which sends a point (x : y : z) ∈ CP2 to the point with projective coordinates uij k = x i y j zk , where i + j + k = n; here   n(n+3) d = n+2 2 . 2 −1= Exercise 2.11 a) Check that if not all of the numbers x, y, z are zero, then at least one of the numbers x i y j zk is not zero. b) Show that the Veronese embedding sends distinct points of CP2 to distinct points of CPd .  The Veronese embedding sends a curve aij k x i y j zk = 0 to the cross-section of  the image of CP2 in CPd by the hyperplane aij k uij k = 0. It is also clear that the  image of CP2 in CPd cannot lie entirely in one hyperplane aij k uij k = 0, since  otherwise all points of CP2 would lie on the curve aij k x i y j zk = 0. One can draw a hyperplane through any d points in CPd . This means, in particular, that one can draw a curve of degree n through any d points in CP2 . Since the image of CP2 does not lie in one hyperplane, one can choose d + 1 points in CP2 whose images do not lie in one hyperplane. This means that one can choose d + 1 points in CP2 through which no curve of degree n can pass. This can be done, for example, as follows. Choose a point A1 in an arbitrary way. If A1 , . . . , Ak are chosen, draw through them a curve of degree n and choose a point Ak+1 outside it. In other words, we draw a hyperplane through the images of A1 , . . . , Ak in CPd and choose a point lying in the image of CP2 but outside the resulting hyperplane. This construction allows us to choose points in CPd not lying in one hyperplane. The corresponding points in CP2 do not lie on a curve of degree n. The Veronese images of the chosen points A1 , . . . , Ad+1 are the vertices of a d-dimensional simplex. Any d of these points correspond to a face of the simplex. This means, in particular, that through A1 , . . . , Ad there is a unique curve of degree n. Let us summarize. Through any d = n(n+3) points in CP2 there is a curve of 2 degree n. One can indicate d points in CP2 through which there is a unique curve of degree n. Moreover, if for given d points such a curve is not unique, then by applying a small perturbation one can ensure that for the perturbed points it is unique (d points in CPd lying in the same (d − 2)-dimensional plane can be transformed by a small perturbation into d points not lying in one (d − 2)-dimensional plane).

2.2 Bézout’s Theorem and Its Applications

19

Exercise 2.12 Let p1 , p2 , and p3 be lines in the plane that intersect the lines q1 , q2 , and q3 in 9 distinct points. Show that the images of these points under the Veronese embedding v3 : CP2 → CP9 lie in the same 7-dimensional plane. Given a point in CP2 , the condition that a curve of degree n passes through this point singles out a projective subspace of codimension 1 in CPd . If we require additionally that the point is singular of multiplicity at least k, then this condition singles out a subspace of codimension k(k+1) 2 . Indeed, we may assume that the given point has the coordinates (0 : 0 : 1). Write the equation of the curve in the form a0 zn + a1 (x, y)zn−1 + . . . + an (x, y) = 0, where as (x, y) is a homogeneous polynomial of degree s. The condition is that the polynomials a0 , . . . , ak−1 are zero. Thus in total we obtain 1 + 2 + . . . + k = k(k+1) linear equations. 2 Exercise 2.13 What is the dimension of the condition “a curve touches a given line”? And if we require that the order of tangency is at least k?

2.2 Bézout’s Theorem and Its Applications Let A be a common point of two curves F = 0 and G = 0 in the projective plane. For each of the two curves, it can be either singular or smooth. Besides, if A is a smooth point for both curves, then they can either touch each other at this point or intersect transversally. A common point of two curves that is smooth for both of them is called a point of transversal intersection (respectively, a point of tangency) if the tangents to the curves at this point are different (respectively, coincide). The possible relative positions of two curves at their common point are shown in Fig. 2.4. Transversality is stable: after a small perturbation of the equations Fig. 2.4 The relative position of curves at a common point: (a) transversal intersection at a smooth point of both curves; (b) tangency at a smooth point of both curves; (c) the common point is smooth for one curve and singular for the other one; (d) the common point is singular for both curves

b)

a)

c)

d)

20

2 Algebraic Curves

a) b)

c)

Fig. 2.5 (a) Transversal intersection is stable under small perturbations of the coefficients of curves; (b) tangency is not stable under small perturbations of the coefficients of curves: it turns into a pair of transversal intersections; (c) triple intersection is not stable either: under a small perturbation of the coefficients it turns into three pairwise transversal intersections

of the curves, a neighborhood of a transversal intersection still contains a point of transversal intersection of the new curves. Tangency is an unstable configuration: a small perturbation of the coefficients of the curves can turn a point of tangency into two (or more) points of transversal intersection, see Fig. 2.5. (Note that a perturbation cannot transform tangent curves into disjoint ones, a picture with real parts in this case is misleading.) Let us proceed to the enumeration of the intersection points of two curves. First, consider the case where one curve is a line, and let it intersect a curve of degree n given by a homogeneous equation F (x, y, z) = 0. The line can be parametrized by a map CP1 → CP2 . In an affine coordinate t on the line, it is given by the formula t → (x0 + x1 t, y0 + y1 t, z0 + z1 t). The composition with F gives an equation on the intersection points of the line and the curve F = 0: F (x0 + x1 t, y0 + y1 t, z0 + z1 t) = 0. This is a polynomial equation of order n in t. The images of n roots of this equation are the n intersection points of the line and the curve. If all of them are distinct, then the line and the curve intersect transversally at all their common points. Some roots or their images may coincide, and in this case we have multiple intersection points. However, the sum of the multiplicities is always the same, it is equal to n, the degree of the curve. Hence a line intersects a curve of degree n in n points, counted with multiplicity. To understand how curves of higher degrees intersect, consider the intersection of a curve of degree 3 and a curve of degree 2. Let F (x, y, z) = 0 and G(x, y, z) = 0 be the homogeneous equations of these curves. The polynomials F and G can be

2.2 Bézout’s Theorem and Its Applications

21

written in the form F (x, y, z) = a0 y 3 + a1 (x, z)y 2 + a2 (x, z)y + a3 (x, z), G(x, y, z) = b0 y 2 + b1 (x, z)y + b2 (x, z), where ak (x, z) and bk (x, z) are homogeneous polynomials of degree k (if not identically zero). A point (x0 : y0 : z0 ) is an intersection point of the curves F = 0 and G = 0 if and only if the polynomials F (x0 , y, z0 ) and G(x0 , y, z0 ) have a common root y0 . Let ϕ = ϕ(y) and ψ = ψ(y) be arbitrary complex polynomials of degrees m and n, respectively. They have a common root over C if and only if there exist polynomials ϕ1 and ψ1 of degrees less than m and n, respectively, such that ϕψ1 = ψϕ1 . Indeed, if ϕ and ψ have a common root over C, then they have a common divisor η over C of degree at least 1. Thus we can take ϕ1 = ϕ/η and ψ1 = ψ/η. Conversely, if ϕψ1 = ψϕ1 , then the polynomial Φ = ϕψ1 = ψϕ1 contains all factors of the polynomials ϕ and ψ, but the degree of Φ is less than the sum of the degrees of ϕ and ψ. Hence ϕ and ψ have a common divisor and, consequently, a common root over C. Thus, curves F (x, y, z) = 0 and G(x, y, z) = 0 of degrees m and n, respectively, have a common point (x0 : y0 : z0 ) if and only if for the polynomials F (x0 , y, z0 ) and G(x0 , y, z0 ) there are polynomials f1 (y) and g1 (y) with deg g1 < n and deg f1 < m such that F g1 = f1 G. But here we assume that the polynomials F (x0 , y, z0 ) and G(x0 , y, z0 ) have the same degrees as the curves, i.e., a0 = 0 and b0 = 0. Below we will see that these conditions can always be satisfied by a change of coordinates. In the case where m = 3 and n = 2, the polynomials f1 and g1 have the form f1 (y) = u0 y 2 + u1 y + u2 , g1 (y) = v0 y + v1 . The coefficients u0 , u1 , u2 , v0 , v1 should be chosen so that (a0 y 3 + a1 y 2 + a2 y + a3 )(v0 y + v1 ) = (b0 y 2 + b1 y + b2 )(u0 y 2 + u1 y + u2 ), i.e., a0 v0 a1 v0 + a0 v1 a2 v0 + a1 v1 a3 v0 + a2 v1 a3 v1

= b 0 u0 , = b 1 u0 + b 0 u1 , = b 2 u0 + b 1 u1 + b 0 u2 , = + b 2 u1 + b 1 u2 , b 2 u2 .

22

2 Algebraic Curves

Thus we obtain a homogeneous system of linear equations in v0 , v1 , u0 , u1 , u2 . It has a nonzero solution if and only if the determinant of the matrix ⎛ a0 ⎜0 ⎜ ⎜ ⎜b0 ⎜ ⎝0 0

a1 a0 b1 b0 0

a2 a1 b2 b1 b0

a3 a2 0 b2 b1

⎞ 0 a3 ⎟ ⎟ ⎟ 0⎟ ⎟ 0⎠ b2

(2.2)

vanishes. For curves in CP2 , the degrees of the homogeneous polynomials ai (x, z) and bi (x, z) are exactly i (unless the polynomials are zero). It is also clear that a0 = F (0, 1, 0) and b0 = G(0, 1, 0). This means that the condition a0 b0 = 0 holds if none of the curves F = 0 and G = 0 passes through the point at infinity (0 : 1 : 0). For curves F = 0 and G = 0 of arbitrary degree, one can construct a matrix similar to (2.2). Its determinant is called the resultant of the polynomials F and G. The computation of the resultant is also called the “elimination of a variable” (in our case, the variable y is being eliminated). For curves in CP2 viewed in the affine chart y = 1, the resultant is a polynomial in x and z. Its roots are the (x : z)coordinates of the intersection points of the curves. To compute the y-coordinate, one should substitute the obtained root (x0 : z0 ) of the resultant into one of the polynomials F or G, find the roots of the resulting polynomial in y, and choose whichever of them is a root of the second polynomial. In general, these computations can be performed only approximately. Theorem 2.1 (Bézout) For curves of degrees m and n in CP2 , the resultant is a homogeneous polynomial of degree mn (or identically zero). Proof We again restrict ourselves to the case m determinant of the matrix ⎛ a0 a1 a2 a3 ⎜0 a a a ⎜ 0 1 2 ⎜ S = ⎜ b0 b1 b2 0 ⎜ ⎝ 0 b0 b1 b2 0 0 b0 b1

= 3 and n = 2. Let R(x, z) be the ⎞ 0 a3 ⎟ ⎟ ⎟ 0 ⎟. ⎟ 0⎠ b2

We must prove that R(λx, λz) = λ6 R(x, z). Indeed, in this case either R is identically zero, or R is a homogeneous polynomial of degree 6. By assumption, ak (λx, λz) = λk ak (x, z) and bk (λx, λz) = λk bk (x, z). Therefore, R(λx, λz) = det Sλ , where

2.2 Bézout’s Theorem and Its Applications

⎛ a0 ⎜0 ⎜ ⎜ Sλ = ⎜b0 ⎜ ⎝0 0

λa1 a0 λb1 b0 0

23

λ2 a2 λa1 λ2 b2 λb1 b0

⎞ λ3 a3 0 λ2 a2 λ3 a3 ⎟ ⎟ ⎟ 0 0 ⎟. ⎟ 2 λ b2 0 ⎠ λb1 λ2 b2

Multiply the first and second rows of the matrix Sλ by 1 and λ, and multiply the third, fourth, and fifth rows by 1, λ, and λ2 , respectively. This results in the matrix S with its columns multiplied by 1, λ, λ2 , λ3 , and λ4 . Hence det Sλ = λp−q−r det S, where p = 1 + 2 + 3 + 4 = 10, q = 1, and r = 1 + 2 = 3. In the general case, p = 1 + 2 + . . . + (m + n − 1), q = 1 + 2 + . . . + (m − 1), and r = 1 + 2 + . . . + (n − 1). Hence p−q −r =

(m + n)(m + n − 1) m(m − 1) n(n − 1) − − = mn. 2 2 2

Exercise 2.14 Let A be a point of multiplicity r of a curve f = 0 and a point of multiplicity s of a curve g = 0. Show that it corresponds to a root of the resultant of f and g of multiplicity at least rs. Here is another, topological, proof of Bézout’s theorem. Assume that the first curve is a collection of m distinct lines passing through one point, the second curve is a collection of n distinct lines passing through one point (different from the first one), and these two collections have no common lines. Clearly, these curves have mn points of transversal intersection. Slightly changing their coefficients, we obtain a pair of curves still having mn points of transversal intersection. Thus, in the space of pairs of curves (i.e., in the product of two projective spaces, that of coefficients of curves of degree m and that of coefficients of curves of degree n) there is an open subset of curves having mn points of transversal intersection. The condition that the intersection of two curves is transversal is an algebraic condition determining a hypersurface in the space of pairs of curves. A hypersurface in a complex manifold does not separate it. The complement to the hypersurface under consideration is a connected open dense subset consisting of the pairs of curves with transversal intersection, and for all such pairs the number of intersection points is the same. It follows that the space of pairs of curves contains an open dense subset of pairs of curves that intersect transversally and have the same number mn of intersection points. Pairs of curves with nontransversal intersection also have mn intersection points counted with multiplicity (if they have no common irreducible components). The theorem is proved. Now let us describe some applications of Bézout’s theorem. Recall that through any d = n(n+3) points in the plane there is a smooth curve of degree n. For almost 2 all collections of points, such a curve is unique. In the case where through given d points there is a unique curve of degree n, we will say that these points are in general position. Subsets of a system of points in general position will also be called points in general position (for a curve of the same degree n).

24

2 Algebraic Curves

Using Bézout’s theorem, one can prove some properties of systems of points of algebraic curves. As a rule, these theorems deal with points in general position. Theorem 2.2 Assume that two curves of degree n have n2 intersection points, with np of these points lying on an irreducible curve of degree p. Then the remaining n(n − p) points lie on a curve of degree n − p. Proof Let F1 = 0 and F2 = 0 be the equations of the curves of degree n and G = 0 be the equation of the curve of degree p containing the np intersection points. For every point of the plane, we can choose λ1 and λ2 such that the curve λ1 F1 + λ2 F2 = 0 contains it. Hence we may assume that the curve λ1 F1 + λ2 F2 = 0 passes through a point of the curve G = 0 different from the given np points. Then the curves λ1 F1 + λ2 F2 = 0 and G = 0 have np + 1 common points; therefore, they have a common component. Since the curve G = 0 is irreducible, this common component necessarily coincides with it. Hence λ1 F1 + λ2 F2 = GH , where H is a homogeneous polynomial of degree n − p. The curve GH = 0 passes through all n2 intersection points, while the curve G = 0 passes through only np of them. Therefore, the curve H = 0 passes through the remaining n(n − p) points. Corollary 2.1 Let a 2n-gon be inscribed in a curve of degree 2 (see Fig. 2.6). Then all intersection points of even-numbered sides with nonadjacent odd-numbered sides lie on some curve of degree n − 2. Proof The lines containing the even-numbered sides form a degenerate curve of degree n, and the lines containing the odd-numbered sides form another curve. The vertices of the 2n-gon form a system of 2n intersection points of the curves under consideration. They lie on a curve of degree 2, hence the remaining n(n − 2) points lie on a curve of degree n − 2.

Fig. 2.6 A hexagon inscribed in a conic. Its sides are divided into two groups, each containing three sides. The intersection points of the sides of the first group with the nonadjacent sides of the second group lie on the same line

2.2 Bézout’s Theorem and Its Applications

25

An example for the case n = 3 is shown in Fig. 2.6. A pencil of curves is a family of curves of the form λF + μG = 0, where F = 0 and G = 0 are two different curves of the same degree and λ and μ are arbitrary numbers that are not simultaneously zero. All curves of a pencil pass through the intersection points of two distinct curves of this pencil. Theorem 2.3 A pencil of curves of degree n passing through given − 1 points in general position has (n−1)(n−2) common points d − 1 = n(n+3) 2 2 more. Proof Let F = 0 and G = 0 be the equations of two curves from the pencil under consideration. Then all other curves of this pencil have equations of the form λF + μG = 0. Hence they all pass through the n2 common points of the curves F = 0 and G = 0. The given points constitute only a part of them; the number of remaining points is n2 −

(n − 1)(n − 2) n(n + 3) +1= . 2 2

In particular, for n = 3 we obtain Corollary 2.2 For any eight points in general position in the plane there exists a ninth point such that any cubic curve passing through the eight points also passes through the ninth point. For intersection points of curves of distinct degrees, a theorem similar to Theorem 2.3 looks as follows. Theorem 2.4 Consider all curves of degree n passing through given np − (p−1)(p−2) points of a curve of degree p with p < n. Then all of them 2 (p−1)(p−2) have common points more, and all these points lie on the degree p curve 2 under consideration. Proof If α points are chosen on a curve F = 0 of degree p and α ≤ n(n + 3)/2, then there exists a curve G = 0 of degree n passing through them. The curve G = 0 intersects the curve F = 0 in np = α + β points. We must prove that if , then the β points do not depend on the choice of a curve α = np − (p−1)(p−2) 2 G = 0. Consider a curve G = 0 of degree n passing through the given α points. The curves G = 0 and G = 0 have n2 intersection points. They include the given α points; choose α  = n(n+3) − 1 − α more of them. Then there is a pencil of curves 2

26

2 Algebraic Curves

of degree n passing through the given α + α  = α ≤

n(n+3) 2

− 1 points. In the case where

(n − p)(n − p + 3) , 2

through the chosen α  points there is a curve H = 0 of degree n − p. Hence the pencil of curves of degree n is generated by the curves G = 0 and F H = 0. In particular, G = λG + μF H . Therefore, the curve G = 0 intersects the curve F = 0 in the same np = α + β points as the curve G = 0. It remains to observe that the condition α  ≤ (n−p)(n−p+3) is equivalent to 2  ≥ np − (p−1)(p−2) . α = n(n+3) − 1 − α 2 2 Theorem 2.4 can be generalized; namely, instead of a family of curves of degree n passing through the intersection points of curves of degrees n and p, one may consider a family of curves of degree k, not necessarily equal to n or p. The corresponding result looks as follows. Theorem 2.5 (Cayley) passing through

Let n, p < k < n + p − 3. Then any curve of degree k

np −

(n + p − k − 1)(n + p − k − 2) 2

intersection points of curves of degrees n and p also passes through all the other intersection points. Proof Let α points be chosen among the np intersection points of a curve F = 0 of degree p and a curve G = 0 of degree n. Consider an arbitrary curve G = 0 of degree k passing through them. Among the intersection points of the curves F = 0 and G = 0, choose α  = (k−n)(k−n+3) points; among the intersection points of the 2   curves G = 0 and G = 0, choose α = (k−p)(k−p+3) points more. This can be 2 done, since, by assumption, k − p + 3 ≥ p and k − p + 3 ≤ n, whence α  < kp and α  < kn. Through α  points there is a curve H  = 0 of degree k − p, and through α  points there is a curve H  = 0 of degree n − p. In our case α = np − (n+p−k−1)(n+p−k−2) , whence α + α  + α  = k(k+3) − 1. 2 2 This means that the chosen points determine a pencil of curves of degree k. The curves G = 0, F H  = 0, and GH  = 0 belong to this pencil. Therefore, G = λF H  + μGH  , and hence the curve G = 0 passes through all intersection points of the curves F = 0 and G = 0.

2.3 Rational Parametrization We have already seen how a line in the plane can be given in parametric form. Next in simplicity is a rational parametrization of a conic, which allows one to represent its points in the form (x(t) : y(t) : z(t)), where x(t), y(t), z(t) are

2.3 Rational Parametrization

27

polynomials. Such a parametrization can be obtained as follows. Let A be a point of a nondegenerate conic. A line passing through it meets the conic also in another point. But all lines passing through a given point form a projective line. Associating with every line passing through A its second intersection point with the curve, we obtain a one-to-one mapping from the projective line to the conic, and it determines a rational parametrization of this conic. To find a coordinate representation of this parametrization, choose affine coordinates in which the conic is given by the equation x 2 + y 2 = 1 and the point A has the coordinates (−1, 0). The equation of a line passing through (−1, 0) has the form y = t (x + 1), and the x-coordinate of its intersection points with the conic is given by the equation x 2 + (t (x + 1))2 = 1, i.e., (x + 1)((1 + t 2 )x − (1 − t 2 )) = 0. The solution x = −1 corresponds to the original point A. The second 2 solution x = 1−t is the coordinate of the second intersection point. Substituting 1+t 2 it into the equation of the line, we obtain a formula for the y-coordinate: 2t y = 1+t 2 . The corresponding polynomial parametrization of the curve is (s : t) → (s 2 − t 2 : 2st : s 2 + t 2 ). All coordinate maps of such a parametrization are homogeneous polynomials of degree 2 in the projective coordinates (s : t) on the projective line. Exercise 2.15 Write down parametrization equations for an arbitrary nondegenerate conic F (x, y, z) = 0 which is not in the standard form. A rational parametrization of a conic has the following remarkable property: if the conic is given by an equation with rational coefficients, and a point (x0 , y0 ) has rational coordinates, then the point (x(t), y(t)) has rational coordinates if and only if t is a rational number (or t = ∞). Indeed, if x(t) and y(t) are rational, )−x0 then t = x(t y(t )−y0 is also rational. The converse is clear from our construction. Thus, having a rational parametrization of a conic, one can describe all rational points on this conic, i.e., find all solutions of the corresponding equation in rational numbers. Some equations of higher degree can also be solved by a similar method. Before describing it, we mention another important application of a rational parametrization, to computation of integrals. An example of such an application is the following exercise. Exercise 2.16 a) Parametrize the curve y 2 = ax 2 + bx + c using the family of lines y = t (x − α), where α is a root of the equation ax 2 + bx + c = 0.

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2 Algebraic Curves

b) Apply the obtained parametrization to compute antiderivatives of the form 

 R(x, ax 2 + bx + c) dx,

where R(x, y) is a rational function. The property of curves of higher degree to admit a rational parametrization is closely related to their having singular points. An irreducible curve of degree 3 can have at most one singular point of multiplicity 2. Indeed, if A and B are two different points of multiplicity 2, then the line AB intersects the curve at least four times. For a cubic curve, this means that the whole line AB belongs to the curve, i.e., the curve is reducible. Exercise 2.17 Show that a cubic curve cannot have singular points of multiplicity 3 and more. For a curve of degree n, one can also estimate the number of singular points from above. Theorem 2.6 An irreducible curve of degree n can have at most N = double points.

(n−1)(n−2) 2

Proof The case n = 3 has been already dealt with. Let us now consider the case n = 4. In this case N = 3. Assume that A1 , A2 , A3 , A4 are double points of a curve of degree 4. To arrive at a contradiction, take an arbitrary point P on this curve different from Ai . Through the points P , A1 , A2 , A3 , A4 there is a curve of degree 2. It intersects the degree 4 curve at least once in P and at least twice in each of the points Ai . Thus the total number of intersection points is at least 1 + 2 · 4, but Bézout’s theorem says that there are exactly 2 · 4 of them, a contradiction. For a curve of degree n, assume that A1 , . . . , AN+1 are its double points. Choose arbitrary points P1 , . . . , Pn−3 on this curve different from Ai . Then the total number of the points Ai and Pj is (n + 1)(n − 2) (n − 1)(n − 2) + 1 + (n − 3) = . 2 2 Hence through these points there is a curve of degree n − 2. The multiplicity of the intersection of this curve with the degree n curve is at least (n − 1)(n − 2) + 2 + (n − 3) = n(n − 2) + 1, which contradicts Bézout’s theorem. We will not prove that for every n there exists an irreducible curve of degree n with exactly N = (n−1)(n−2) double points. Instead, we will prove that a curve of 2 degree n with N double points admits a rational parametrization. This means that there exist homogeneous polynomials x(s, t), y(s, t), and z(s, t) of degree n such

2.3 Rational Parametrization

29

Fig. 2.7 A cubic with a double point

that the image of the mapping (s : t) → (x(s, t) : y(s, t) : z(s, t)) ∈ CP2 coincides with the given curve. In general, different values of t yield different points of the curve. Double points are an exception, each of them having exactly two preimages. From the topological point of view, a curve in CP2 that admits a rational parametrization is so structured that puncturing it at the double points results in a sphere with several punctures. Theorem 2.7 An irreducible curve of degree n with N = admits a rational parametrization.

(n−1)(n−2) 2

double points

Proof First, consider a cubic curve with the double point O = (0 : 0 : 1). In appropriate coordinates, in the affine chart z = 1 it has the equation y 2 = (x + 1)x 2 , see Fig. 2.7. A line passing through O is given by an equation of the form y = tx. It has a double intersection with the cubic curve at O, and hence meets it in exactly one other point (x(t), y(t)). To find this second intersection point, substitute the equation of the line into that of the curve. We obtain t 2 x 2 = (x + 1)x 2 .

(2.3)

Dividing by x 2 yields the x-coordinate of the second intersection point x = t 2 − 1, and the y-coordinate is equal to y = t (t 2 − 1). Thus, to every point of the curve except (0, 0) there corresponds exactly one value of t. To the point (0, 0) there correspond two tangents y = t1 x and y = t2 x to the two branches of the curve at this point. Hence the exceptional point of the cubic curve is the double point, and the exceptional values of the parameter t are t1 and t2 .

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2 Algebraic Curves

Fig. 2.8 A lemniscate

For n > 3, given a curve of degree n with double points A1 , . . . , AN , one should additionally fix arbitrary points P1 , . . . , Pn−3 . There is a pencil of curves of degree n − 2 passing through A1 , . . . , AN , P1 , . . . , Pn−3 . Each of these curves has n(n − 2) − 1 common points with the given curve, hence they have exactly one other common point. It is also clear that one can draw a curve of degree n−2 from our pencil through every point of the curve and the points A1 , . . . , AN , P1 , . . . , Pn−3 . Thus we have parametrized all the points of our curve except A1 , . . . , AN , P1 , . . . , Pn−3 by the parameter of the pencil. We leave the reader to check that this parametrization can be extended to the points P1 , . . . , Pn−3 and to a parametrization of each of the two branches at the double points A1 , . . . , AN of the curve. The proof of Theorem 2.7 allows one to construct a rational parametrization explicitly. As an example, consider the family of lemniscates (x 2 + y 2 )2 = a 2 (x 2 −y 2)z2 , which are curves of degree 4, see Fig. 2.8. A lemniscate has three double points (0 : 0 : 1) and (1 : ±i : 0). As an additional point P1 , choose (0 : 0 : 1), i.e., consider the pencil of curves of degree 2 passing through the points (1 : ±i : 0) and touching one of the two branches of the lemniscate at the point (0 : 0 : 1). In the affine chart z = 1, such curves (for one of the branches) have the form x 2 + y 2 = t (x − y). Substituting this into the equation of the lemniscate yields a 2(x 2 − y 2 ) = (x 2 + y 2 )2 = t 2 (x − y)2 , whence y=

t 2 − a2 x; t 2 + a2

in turn, substituting this expression into the equation of the lemniscate yields the parametrization x=

ta 2 (t 2 + a 2 ) , t 4 + a4

y=

ta 2 (t 2 − a 2 ) . t 4 + a4

Exercise 2.18 Show that the curve (x 2 − y)2 = y 3 admits a rational parametrization.

2.3 Rational Parametrization

31

Exercise 2.19 a) Show that the curve Pn (x, y) + Pn−1 (x, y) = 0, where Pn and Pn−1 are homogeneous polynomials of degrees n and n − 1, respectively, admits a rational parametrization. b) Show that a curve of degree n having a singular point of multiplicity n−1 admits a rational parametrization. Exercise 2.20 Show that the points (x(t), y(t)) with x(t) = a0 + a1 t + . . . + an t n , y(t) = b0 + b1 t + . . . + bn t n lie on a curve of degree at most n. We mention without proof that if a curve admits a rational parametrization (x(t), y(t)), then the parameter t can be written as a rational function of x and y. This property allows one to use a rational parametrization of a curve F (x, y) = 0 to compute antiderivatives of the form  R(x, y) dx, F (x,y)=0

where R is a rational function. Indeed, let a curve F (x, y) = 0 admit a rational parametrization. Then its points can be written in the form (x(t), y(t)) with x and y rational functions. Hence R(x(t), y(t)) = R1 (t) and dx(t) = r(t) dt with R1 and r rational functions. Therefore,   R(x, y) dx = Q(t) dt, F (x,y)=0

 where Q is a rational function. It is well known that Q(t)dt can be expressed in terms of elementary functions of t. Using the fact that t = t (x, y) is a rational function of x and y, one can obtain an expression for R(x, y)dx in terms of elementary functions of x and y, cf. Exercise 2.16.

Chapter 3

Complex Structure and the Topology of Curves

Every complex algebraic curve is a two-dimensional oriented surface. As we already know, the topology of such surfaces is very simple: for a compact surface, the topology is uniquely determined by its genus (or, equivalently, its Euler characteristic). However, along with a topological structure, a curve has a complex structure. It singles out analytic functions among all the functions on the curve. If a surface has a simple structure, for instance, if it is a disk or a sphere, then any two complex structures on this surface are equivalent. However, a surface of positive genus admits many different complex structures. Moreover, the various complex structures on surfaces of a given genus constitute a space. The geometry of these spaces becomes more complicated as the genus grows. We will study the simplest properties of this geometry throughout the whole course.

3.1 A Complex Structure on a Curve Locally, a smooth curve in the n-dimensional projective space is given by a system of n − 1 polynomial equations, hence, by the implicit function theorem, we can identify a neighborhood of every point of such a curve C with the unit disk in C1 . Thus, for every point A of C there is a one-to-one mapping mU from a neighborhood U = U (A) ⊂ C onto the unit disk. It is called a local coordinate1 in U . Of course, there are many such neighborhoods and many such mappings. If two such neighborhoods have a nonempty intersection, then the mapping mU m−1 V , defined in a domain contained in the unit disk, is holomorphic, i.e., complex analytic and invertible, and the inverse mapping is also complex analytic. Such a collection of neighborhoods and local coordinates is called a complex structure on C.

1 Sometimes, in the definition of a local coordinate, one requires that the point A is mapped to the center of the disk.

© Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_3

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3 Complex Structure and the Topology of Curves

The notion of a complex structure allows one to introduce also the notion of a holomorphic mapping between complex curves. A mapping f : C → C  between complex curves endowed with complex structures {(U, mU )} and {(U  , mU  )}, respectively, is said to be holomorphic if for every point A ∈ C there is a neighborhood U  ⊂ C  of f (A ) with a mapping mU  : U  → C and a neighborhood U of A with a mapping mU : U → C such that the mapping mU  f m−1 U is −1  holomorphic where it is defined (its domain of definition mU (f (U ) ∩ U ) is open and contains the point mU (A), hence it is necessarily nonempty). It is clear that if the composed mapping fU = mU  f m−1 U is holomorphic for some pair of neighborhoods U, U  and local coordinates mU , mU  , then for any other pair of neighborhoods V , V  and local coordinates mV , mV  the mapping fV = mV  f m−1 V is holomorphic on the set mV (f −1 (U  ∩ V  ) ∩ U ∩ V ). Indeed, fV = mV  mU 

−1

fU mU m−1 V .

Example 3.1 The set of lines passing through a given point of the projective plane is a projective line. Hence, fixing a point A in CP2 , we can construct a mapping from an arbitrary plane curve C to the projective line, by associating with every point of C the line passing through this point and the point A. Show that this mapping is holomorphic. A one-to-one holomorphic mapping between a pair of curves for which the inverse mapping is also holomorphic is called a biholomorphic mapping, or an isomorphism of curves. The existence of a complex structure can be taken as a definition of a complex curve. A Riemann surface is a two-dimensional oriented surface endowed with a complex structure. Every smooth complex curve in a projective space, and also every open subset of such a curve, is a Riemann surface. It turns out that these examples exhaust all Riemann surfaces: every Riemann surface can be realized as an open subset of a smooth projective curve. However, the proof of this fact is not easy, see, e.g., [16]. To prove it, one must show that every Riemann surface has sufficiently many meromorphic functions (which allows one to construct an embedding of this surface into a projective space). We will not prove this theorem; it can be assumed that we consider only Riemann surfaces that admit a projective embedding.

3.2 The Genus of a Smooth Plane Curve A nonsingular algebraic curve in CP2 is a Riemann surface. A complex structure determines an orientation on this surface: the multiplication by the imaginary unit i in any local coordinate defines the positive direction of rotation in the vicinity of every point. Every smooth algebraic curve is compact, since it is a closed subset in

3.2 The Genus of a Smooth Plane Curve

35

a compact space. From the topological point of view, a curve in CP2 is a surface of some genus g, i.e., a sphere with g handles. To compute the genus of a curve and to study some other properties of curves, it is sometimes useful to consider a projection of the plane CP2 to CP1 and the induced mapping from a curve C ⊂ CP2 onto CP1 . By a projection we mean an ordinary projection of a plane onto a line. An example of such a projection is the mapping (x : y : z) → (x : y), defined on the complement to the point (0 : 0 : 1) ∈ CP2 . Under a projection of the (punctured) plane CP2 onto CP1 , the preimage of a point is a line. In general, a line in CP2 intersects a curve C of degree n in exactly n points. Hence for all points of CP1 except finitely many, the preimage under the projection of the curve C ⊂ CP2 onto CP1 consists of exactly n points. If we exclude from consideration projections along lines that are components of C, or consider only irreducible curves of degree greater than 1, then the preimage of every point will contain at most n points. From the topological point of view, the mapping p : C → CP1 is a ramified covering. Points of CP1 whose preimages contain less than n points are ramification points of this covering. We have established that every plane curve can be represented as a ramified covering of the projective line. The latter is obtained by adding a point at infinity to the one-dimensional complex space C, i.e., is homeomorphic to the sphere S 2 . Thus we have given another proof of the fact that every algebraic curve C in CP2 is an oriented surface. The study of the ramified covering allows one to find the genus of a curve. Example 3.2 The genus of the Fermat curve x n + y n + zn = 0 is

(n−1)(n−2) . 2

Proof First observe that the Fermat curve C is nonsingular. Besides, the projection p : CP2 \(0 : 0 : 1) → CP1 induces a mapping p : C → CP1 , since (0 : 0 : 1) ∈ / C. The preimage of a point (x0 : y0 ) ∈ CP1 consists of the points (x0 : y0 : z) ∈ CP with zn = −(x0n + y0n ). If x0n + y0n = 0, then it consists of exactly n points, and if x0n + y0n = 0, then it consists of one point. Hence p is an n-sheeted ramified covering with the ramification points (1 : εn ) where εn is an nth root of −1. There are n such points, and the ramification index at each of them is equal to n. The Riemann–Hurwitz theorem implies that χ(C) = n(χ(S 2 ) − n) + n = n(2 − n) + n = −n2 + 3n. Therefore, g(C) =

n2 − 3n + 2 (n − 1)(n − 2) 2 − χ(C) = = . 2 2 2

Exercise 3.1 Define a mapping from the Fermat curve C to CP1 by setting, in the chart z = 1, its value at a point A of C equal to the nth power of the x-coordinate of A.

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3 Complex Structure and the Topology of Curves

a) Show that this mapping can be extended to a holomorphic mapping from the Fermat curve to CP1 . b) Find the ramification points of this mapping and the ramification indices at their preimages. c) Show that the preimage of the real line under this mapping is the complete tripartite graph Kn,n,n (its vertices are the preimages of the critical values, and its edges are the preimages of the real segments connecting them). Remark 3.1 This construction gives an embedding of the graph Kn,n,n into a surface of the smallest possible genus. The above example is a special case of the following more general result. Theorem 3.1 The genus of a nonsingular plane curve of degree n is

(n−1)(n−2) . 2

In particular, the genus of a curve of degree 3 is 1, the genus of a curve of degree 4 is 3, and the genus of a curve of degree 5 is 6. The First Proof Let F (x, y, z) = 0 be the equation of a nonsingular curve C ⊂ CP2 of degree n. We may assume that the point (0 : 0 : 1) does not belong to C. Then we may consider the projection p : C → CP1 that sends (x : y : z) to (x : y). The preimage of a point (x0 : y0 ) consists of the points (x0 : y0 : z) where z is a root of the polynomial P (z) = F (x0 , y0 , z). The ramification points correspond to the multiple roots of this polynomial, i.e., the intersection points of the curves F = 0 and ∂F ∂z = 0. The degrees of these curves are equal to n and n − 1, respectively. First consider the case where these curves have no multiple intersections, i.e., there are n(n − 1) intersection points. Then the preimage of each intersection point consists of exactly n − 1 points, whence χ = n(2 − n(n − 1)) + n(n − 1)2 = 2n − n(n − 1) = 3n − n2 and, consequently, g=

(n − 1)(n − 2) . 2

Now assume that there are multiple intersection points. Recall that χ(C) = n(2 − k) + n1 + . . . + nk = 2n −

k 

(n − ni ).

i=1

For ordinary intersection points, n − ni = 1. If d ordinary intersection points merge into one point of multiplicity d, then the d terms n − ni = 1 are replaced by a single term equal to d, and the sum does not change.

3.2 The Genus of a Smooth Plane Curve

37

The Second Proof Associating with a curve of degree n the coefficients of its defining equation (up to proportionality), we can identify the set of all curves of degree n with CPd , where d = n(n+3) 2 . A curve F = 0 is singular if the system of equations F = 0, Fx = 0, Fy = 0, Fz = 0 has a nonzero solution. In this system, the equation F = 0 is superfluous, since nF = xFx +yFy +zFz . From the three equations Fx = 0, Fy = 0, Fz = 0 we can exclude x, y, and z. This results in an algebraic equation on the coefficients of the function F . Thus, singular curves belong to a set of complex codimension 1, i.e., real codimension 2. It does not separate CPd , hence the set of nonsingular curves is connected. A small perturbation of a nonsingular curve cannot change its genus, hence all nonsingular curves of degree n have the same genus. Now we could use Example 3.2, where we have computed the genus of one nonsingular curve of degree n. But we can also compute the genus of another curve of degree n, for instance, l1 · . . . · ln = ε, where l1 , . . . , ln are lines in general position and ε is a sufficiently small nonzero number. The curve li = 0 is a sphere S 2 ⊂ CP2 , and the curve l1 · l2 · . . . · ln = 0 is the union of such spheres pairwise intersecting in one point. For the curve l1 · . . . · ln = ε, the intersection point of two curves is replaced by a tube connecting the spheres corresponding to these lines. Thus the curve under consideration looks as n spheres pairwise connected by tubes (see Fig. 3.1 for n = 4). A curve of degree n can be obtained from a curve of degree n − 1 by adding n − 2 handles, and the genus of a curve of degree 1 is 0. Hence the genus of a curve of degree n is 1 + 2 + . . . + (n − 2) =

(n − 1)(n − 2) . 2

Exercise 3.2 Let Y be the plane curve given by the equation y 2 = x − x 3 in the chart z = 1. a) Draw the real part of this curve. b) Find the ramification points of the projection of Y to the x axis and to the y axis, and the ramification indices at their preimages. Exercise 3.3 Let Y be a generic curve in CP1 × CP1 whose projections to the first and second factors have degrees m and n, respectively. Find the genus of Y . Fig. 3.1 A smooth curve close to a quadruple of lines

38

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3 Complex Structure and the Topology of Curves

b)

Fig. 3.2 (a) A loop shrinks down to a singular point; (b) an irreducible curve of degree 4 possessing the maximum possible number of double points

Theorem 3.1 allows one to give another proof of Theorem 2.6 on the number of double points of an irreducible curve of a given degree. Indeed, a smooth curve of degree n is a two-dimensional orientable surface of genus (n − 1)(n − 2)/2. The appearance of a double point in the deformed curve, provided that it remains irreducible, means that a loop near the surface shrinks down to a point, see Fig. 3.2a. Locally, a neighborhood of a singular point looks like a transversal intersection of two real two-dimensional planes in a four-dimensional real space. Separating the sheets at the intersection point, we obtain a new surface. The irreducibility of the curve is equivalent to the connectedness of the resulting surface. If it is connected, then its genus is one less than the genus of the original surface. Thus the possible number of double points does not exceed the genus of the curve: the appearance of each additional double point decreases the genus by 1. If the number of double points is exactly equal to the genus (Fig. 3.2b), then splitting all these points yields a sphere, i.e., the singular curve admits a rational parametrization, see Theorem 2.7. The Riemann–Hurwitz formula counts the number of singular points in a ramified covering of a line. Applying it to the mapping that projects from a point in the plane, one can find the number of tangents to a given curve C passing through a given point O. Theorem 3.2 There are exactly n2 − n distinct tangents from a point in general position to a nonsingular curve of degree n. Proof We have essentially proved this assertion in the proof of Theorem 3.1. Consider the projection p of the curve C to the projective line CP1 formed by the lines passing through O. The mapping p is a ramified covering. The tangents to C passing through O correspond to the ramification points (Fig. 3.3). Assume that C is a nonsingular curve of degree n and O lies neither on C, nor on double tangents, nor on tangents at inflection points. (A double tangent is a line that touches C at two different points. A point of C is an inflection if the multiplicity of the intersection of C and the tangent to C at this point is greater than 2.) Then for all ramification

3.2 The Genus of a Smooth Plane Curve

39

Fig. 3.3 Projecting a cubic curve from a point. Two of the six tangents lie in a complex domain and are not shown in the figure

Fig. 3.4 A pair of tangents to a curve merge into one as the intersection point tends to a point on the curve

points n − ni = 1, whence χ(C) = 2n −

k 

(n − ni ) = 2n − k,

i=1

where k is the number of ramification points. In our case χ(C) = 3n − n2 , whence k = n2 − n. Corollary 3.1 From a point in general position on a nonsingular curve C of degree n there are exactly n2 − n − 1 distinct tangents to C. Proof As a point O tends to a point O1 on C, the two tangents passing through O merge into one (Fig. 3.4). For a point O on a curve C, one of the n2 − n − 1 tangents is the tangent to C at O. If we exclude it, there remain n2 − n − 2 tangents. The situation with these tangents is most interesting in the case of cubic curves. In this case n2 − n − 2 = 4. A cubic curve has no double tangents: the multiplicity of the intersection of a double tangent with a curve is at least 4, while the multiplicity of the intersection of a line

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3 Complex Structure and the Topology of Curves

with a cubic curve cannot exceed 3. Hence points not in general position are only inflection points. But at such a point nothing bad happens, just one of the four tangents coincides with the tangent to the curve at this point. Thus through each point of a cubic curve C there are four pairwise distinct tangents to C (which differ from the tangent at this point for all points except inflections). This quadruple of lines determines four points of the projective line formed by all lines passing through the given point. One can consider their cross ratio d−a [a, b, c, d] = c−a c−b : d−b , where a, b, c, and d are the coordinates of these points in some coordinate on the line (the number does not depend on the choice of a coordinate). The cross ratio depends on the order of points. Exercise 3.4 a) Show that [a, b, c, d] = [b, a, c, d]−1 = [a, b, d, c]−1. b) Show that [a, b, c, d] = 1 − [a, c, b, d]. c) Show that by permuting the four numbers a, b, c, d we can obtain at most six different values of their cross ratio, and give examples of quadruples for which the number of different ratios is equal to 1, 2, 3, and 6. Exercise 3.5 a) Show that the function J (λ) =

(1 − λ + λ2 )3 λ2 (1 − λ)2

of a complex variable λ is invariant under replacing λ with λ−1 and with 1 − λ. b) Show that if λ = [a, b, c, d] is the cross ratio of four points, then J (λ) does not depend on the order of the points. Let x be a point of a cubic curve C and λ(x) be the cross ratio of the four tangents from x to C. These tangents are always pairwise distinct, hence λ(x) does not take the values 0, 1, and ∞. This means that J (λ(x)) is a one-valued function that does not take the value ∞, i.e., a bounded function. But a cubic curve C in CP2 is compact, and a bounded meromorphic function on a compact surface without boundary is constant. Hence the value J (λ(x)) = J (C) does not depend on the choice of a point on C, but depends only on the curve itself. For some reason, instead of the invariant J it is more convenient to consider the invariant j = 28 J . Under a projective transformation of the plane, tangents to a given curve C go to tangents to its image. The quadruple of tangents from a given point also undergoes a projective transformation, hence their cross ratio remains the same. Thus the value J (C) is a projective invariant of a cubic curve. As we will see later, this means that it is also a biholomorphic invariant. Exercise 3.6 Let C be the curve given by the equation y 2 = x(x − 1)(x − λ) in the affine chart z = 1. Show that the invariant J (C) equals J (λ).

3.3 Hessian and Inflection Points

41

This exercise means that there are different cubic curves: if J (λ1 ) = J (λ2 ), then the curves y 2 = x(x − 1)(x − λ1 ) and y 2 = x(x − 1)(x − λ2 ) cannot be projectively equivalent and biholomorphic to each other.

3.3 Hessian and Inflection Points Let A be a smooth point of a plane curve C given by a homogeneous equation F (x, y, z) = 0. The restriction of the function F to the tangent A + tP to C at the point A has a zero of order at least 2. For a curve C that does not contain a line as an irreducible component, this order is exactly 2 for all points except finitely many. As we have already mentioned, points at which this order is greater than 2 are called inflection points of the curve C. An inflection point is said to be ordinary if this order is exactly 3, see Fig. 3.5. To find the inflection points of a curve, one uses the Hessian of its defining polynomial. The Hessian of a homogeneous polynomial is the determinant of the matrix of its second-order partial derivatives:  2  ∂ F  ∂x 2  ∂2F H (F ) =  ∂y∂x  ∂2F  ∂z∂x

∂2F ∂x∂y ∂2F ∂y 2 ∂2F ∂z∂y

∂2F ∂x∂z ∂2F ∂y∂z ∂2F ∂z2

    .   

Exercise 3.7 Show that a point A of a curve is an inflection if and only if the Hessian of the defining polynomial of this curve vanishes at A. This exercise allows one to count the number of inflection points for a curve of a given degree. Equating the Hessian of the polynomial F to zero determines a new algebraic curve in the plane, and the inflection points of the original curve C are exactly the intersection points of C with the curve H (F ) = 0. The number of intersection points (under the assumption that the intersection is transversal) can now be found by Bézout’s theorem. For instance, if F has degree 3, then its Hessian Fig. 3.5 A curve with an ordinary inflection point and the tangent at this point

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also has degree 3 (second-order derivatives of a cubic polynomial have degree 1, hence the degree of the 3×3 determinant composed of them is 3). Therefore, a cubic curve has 9 inflection points. Unfortunately, among them at most 3 are real, hence there exist no real cubic curves with 9 inflection points, and we cannot depict all inflection points of a cubic in a figure. Exercise 3.8 Show that if all inflection points of a curve of degree n are ordinary, then there are 3n(n − 2) of them. In particular, quadrics have no inflection points. As the exercises below show, nine inflection points of a cubic curve form a very interesting configuration of points in the complex projective plane. Exercise 3.9 Show that the third intersection point of the line connecting two inflections of a cubic curve with this curve is an inflection point. Find the total number of lines each of which contains a triple of inflection points of a given cubic curve. How many of these lines pass through a given inflection point? Exercise 3.10 Show that all cubics from the pencil λF + μH (F ) = 0, λ, μ ∈ C, have the same inflection points. Exercise 3.11 Show that every pencil of the form λF + μH (F ) = 0, λ, μ ∈ C, containing a smooth cubic can be transformed into the pencil λ(x 3 + y 3 + z3 ) + μxyz = 0 of the Fermat curve by an appropriate projective change of coordinates. In particular, all collections of nine inflection points of cubic curves are projectively equivalent. Find the inflection points for curves of this pencil.

3.4 Hyperelliptic Curves Let Pn = Pn (x) be a polynomial of degree n without multiple roots. Consider the plane curve given by the equation y 2 = Pn (x) in the affine chart z = 1. In this affine chart, it has no singular points. Indeed, it is given by the equation F (x, y) = 0,

F (x, y) = y 2 − Pn (x).

The differential of F equals ∂F ∂F dx + dy = −Pn (x) dx + 2y dy. ∂x ∂y It vanishes if y = 0 and Pn (x) = 0. If a point (x, 0) lies on the curve under consideration, then Pn (x) = 0. At this point the differential does not vanish, since Pn (x) has no multiple roots by assumption.

3.4 Hyperelliptic Curves

43

CP2 , the above for n ≥ 3 is given by the equation y 2 zn−2 =  curve  In n−i i i ai z x , where ai x = Pn (x). It has exactly one point at infinity (0 : 1 : 0), nonsingular for n = 3 and singular for n > 3. The curve under consideration can be mapped to CP1 by sending (x : y : z) to (x : z). To find the preimage of a point x0 = (x0 : 1) from CP1 , one should solve the equation y 2 = Pn (x0 ). If x0 is not a root of Pn , then this equation has exactly two roots. Hence the ramification points are the roots of Pn and, possibly, the point (1 : 0). Let us check that (1 : 0) is a ramification point if and only if n is odd. For small z, the preimage of (1 : z) consists of the points (1 : y : z) where y 2 ≈ an z2−n . Let z = ρeiϕ . As ϕ varies from 0 to 2π (i.e., when the point (1 : z) turns around the point (1 : 0) ∈ CP1 ), the argument of y changes by (2 − n)π. Hence if n is odd, then y changes the sign, i.e., we pass to the other branch; and if n is even, then y remains the same, i.e., we return to the original branch. Puncturing the curve under consideration at the point at infinity (0 : 1 : 0) and adding instead two points for even n and one point for odd n, we obtain an associated smooth curve. It is punctured at two points for even n and at one point for odd n. This smooth curve is a two-sheeted covering of CP1 , with the number of  ramification points equal to 2 n+1 2 . Hence, by the Riemann–Hurwitz theorem,   n + 1  n + 1 2 − 2g = 2 2 − 2 +2 , 2 2     n−1 − 1 = i.e., g = n+1 2 2 . The smooth curve associated with a curve of the form y 2 = Pn (x) for n = 3 and n = 4 has genus 1. Such a curve is said to be elliptic. One can show that every smooth curve of genus 1 is isomorphic to a curve given by an equation of the form y 2 = P3 (x), i.e., is elliptic (see Sect. 9.3). A curve C given by an equation of the form y 2 = Pn (x) with n > 4 is said to be hyperelliptic (Fig. 3.6).

a)

b)

Fig. 3.6 The real part of a hyperelliptic curve given by an equation of the form (a) y 2 = P2k+1 (x); (b) y 2 = P2k (x); in both cases, the roots of the polynomials in the right-hand side are real and distinct

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3 Complex Structure and the Topology of Curves

Remark 3.2 To compute the genus of a hyperelliptic curve, we cannot directly use the formula for the genus in terms of the degree. Indeed, this formula holds only for smooth curves in the projective plane, while a hyperelliptic curve is not smooth. More generally, a Riemann surface is said to be hyperelliptic if its genus is greater than 1 and it is a two-sheeted ramified covering of the Riemann sphere. All ramification points of a two-sheeted covering are necessarily simple. By the Riemann–Hurwitz formula, the number of these ramification points is even; if it is equal to 2k, then the genus of the covering surface is g = k − 1. Below we will see that every hyperelliptic Riemann surface is given (though not uniquely) by an equation of the form y 2 = Pn (x), where n = 2g + 1 or n = 2g + 2. Given a hyperelliptic curve C, one can define the so-called hyperelliptic involution σ : C → C, which permutes the sheets of the ramified covering. (By definition, an involution is an arbitrary transformation whose square is the identity.) For a plane hyperelliptic curve given by an equation y 2 = Pn (x), it is the restriction to the curve of the involution in the plane given by the formula (x : y : z) → (x : −y : z). Exercise 3.12 Show that the hyperelliptic involution on a hyperelliptic curve is unique: all two-sheeted ramified coverings of the Riemann sphere by a given curve of genus g ≥ 2 determine the same involution. Remark 3.3 One should not confuse the hyperelliptic involution with the complex conjugation involution. The latter is defined on real curves, i.e., on curves given by polynomial equations with real coefficients. In contrast to the hyperelliptic involution, the real involution does not preserve the complex structure of curves, so it is not a holomorphic mapping.

3.5 Lifting a Complex Structure Let C be a complex curve, Y be a two-dimensional surface, and f : Y → C be a covering. Then we can “lift” the complex structure from C to Y . This can be done as follows. Take a point y ∈ Y and its image t = f (y) under f . The point t ∈ C has a neighborhood U = U (t) such that the restriction of the covering map to the connected component of f −1 (U ) containing y is one-to-one. A complex coordinate on such a neighborhood is defined as the composition mU ◦ f (if U is not among the neighborhoods that define the complex structure on C, then one should take a defining neighborhood and restrict the coordinate map from this neighborhood to its intersection with U , considering this intersection as a new coordinate neighborhood). Exercise 3.13 Verify that the local coordinates thus defined do indeed determine a complex structure on Y . This means that changes of local coordinates on intersections of defining neighborhoods are holomorphic.

3.6 Quotient Curve

45

Unramified coverings of complex curves are rare, hence it is crucially important to extend the above construction to the case of ramified coverings. To do this, we should learn to extend a complex structure to neighborhoods of preimages of ramification points. Let f : Y → X be a ramified covering of a smooth irreducible complex curve X by a surface Y , and let y ∈ Y be the preimage of a ramification point f (y) = x. Then y has a punctured circular neighborhood that covers a punctured neighborhood of x with multiplicity k (which is called the ramification index at y). Choosing a coordinate z in the neighborhood of x ∈ X, we can introduce the coordinate t = z1/ k on the punctured neighborhood of y, which, as one can easily see, can be extended to the point filling the puncture. In the local coordinate t, our ramified covering has the form t → t k . As we will see, every smooth compact complex curve can be obtained as a ramified covering of the projective line CP1 . Thus we have obtained another extremely powerful tool for constructing complex curves. Namely, to construct a curve, one should take a collection of points in the complex projective line, build a star with rays going to these points (a star is determined up to homotopic equivalence), and take a collection of permutations of a set of d elements satisfying two properties: the action they determine is transitive, and their product is equal to the identity permutation. Such a collection determines a ramified covering of the sphere (see Sect. 1.6), and, lifting a complex structure from the sphere to the covering surface, we turn it into a Riemann surface. The topology of this Riemann surface is determined by the collection of permutations, while the complex structure is determined by the positions of the vertices of the star.

3.6 Quotient Curve Another method of constructing complex curves is, conversely, to descend a complex structure from a covering curve to a covered curve. Let Y be a complex curve with a discrete fixed-point-free action of a group G of transformations preserving the complex structure, i.e., biholomorphic transformations of Y . Then the quotient by this action, i.e., the set of G-orbits, is endowed with a natural structure of a complex curve. Example 3.3 Let Y = C be the complex line. Consider the lattice on Y generated by the vectors 1 and τ , where τ ∈ C is a complex number with positive imaginary part, see Fig. 3.7. The translations by 1 and τ generate a group of transformations of C preserving its complex structure. This is the free commutative group with two generators. Its orbits are the images of the lattice under various translations. The quotient by the action of this group is a two-dimensional torus, which can be identified with the parallelogram spanned by the vectors 1 and τ , with opposite edges identified. One can prove that such a torus is a smooth curve given by an equation of the form y 2 = P3 (x) (see Sect. 9.3), i.e., an elliptic curve. Conversely, every curve of the form y 2 = P3 (x) can be obtained as a quotient by some lattice. So, in what follows, a torus obtained as a quotient by a lattice will also be called an elliptic curve.

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3 Complex Structure and the Topology of Curves

b) a) Fig. 3.7 (a) A lattice on the complex line; (b) the fundamental parallelogram of this lattice

Exercise 3.14 Give an example of two nonisomorphic elliptic curves. Since the complex line is an Abelian group under addition, and a lattice is a subgroup of this group, every elliptic curve is endowed with a structure of an Abelian group, which is the quotient of the line by a lattice. The image of the lattice is the zero of this group. Note, however, that by an appropriate translation the zero can be moved to any point of the curve (of course, the group structure will undergo corresponding changes). Exercise 3.15 Show that if the zero of an elliptic curve group is chosen to be an inflection point of a plane cubic, then all nine inflection points form the 3-torsion subgroup in this group.

3.7 Meromorphic Functions Given a curve C, a holomorphic mapping f : C → CP1 is called a meromorphic function on C. Here it is understood that the Riemann sphere CP1 is represented as C∪∞; in particular, it contains two distinguished points, 0 and ∞. The preimages of 0 are the zeros of f , and the preimages of ∞ are its poles. We assume that zeros have positive degrees and poles have negative degrees. A common zero of two functions is a zero of their product with order equal to the sum of the orders of the zeros of the factors; a common pole of two functions is a pole of their product with order equal to the sum of the orders of the poles of the factors. More generally, assuming that the order of a function at a point where it takes a value different from 0 and ∞ is zero, the order of the product of functions at every point is equal to the sum of the orders of the factors at this point. An example of a meromorphic function is any polynomial z → P (z) = p0 zn + p1 zn−1 + . . . + pn ,

pi ∈ C, p0 = 0, n ≥ 1,

on the Riemann sphere. Indeed, the mapping determined by a polynomial is obviously holomorphic at every finite point of the projective line. To see how it

3.7 Meromorphic Functions

47

behaves at infinity, make the change of coordinates z = 1/y. In the coordinate y, the mapping in a neighborhood of ∞ takes the form y → P (1/y) =

p0 p1 1 + n−1 + . . . + pn = n (p0 + p1 y + . . . + pn y n ). n y y y

The expression 1/P (1/y), which is the coordinate representation of the function in a neighborhood of infinity on the image curve, has the form 1 yn 1 n p1 n+1 p2 − p1 p0 n+2 = = y − 2y + y + ..., P (1/y) p0 + p1 y + . . . + pn y n p0 p0 p03 i.e., it determines a holomorphic function with zero of order n (recall that p0 = 0). This means that the function P has a pole of order n at infinity. Theorem 3.3 On the Riemann sphere, every meromorphic function is a rational function, i.e., the ratio P (z)/Q(z) of two polynomials. Proof Let f be a meromorphic function on the Riemann sphere CP1 . The Riemann sphere is compact, hence the number of zeros and poles of f is finite. Let z1 , . . . , zn be the zeros and poles of f , and let e1 , . . . , en be their orders. Consider the rational function r(z) = (z − z1 )e1 . . . (z − zn )en ; in the finite plane, it has the same zeros and poles as f . Consider the function g(z) = f (z)/r(z). It is meromorphic on the Riemann sphere and has no zeros and poles in the finite plane. In particular, regarded as a function on C, it is holomorphic, and hence can be represented by a ∞  Taylor series: g(z) = ak zk for all z ∈ C. Also, the function g(z) is meromorphic k=0

at ∞. For the local coordinate w = 1/z at ∞, this means that the function g(w) = ∞  ak w−k is meromorphic at w = 0. Therefore, g(z) is a polynomial, and, by k=0

construction, it has no zeros in the whole complex plane. Thus g(z) = const, and f (z) is a rational function. Corollary 3.2 The sum of the orders of all zeros and poles of a meromorphic function on the Riemann sphere is 0. Proof This is obvious for a polynomial (its degree is equal, on the one hand, to the order of the pole at infinity and, on the other hand, to the number of its roots), and hence for any rational function.

Remark 3.4 Later we will show that the sum of the orders of all zeros and poles of a meromorphic function f on any smooth curve is 0 (Theorem 7.4). This is quite obvious from topological considerations. Indeed, the sum of the orders of all zeros of a meromorphic function is equal to the degree of the corresponding ramified covering of the projective line. The sum of the orders of all poles is the same degree but taken with the minus sign. Hence the sum of these numbers is 0. Topologically, zeros and poles of a meromorphic function are not distinguished in any way among

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preimages of points of CP1 : the sum of the multiplicities of the preimages of every point is equal to the degree of the function. In homogeneous coordinates, a polynomial of degree n on the complex projective plane can be written  as the ratio of two homogeneous polynomials of degree n, namely, P (z/w) = ( ai zi wn−i )/wn . Hence every meromorphic function on the complex plane can be written in homogeneous coordinates as the ratio of two homogeneous polynomials of the same degree. Theorem 3.4 If a meromorphic function on a smooth compact curve C has only one simple pole, then C is the Riemann sphere CP1 . Proof Let p be the pole of the meromorphic function in question. Consider the mapping f : C → CP1 it determines. By assumption, p is the unique point mapped to ∞ ∈ CP1 and f has multiplicity 1 at p. Therefore, f is a one-sheeted covering, i.e., a homeomorphism. Using the description of meromorphic functions on the sphere (Theorem 3.3) and the fact that a hyperelliptic curve is a two-sheeted ramified covering of the sphere, one can obtain the following description of meromorphic functions on hyperelliptic curves. Theorem 3.5 Let C be a hyperelliptic curve given by an equation y 2 = P (x). Then every meromorphic function f on C can be uniquely written in the form f = R(x) + yS(x), where R and S are rational functions of x. Proof Let σ be the hyperelliptic involution on the curve C and p : C → CP1 be the mapping (two-sheeted ramified covering) given by the formula (x : y : z) → (x : z). Note that σ is a holomorphic mapping. It is also clear that p ◦ σ = p. The hyperelliptic involution acts on the space of meromorphic functions on C: with a meromorphic function f on C we can associate the meromorphic function σ ∗ f = f ◦ σ . The function f + σ ∗ f is σ ∗ -invariant, since σ ∗ (f + σ ∗ f ) = σ ∗ f + f (here we have used the fact that σ 2 = id). With a meromorphic function r on the Riemann sphere CP1 we can associate the meromorphic function g = p∗ r = r ◦ p on the curve C. The function g is σ ∗ -invariant, since σ ∗ g = g ◦ σ = r ◦ p ◦ σ = r ◦ p = g. Let us show that in fact every σ ∗ -invariant meromorphic function g on C can be obtained in this way from a unique meromorphic function r on the Riemann sphere. Namely, for ξ ∈ C set r(ξ ) = g(ξ ); this function is well defined, as follows from the σ ∗ -invariance of g. The function f + σ ∗ f is σ ∗ -invariant. The functions f − σ ∗ f and y are both ∗ σ ∗ -antiinvariant, hence the function f −σy f is σ ∗ -invariant. Therefore, f + σ ∗ f = ∗

and f −σy f = 12 S(x), where R(x) and S(x) are meromorphic (i.e., rational) functions on the sphere. The theorem follows. 1 2 R(x)

Exercise 3.16 Show that for every meromorphic function on an elliptic curve, the sum of the orders of the zeros coincides with the sum of the orders of the poles.

3.7 Meromorphic Functions

49

Given an arbitrary curve C ⊂ CPN in a projective space, meromorphic functions on C can be constructed as follows. Consider homogeneous coordinates (x0 : . . . : xN ) in CPN . Let P , Q be two homogeneous polynomials of the same degree d in N + 1 variables (x0 , . . . , xN ) such that the curve C does not lie entirely in the hypersurface Q = 0, and the polynomials P and Q have no common divisors of positive degree. Then the ratio P /Q defines a meromorphic function on CPN whose restriction to the curve C is a meromorphic function on this curve. The zeros and poles of this function are contained in the union of the zeros of P and Q on C. In fact, every meromorphic function on a given curve can be realized in this way for an appropriate embedding and an appropriate pair of polynomials. Exercise 3.17 Find the degree of the meromorphic function that is the restriction Pk to a plane curve of degree d of an irreducible rational function Q , where the k numerator and denominator are homogeneous polynomials of degree k.

Chapter 4

Curves in Projective Spaces

As we will see below, the projective plane (like any other two-dimensional surface) is too narrow to accommodate every smooth curve. In the three-dimensional space there is much more freedom. However, to define curves in CP3 and higher dimensional projective spaces is more difficult than in the plane. In this chapter, we discuss methods of defining such curves.

4.1 Definition and Examples We start with an example. Example 4.1 Let us construct a mapping from the projective line CP1 to the projective space CP3 . For this, we • introduce homogeneous coordinates (t0 : t1 ) in CP1 ; • introduce homogeneous coordinates (z0 : z1 : z2 : z3 ) in CP3 ; • consider the mapping from CP1 to CP3 given by the formula (t0 : t1 ) → (t03 : t02 t1 : t0 t12 : t13 ). Exercise 4.1 Verify that this mapping is nondegenerate and one-to-one onto its image. The image of CP1 under the constructed mapping lies on each of the following quadrics: z0 z3 − z1 z2 = 0, z12 − z0 z2 = 0, z22 − z1 z3 = 0, © Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_4

51

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and it can be defined as the intersection of these quadrics. Exercise 4.2 Verify that the intersection of any two quadrics from the above example does not coincide with the image of the constructed mapping (i.e., contains extra points). Example 4.1 shows that one can hardly expect a curve in CP3 to be representable as the intersection of two smooth hypersurfaces (in the general case of CPn , as the intersection of n − 1 smooth hypersurfaces). Hence it is natural to give the corresponding definition only locally. A smooth curve in CPn is a set C of points such that for every point A ∈ C there exists a neighborhood of A and a collection of homogeneous polynomials F1 , . . . , Fn−1 such that in the chosen neighborhood C coincides with the intersection of the hypersurfaces F1 = 0, . . . , Fn−1 = 0 and the differentials dF1 , . . . , dFn−1 are linearly independent at A. From the topological point of view, a smooth curve is an orientable two-dimensional surface. Exercise 4.3 Verify that the image of CP1 in the above example is a smooth curve: in a neighborhood of every point it can be defined as the intersection of two out of the three quadrics. The degree of a smooth plane algebraic curve is defined as one of the two coinciding positive integers: either as the degree of the homogeneous polynomial defining this curve, or as the number of intersection points of the curve with a generic projective line. In the three-dimensional space, the first definition does not work, but the second one goes over without essential change. The degree of a smooth curve C in a projective space is the number of intersection points of C with a generic hyperplane. Example 4.2 Considering the intersection of the image of CP1 in the above example with an arbitrary plane, we see that it is a curve of degree 3. It follows, in particular, that this curve cannot be defined as a transversal intersection of two surfaces F = 0 and G = 0. Indeed, the degree of such a transversal intersection is the product of the degrees of the homogeneous polynomials F and G. For the intersection to be a curve of degree 3, the polynomials F and G should have degrees 3 and 1. This means that the intersection curve lies in the two-dimensional projective plane determined by the second equation. However, as one can easily see, no four points of our curve lie in the same plane. An interesting example of a curve is the intersection of two quadrics in general position in CP3 . The cross section of two quadrics in CP3 by a hyperplane in general position is a pair of conics in general position. They meet at four points, hence the curve under consideration has degree 4. Let us show that its genus is 1, i.e., it is homeomorphic to the torus. Let our quadrics be given by the equations F (x, y, z, w) = x 2 + y 2 + z2 + w2 = 0 and G(x, y, z, w) = ax 2 + by 2 + cz2 + dw2 = 0.

4.1 Definition and Examples

53

Exercise 4.4 Show that in the case of a generic collection of coefficients a, b, c, d these two quadrics intersect transversally, i.e., the differentials dF and dG at their intersection points are linearly independent. Consider the mapping p : CP3 \(0 : 0 : 0 : 1) → CP2 that sends a point (x : y : z : w) to (x : y : z), i.e., the projection from the point (0 : 0 : 0 : 1). This mapping sends the intersection of the quadrics to the conic ax 2 + by 2 + cz2 = d(x 2 + y 2 + z2); it is a two-sheeted ramified covering whose ramification points are the intersection points of the conics ax 2 + by 2 + cz2 = 0 and x 2 + y 2 + z2 = 0. We obtain a two-sheeted ramified covering of the sphere S 2 with four ramification points (which are simple, since the covering is two-sheeted). Hence the covering surface is a torus. Since the transversality condition for the intersection of two quadrics is an algebraic condition on the space of pairs of quadrics, we can conclude that the intersection of any two transversally intersecting quadrics in CP3 is a smooth curve of genus 1. Exercise 4.5 a) Show that the mapping (x1 : x2 ) → (x12 : x1 x2 : x22 ) is a biholomorphic mapping from the projective line CP1 onto the smooth conic z22 = z1 z3 in CP2 . b) Show that the mapping {(x1 : x2 ), (y1 : y2 )} → (x1 y1 : x1 y2 : x2 y1 : x2 y2 ) is a biholomorphic mapping from the product CP1 × CP1 onto the smooth quadric z1 z4 = z2 z3 in CP3 . Exercise 4.6 Show that the transversal intersection of a quadric and a cubic in CP3 is a smooth curve of genus 4. As another example of a curve in CPn , which generalizes the above example of a curve in CP3 , consider the image of the projective line CP1 under the mapping (x : y) → (x n : x n−1 y : . . . : y n ).

(4.1)

This curve lies in the intersection of the hypersurfaces z12 = z0 z2 ,

z22 = z1 z3 ,

...,

2 zn−1 = zn−2 zn ,

but, as we have seen in the example for n = 3, does not coincide with this intersection. Denote the obtained curve by Γn . Since the mapping (4.1) is one-to-one, the genus of Γn is 0. One can easily check that the degree of Γn is n. Indeed, the

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hyperplane a0 z0 + a1 z1 + . . . + an zn = 0 meets the curve Γn at points such that a0 t n + a1 t n−1 + . . . + an = 0,

where t = x/y.

Thus the intersection points of Γn and the hyperplane correspond to the roots of a polynomial of degree n, and such a polynomial has exactly n roots counted with multiplicity. For this reason, Γn is called the rational normal curve of degree n. The special case Γ3 of this curve is known as the twisted cubic. Exercise 4.7 Show that a) every smooth rational curve of degree n in CPn coincides, for an appropriate choice of coordinates, with the rational normal curve of degree n; b) any n + 1 distinct points on the rational normal curve of degree n are linearly independent; c) for any n + 3 distinct points in CPn that are in general position (i.e., such that no n + 1 of them lie in one hyperplane), there is exactly one rational normal curve passing through them. Assertion a) shows that a rational normal curve of every degree is essentially unique. It is an important geometric tool in the study of properties of polynomials in one variable. Exercise 4.8 Assume that the intersection of two different quadrics in CP3 contains a twisted cubic. Show that then it is the union of this cubic and a line. Verify that this is the case for the pairs of quadrics described after Example 4.1. Exercise 4.9 Describe the family of all quadrics in CP3 containing a given twisted cubic.

4.2 Embeddings and Immersions of Curves When one projects a projective space to a subspace of smaller dimension, curves in the original space go to curves in the subspace. This property can be used to embed curves into spaces of smallest possible dimension. Theorem 4.1 Every curve can be embedded into the three-dimensional projective space. Proof If the curve is a projective line, or it is already embedded into CP2 or CP3 , then there is nothing to prove. Now we assume that the curve is embedded into a projective space of dimension greater than 3 and prove that it can be projected to a space of one dimension less in such a way that the image is a smooth curve and the projection is one-to-one onto its image.

4.2 Embeddings and Immersions of Curves

55

Fig. 4.1 Forbidden projection directions for a curve

Consider a curve C ⊂ CPn with n ≥ 4. We associate with C its secant variety, the subvariety in CPn defined as the closure of the union of all lines joining two points of C (the closure also contains the points of the tangents to C), see Fig. 4.1. The secant variety is at most three-dimensional, since its points are determined by three parameters: two points of the curve and a point of the line passing through them. Hence it does not coincide with the whole ambient projective space CPn . Thus, there is a point in CPn that does not lie in the secant variety. The projection from this point maps C to CPn−1 , the mapping being one-to-one onto the image of C. The following exercise completes the proof. Exercise 4.10 Show that the image of C under such a projection is a smooth curve. Remark 4.1 Both the statement and the proof of the theorem essentially coincide with those of Whitney’s theorem on the embeddability of a k-dimensional manifold into the (2k + 1)-dimensional space. Exercise 4.11 Show that for n > 3, given a curve C in CPn that does not lie in any hyperplane, one can find a point on C such that the projection from this point is a biholomorphic mapping of C. Exercise 4.12 Show that projecting a curve C from a generic point outside C preserves its degree. Show that projecting a curve C from a generic point on C decreases its degree by 1. For a smooth curve in CP3 , we can no longer choose a projection center so that the image of the curve is necessarily smooth. However, we can ensure that the image has only singularities of simplest form: points of transversal self-intersection. Such a mapping is called an immersion. Theorem 4.2 Every complex curve C can be mapped to the projective plane CP2 so that all nonsmooth points of the image are points of simple (i.e., double) transversal self-intersection, the mapping is locally biholomorphic on the complement to these points, and every double point has two preimages. Proof By the previous theorem, the curve C can be embedded into the complex projective space CP3 . Choose a point in CP3 such that the restriction to C of the projection from this point is one-to-one and biholomorphic onto the image except finitely many points of the image which have two preimages, the two branches of the image at these points intersecting transversally. Such a point does indeed exist:

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it suffices to take it outside the two-dimensional subvariety in CP3 consisting of the following components: • the union of the tangents to C; • the union of the “singular” secants, i.e., secants that contain more than two points of C, as well as secants passing through two points of C such that the subspace spanned by the tangent vectors at these points and the secant itself is two-dimensional. The projection from such a point satisfies the desired properties. Now assume that the image of a curve C under an immersion to CP2 is a curve of degree d with δ double points. Let us find the genus of C from these data. Assume that the immersed curve is given by an equation f = 0 in the chart z = 1 and both tangents at each double point are not vertical. By Bézout’s theorem, the number of intersection points of the immersed curve with the curve given by the equation ∂f/∂y = 0 is equal to d(d − 1). Exercise 4.13 State and prove Bézout’s theorem for curves with simple self-intersections. On the other hand, each of the double points is a common zero of the curves f = 0 and ∂f/∂y = 0, hence the ramification points of the projection to the x axis (i.e., the points with vertical tangents) are only those intersection points that are different from double points. There are d(d − 1) − 2δ of them (each intersection point is counted twice, since at such a point the curve ∂f/∂y = 0 meets two branches of the immersed curve). By the Riemann–Hurwitz formula, d(d − 1) − 2δ = 2d + 2g − 2, whence g=

(d − 1)(d − 2) − δ. 2

Exercise 4.14 Find the dimension of the space of plane curves of degree 4 with one ordinary double point. Exercise 4.15 Show that every curve of genus 2 can be immersed into the projective plane as a curve of degree 4 with one ordinary double point. Considering the projection from this point, deduce that every curve of genus 2 admits a holomorphic mapping of degree 2 onto the projective line, i.e., is hyperelliptic. Exercise 4.16 Estimate the degree of plane curves (with double points) needed to represent every curve of a given genus g. A mapping C → C  where C is a smooth curve and C  is a plane curve which may have finitely many points of simple self-intersection is called a normalization mapping, and the curve C is called a normalization of C  . Actually, every plane

4.2 Embeddings and Immersions of Curves

57

curve, whatever singularities it may have, admits a normalization, but we will need neither this fact, nor even its rigorous statement. In the generic case, the projection of a curve C in CPn onto a curve C  in CPm with m < n is a birational isomorphism, i.e., if a point (x0 : . . . : xn ) ∈ C goes to (y0 : . . . : ym ) ∈ C  , then x0 , . . . , xn can be rationally expressed in terms of y0 , . . . , ym , and y0 , . . . , ym can be rationally expressed in terms of x0 , . . . , xn . First of all, observe that coordinates can be chosen so that the projection has the form (x0 : . . . : xn ) → (x0 : . . . : xm ). Hence we should only check that xm+1 , . . . , xn can be rationally expressed in terms of x0 , . . . , xm (for points of C). Let C be given by a system of equations fi (x0 , . . . , xn ) = 0, i = 1, . . . , r. For a point (a0 : . . . : am ) ∈ C we obtain the system of equations fi (a0 , . . . , am , xm+1 , . . . , xn ) = 0, i = 1, . . . , r. To every point of C  there corresponds one point of C, hence xm+1 , . . . , xn can be uniquely expressed in terms of a0 , . . . , am . Therefore, xm+1 (a0 , . . . , am ), . . . , xn (a0 , . . . , am ) are single-valued algebraic functions. But an algebraic function is single-valued only if it is rational. Exercise 4.17 Study the singular points of the plane curve given by the equation y 2 z2 − x 2 (z2 − x 2 ) = 0 and construct its normalization.

Chapter 5

Plücker Formulas

The plane curves break into pairs: with each plane curve one can associate a dual curve, the dual of the dual coinciding with the original curve. Usually, the dual of a smooth curve turns out to be singular, hence, studying duality, we cannot content ourselves with considering only smooth curves. Moreover, it does not suffice to consider only curves with singularities of simplest form, points of transversal selfintersection. However, the pairs of dual curves having only points of transversal self-intersection and cusps form an open subset in the space of pairs of dual curves of given degrees, which makes it natural to study such pairs. Plücker formulas are relations on the number of singularities of various types for a pair of dual curves of given degrees.

5.1 Projective Duality For every vector space V there is a dual vector space V ∨ , which consists of the linear functionals f : V → C on V . We will consider only finite-dimensional vector spaces. The dual space has the same dimension as the original one, and the dual of V ∨ is naturally isomorphic to V : every point v ∈ V determines a linear functional on V ∨ , which is given by the formula v : f → f (v) for f ∈ V ∨ . The projectivization P V ∨ is the dual projective space of the projectivization P V of V . We will show how to associate with a plane curve C ⊂ CP2 a dual plane curve C ∨ in the dual projective plane (CP2 )∨ . With a line l in CP2 we associate a point l ∨ ∈ (CP2 )∨ : this is a nonzero linear functional vanishing on l. Such a functional is unique up to multiplication by a nonzero constant. Thus it defines a point of (CP2 )∨ . Conversely, with a line in the dual plane we associate a point in the original plane. If a line in CP2 is given by a linear equation ax + by + cz = 0, then the corresponding point in (CP2 )∨ has the coordinates (a : b : c). Projective duality has the following important property, © Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_5

59

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5 Plücker Formulas

which immediately follows from the definition: points A and B lie on a line l if and only if the lines A∨ and B ∨ meet at the point l ∨ . Given a smooth curve C ⊂ CP2 , its dual C ∨ ⊂ (CP2 )∨ is constructed as follows. With each point of C we associate the line l that touches C at this point and then consider the point l ∨ . The set of all points l ∨ obtained in this way is the dual curve C ∨ . Exercise 5.1 2

2

a) Show that the dual of an ellipse xa 2 + yb2 = 1 is the ellipse a 2 x 2 + b 2 y 2 = 1. b) Show that the dual of any nonsingular curve of degree 2 has degree 2. Exercise 5.2 Consider the case of real dual curves. Show that the dual of a convex curve is convex. (A closed curve in R2 is said to be convex if it bounds a convex set.) The dual of a singular curve C is defined as the closure in (CP2 )∨ of the set of points l ∨ corresponding to the tangents to C at smooth points. The complex curve dual to a smooth curve of degree greater than 2 is never smooth. However, it is still an algebraic curve. Theorem 5.1 If C is an algebraic curve, then the curve C ∨ is also algebraic. Proof To every line in CP2 there corresponds a point in (CP2 )∨ . Hence with a smooth curve C in CP2 one can associate a curve Cˆ in CP2 ×(CP2 )∨ : it consists of the pairs (a point of C, the tangent to C at this point) and is called the conormal unfolding of C. As one can easily see, the curve Cˆ is smooth. The direct product CP2 ×(CP2 )∨ can be projected to each of the factors; the projection to the first factor sends the curve Cˆ to C, while the projection to the second factor sends Cˆ to the dual curve C ∨ . Let C be given in CP2 by an equation F (x, y, z) = 0. The equation of the tangent to C at a point (x0 : y0 : z0 ) ∈ C is xFx + yFy + zFz = 0, where the derivatives Fx , Fy , and Fz are taken at the point (x0 , y0 , z0 ). Therefore, the conormal unfolding can be determined explicitly by the algebraic equations x1 = Fx (x0 , y0 , z0 ), z1 = Fz (x0 , y0 , z0 ),

y1 = Fy (x0 , y0 , z0 ), F (x0 , y0 , z0 ) = 0,

where (x0 : y0 : z0 ) ∈ CP2 and (x1 : y1 : z1 ) ∈ (CP2 )∨ . The last equation in this system can be replaced by the equivalent (by the homogeneity of F ) equation x1 x0 + y1 y0 + z1 z0 = 0.

5.1 Projective Duality

61

The theorem now follows from the general fact that the image of an algebraic variety under an algebraic mapping is an algebraic variety. To deduce an explicit equation of the curve C ∨ , one must successively eliminate the variables x0 , y0 , z0 from the above equations of the curve Cˆ ⊂ CP2 ×(CP2 )∨ . This can be done as follows. Assume that we have four equations f1 = 0, . . . , f4 = 0 on variables x0 , x1 , y0 , y1 , z0 , z1 . Replace each pair of polynomials (f1 , f4 ), (f2 , f4 ), (f3 , f4 ), regarded as polynomials in x0 , by the resultant of this pair. We obtain a triple of polynomials g1 , g2 , g3 , which no longer depend on x0 . Now replace each of the two pairs of polynomials (g1 , g3 ), (g2 , g3 ), regarded as polynomials in y0 , by the resultant of this pair. Then eliminate, in the same way, the variable z0 from the obtained pair of polynomials (h1 , h2 ). Equating the resulting homogeneous polynomial in the variables x1 , y1 , z1 to zero yields exactly the desired equation of C ∨ . The degree of the curve C ∨ is called the class of C. In other words, the class of C is the number of intersection points of C ∨ with a generic line, i.e., the number of tangents to C passing through a given generic point. The class of a nonsingular curve depends only on its degree. Example 5.1 Let us find the duals of cubic curves from the family y 2 z − x 3 + axz2 = 0.

(5.1)

The equations of the conormal unfolding are x1 + 3x 2 − az2 = 0, y1 − 2yz = 0, z1 − y 2 − 2axz = 0, x1 x + y1 y + z1 z = 0. Eliminating x from the first and third equations paired with the fourth one (the second equation does not contain this variable) yields the system x13 − ax12z2 + 3y12 y 2 + 6y1 yz1 z + 3z12 z2 = 0, y1 − 2yz = 0, x1 y 2 − x1 z1 − 2ay1yz − 2az1z2 = 0. Eliminating y from the first and third equations paired with the second one yields the system 4x13 z2 − 4ax12z4 + 12z12 z4 + 3y14 = 0, x1 y12 − 4x1 z1 z2 − 4ay12z2 − 8az1z4 = 0.

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Fig. 5.1 The intersection point A of two tangents to a curve C, its limiting position, and the corresponding dual objects

Finally, eliminating z yields the equation of the dual curve: 4x13 z13 + 27y12 z14 − a(ax14y12 + 24ax1y14 z1 + 30x12 y12 z12 + 4x15z1 + 4a 2y16 ) = 0. (5.2) Thus, for a generic value of the parameter a, the dual of a cubic has degree 6, i.e., the class of a cubic is 6. For a = 0, the curve (5.1) degenerates into the semicubical curve y 2 z − x 3 = 0, while the dual degenerates into the semicubical curve 4x13 + 27y12z1 = 0. Thus the class of a semicubical curve is 3. In turn, one can construct the dual (C ∨ )∨ ⊂ CP2 of the dual curve C ∨ ⊂ (CP2 )∨ . This can be done as follows. The tangent to C ∨ at each nonsingular point determines a point in CP2 . The closure of the set of these points in CP2 forms the curve (C ∨ )∨ . Dual curves satisfy the relation (C ∨ )∨ = C. Indeed, let l1∨ and l2∨ be two close smooth points of C ∨ . They correspond to tangents l1 and l2 to C. If A is the intersection point of l1 and l2 in the original plane, then the corresponding line A∨ in the dual plane passes through the points l1∨ and l2∨ (Fig. 5.1). As l2∨ tends to l1∨ , the line A∨ approaches the tangent to C ∨ at l1∨ , while the point A approaches the point at which l1 touches C. Therefore, to tangents to C ∨ at smooth points there correspond points of C, as desired. The above arguments show also that the conormal unfolding of the dual curve C ∨ coincides with the conormal unfolding Cˆ of the original curve, with the projective plane CP2 and the dual plane (CP2 )∨ switching places. Exercise 5.3 Find the dual of the curve (5.2) and verify that it coincides with the curve (5.1). If C is a nonsingular curve, then the curve C ∨ is not necessarily nonsingular. Actually, a double tangent of C gives rise to a self-intersection point of C ∨ , and a tangent at an inflection point gives rise to a cusp (Fig. 5.2). The first of these properties is obvious, so we will discuss only the second one. It can be studied locally, in a neighborhood of an inflection point.

5.1 Projective Duality

63

Fig. 5.2 (a) A double tangent and a tangent at an inflection point; (b) singularities of the dual curve

Consider the curve y = x 3 , for which (0, 0) is an inflection point. The equation of the tangent to this curve at a point with coordinates (x0 , y0 ) is y − y0 = 3x02 (x − x0 ), i.e., −3x02x + y + 2x03 = 0. Hence the point of the dual curve associated with a point (x0 : x03 : 1) ∈ CP2 is  −3x0−1 x0−3 : 2 : 1). Hence the dual curve consists of the points (−3x02 : 1 : 2x03 ) = 2  3 3x0−1 x −3 (x1 : y1 : 1) with x1 = − 2 , y1 = 02 , and its equation is − 2x31 = 2y1. Thus we have obtained almost the same curve, but to the point (0 : 0 : 1) of the original curve there corresponds the point (0 : 1 : 0) of the dual curve. In a neighborhood of the latter point, it is convenient to use the coordinates (x1 : 1 : z1 ). Then x1 = −3x0 and z1 = 2x03 , i.e., in these coordinates the dual curve is given by the equation 27z12 + x13 = 0. Thus the dual of an inflection point (Fig. 5.2a) is a cusp (Fig. 5.2b). 2

2

The ellipses xa 2 +a 2 y 2 = 1 and a 2 x 2 + ya 2 = 1 are dual to each other, hence the curve consisting of this pair of ellipses is self-dual. Using this property, and also the fact that the dual of an inflection point is a cusp and the dual of a double tangent is a self-intersection point, one can easily check that the curves shown in Fig. 5.3 are dual.

Fig. 5.3 A tangent to the original curve is a point of the dual curve

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5 Plücker Formulas

5.2 Plücker Formulas for Nonsingular Curves We already know some numerical characteristics of duals to smooth curves. Let C be a smooth projective curve. The class of C is the degree of the dual curve C ∨ , i.e., the number of intersection points of C ∨ with a generic line in the dual projective plane. Such a line corresponds to some point of the original plane and consists of all lines passing through it. In the generic case, intersection points are lines touching C. If C is a smooth curve of degree n, then, as we know, through a generic point outside C one can draw n(n − 1) tangents to C (see Theorem 3.2). Therefore, we obtain the following result. Proposition 5.1 The class n∨ of a nonsingular plane curve of degree n is n(n − 1). This proposition immediately implies that for n > 2 the dual C ∨ of a smooth curve C of degree n is singular. Indeed, otherwise the degree of C ∨∨ would be n(n − 1)(n(n − 1) − 1), which is greater than n, so C ∨∨ could not coincide with C. One can also easily find the number of cusps of C ∨ . The cusps of the dual curve are in a one-to-one correspondence with the inflection points of the original curve. As we know, a generic smooth curve of degree n has 3n(n − 2) inflection points (see Exercise 3.8). Hence the following result holds. Proposition 5.2 The curve dual to a generic nonsingular plane curve of degree n has k ∨ = 3n(n − 2) cusps. Now we have all data needed to find the number of double points on the dual of a smooth curve. The curve C ∨ is the image of a smooth curve C under the holomorphic projective duality mapping, which is one-to-one almost everywhere and nondegenerate. This means that the smooth curve C is a normalization of the singular curve C ∨ . Let us define the genus of a singular curve as the genus of its normalization. If C ∨ were a smooth curve of degree n(n − 1), then its genus would be equal to (n(n − 1) − 1)(n(n − 1) − 2)/2. The existence of singular points causes the genus to decrease, the decrease being the same for singular points of the same type. The dual of a generic smooth curve has singular points of two types: cusps and points of transversal self-intersection. The semicubical parabola y 2 = x 3 z has a unique singular point, which is a cusp, and its normalization is a rational curve; while a smooth curve of degree 3 has genus 1. Thus the appearance of a cusp on a curve of given degree reduces its genus by 1. On the other hand, as we already know (see Sect. 2.3), the appearance of a self-intersection point also reduces the genus by 1. Thus we obtain the following expression for the genus ˆ of a curve C which is a normalization of C ∨ and which is g = g(C) = g(C) ˆ isomorphic to its own conormal unfolding C: g=

1 ∨ (n − 1)(n∨ − 2) − δ ∨ − k ∨ . 2

5.3 Plücker Formulas for Singular Curves

65

Therefore, the number of self-intersection points on the dual curve is equal to 1 ∨ 1 (n − 1)(n∨ − 2) − (n − 1)(n − 2) − k ∨ 2 2 1 1 = (n(n − 1) − 1)(n(n − 1) − 2) − (n − 1)(n − 2) − 3n(n − 2) 2 2 1 = n(n − 2)(n − 3)(n + 3). 2

δ∨ =

Thus we have proved the following result. Proposition 5.3 The curve dual to a generic smooth curve of degree n has δ∨ =

1 n(n − 2)(n − 3)(n + 3) 2

points of self-intersection, i.e., a generic smooth curve of degree n has 1 2 n(n − 2)(n − 3)(n + 3) double tangents. For n = 3, the obtained expression for δ ∨ vanishes, which agrees with the fact that the dual of a smooth cubic has no points of self-intersection (but has 9 cusps). For n = 4, we have δ ∨ = 28: a generic smooth curve of degree 4 has 28 double tangents.

5.3 Plücker Formulas for Singular Curves The symmetry between a plane curve and its dual makes it natural to extend the formulas deduced in the previous section to the case where the curve C also has singularities: points of double self-intersection and semicubical cusps. For a generic such curve C of given degree, the dual curve also has only such singularities. The singularities of the conormal unfolding Cˆ of C are determined by the local singularities of C. As we have seen, the only local singularities of the curve C ∨ dual to a generic smooth curve C are self-intersection points and semicubical cusps. But the conormal unfolding Cˆ = Cˆ∨ is a smooth curve, hence the conormal unfolding of any generic curve whose singularities are all double points and semicubical cusps is also a smooth curve. In particular, this means that both curves C and C ∨ are irreducible. Thus the conormal unfolding is a normalization both for C and C ∨ . Given a curve C, denote by n its degree, by δ the number of self-intersection points, and by k the number of cusps; by n∨ , δ ∨ , and k ∨ we denote the same quantities for C ∨ .

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The generalizations of the formulas deduced above to the case of singular curves have the following form and are called the Plücker formulas: n∨ = n(n − 1) − 2δ − 3k, n = n∨ (n∨ − 1) − 2δ ∨ − 3k ∨ ,

3n(n − 2) = k ∨ + 6δ + 8k, 3n∨ (n∨ − 2) = k + 6δ ∨ + 8k ∨ .

The formula for the genus of Cˆ deduced in the previous section carries over without change to the case where C is a singular curve: ˆ = g(C)

1 1 (n − 1)(n − 2) − δ − k = (n∨ − 1)(n∨ − 2) − δ ∨ − k ∨ . 2 2

The above five relations are not independent. It is easy to check that any two of them imply the three remaining ones. Besides, if we prove, for example, the formula for n∨ , then replacing C with C ∨ yields the similar formula for n. Nevertheless, we will give independent proofs of each of the Plücker formulas, which reveal their geometric meaning. Consider the defining equation of the curve C and a one-parameter deformation of its coefficients depending on a parameter t such that for small values t = 0 the curve Ct defined by the deformed equation is a generic curve of the same degree n and C0 = C. By the results of the previous section, from a generic point P ∈ CP2 one can draw n(n − 1) tangents to Ct , t = 0, and Ct has exactly 3n(n − 2) inflection points. As t → 0, some tangents to Ct passing through P approach tangents to C = C0 passing through P , while the other ones approach lines that join P with singular points of C. In a similar way, some inflection points of Ct approach inflection points of C = C0 , while the other ones approach singular points of C = C0 . The number of tangents to Ct that pass through P and approach the line joining P with a given singular point of C depends only on the type of this singular point: whether it is a self-intersection or a semicubical cusp. In a similar way, the number of inflection points of Ct that “disappear” at a singular point of C is determined by the type of this point. Thus the Plücker formulas we want to prove are equivalent to the following proposition. Proposition 5.4 Assume that a polynomial equation f (x, y) = 0 determines a plane curve that has a singularity at the origin which is a self-intersection or a semicubical cusp, the equation being generic for curves satisfying this condition. Consider the curves ft (x, y) = 0 obtained from the curve f (x, y) = 0 by a generic small perturbation preserving the degree of f , with f0 (x, y) = f (x, y). Then for small nonzero values of t, on the curve ft (x, y) = 0 near the origin • there are 2 points at which the tangent has a given direction and 6 inflection points whenever the origin is a double point of the original curve; • there are 3 points at which the tangent has a given direction and 8 inflection points whenever the origin is a semicubical cusp of the original curve.

5.3 Plücker Formulas for Singular Curves

67

Proof We have essentially proved this when deducing the Plücker formulas for smooth curves; however, we will give an explicit proof. First, consider disappearing tangents. Choosing an appropriate coordinate system, we can put the equation of a curve with a double point at the origin into the form f (x, y) = y 2 − x 2 + . . . , where the dots stand for monomials of degree greater than 2 and we assume that the tangent is vertical. The derivative ∂f/∂y has a nonzero linear part, hence the curve given by the equation ∂f/∂y = 0 is smooth. The restriction of f to this curve has a zero of order 2 at the origin, hence the perturbed equation ft (x, y) =

∂ft =0 ∂y

on points with vertical tangents has two solutions near the origin. In a similar way, in the case of a semicubical cusp the equation of the curve can be put into the form f (x, y) = y 2 − x 3 + . . . , where the dots stand for monomials of degree greater than 3. In this case, the equation ∂f/∂y = 0 also defines a curve that is smooth at the origin, and the restriction of f to this curve has a zero of order 3. Therefore, the perturbed equation on points with vertical tangents has three solutions near the origin. Now we proceed to enumerate the disappearing inflection points. As we know, inflection points are intersection points of the curve with the zero set of the Hessian. The zero set of the Hessian of a curve of degree 3 or higher has singularities also at singular points of the curve. Hence the enumeration of inflection points disappearing at a singular point is not so easy as the enumeration of disappearing tangents. Nevertheless, the result is universal and can be obtained by considering an arbitrary model example. As a model example of a curve with a semicubical cusp, one can take the semicubical parabola y 2 z = x 3 , whose dual is also a semicubical parabola. This curve has one inflection point, while its perturbation, which is a generic cubic curve, has nine inflection points. Hence eight inflection points disappear at a semicubical cusp. As a model example of a curve with a double point, we take the rational cubic curve y 2 z = x 2 (x − z). It has three inflection points: the dual curve, the so-called deltoid, is a rational curve of degree 4 with three semicubical cusps. Therefore, 9 − 3 = 6 inflection points disappear at a double point. Exercise 5.4 Check that the curve y 2 z = x 3 has one inflection point and the curve y 2 z = x 2 (x − z) has three inflection points.

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5.4 Newton Polygons Assume that the plane curve given by a homogeneous equation P (x, y, z) = 0 is irreducible. This means that the polynomial P cannot be written as a product of two polynomial factors of positive degree. However, it may happen that in a neighborhood of a chosen point (we will assume that it has homogeneous coordinates (0 : 0 : 1)), the equation of the curve can indeed be written as a product of nontrivial factors. In this case we will say that each of these factors determines a local branch of the curve. Definition 5.1 Let p(x, y) = 0 be the equation of a curve P (x, y, z) = 0 in a neighborhood of (0, 0), that is, p(x, y) = P (x, y, 1). We say that the curve is locally irreducible in a neighborhood of (0, 0) if the function p(x, y) cannot be written as a product of two analytic functions p(x, y) = p1 (x, y)p2 (x, y) vanishing at (0, 0). If p admits a factorization into locally irreducible factors p(x, y) = p1 (x, y) . . . pd (x, y), then the curves locally given by the equations pi (x, y) = 0, i = 1, . . . , d, are called the local branches of the curve at the point (0, 0). Note that since the polynomial p(x, y) = P (x, y, 1) is irreducible, the factors in its decomposition cannot be polynomials: in the affine coordinates x, y they are defined by infinite series. Exercise 5.5 Prove the following properties of local branches: a) at a smooth point, every curve is locally irreducible, i.e., a curve can be locally reducible only at a singular point; b) at a singular point, a curve is not necessarily locally reducible; for instance, the semicubical parabola x 2 = y 3 is locally irreducible at the origin; c) every local branch admits an analytic parametrization, i.e., for every analytic curve pi (x, y) = 0 irreducible at the origin there exist nonzero analytic functions f1 (t), f2 (t) with f1 (0) = f2 (0) = 0 such that in a neighborhood of the origin pi (x, y) = 0 if and only if p(f1 (t), f2 (t)) = 0. Now we explain how one can find the branches of a curve passing through the singular  point (0, 0) using Newton diagrams (polygons, see Fig. 5.4). Consider a curve ai x mi y ni = 0. Its branches passing through (0, 0) are given by equations Fig. 5.4 The Newton polygon for the polynomial a + bx + cy + dxy + ex 2 y + f xy 2

5.4 Newton Polygons

69

of the form y = x α ϕ(x), where ϕ(x) → a = 0 as x → 0. Substituting this expression into the equation of the curve yields 

ai x mi +αni ϕi (x) = 0.

(5.3)

Assume that one of the numbers mk + αnk is less than the other ones. Reducing Eq. (5.3) by x mk +αnk , we obtain ak ψ1 (x) + ψ2 (x) = 0, with lim ψ1 (x) =0 and x→0

lim ψ2 (x) = 0. We arrive at a contradiction, hence for given α at least two of the

x→0

numbers mi +αni and mj +αnj must coincide and be less than all the other numbers of this form. If the minimum is mi + αni = mj + αnj = . . . = ms + αns , then the corresponding branch is (approximately) given by the equation ai x mi y ni + aj x mj y nj + . . . + as x ms y ns = 0; the number of terms in this equation is equal to the number of marked points on the corresponding side of the Newton polygon. The geometric meaning of the fact that the minimum is mi + αni = . . . = ms + αns is rather simple. For given α, the equations m + αn = k with various k determine a family of parallel lines in the (m, n) plane, and the minimal value of k corresponds to the line that is closest to the origin. This means that to the branches of the curve f = 0 there correspond only those lines that contain sides of the Newton polygon and separate it from the origin. Exercise 5.6 Using the Newton diagram method, find the branches of the following curves at (0, 0): a) x 3 + y 3 − 3xy = 0; b) (x + y)(x 2 + y 2 ) + x(x − y) = 0; c) y 2 + x 2 (x − y) = 0.

Chapter 6

Mappings of Curves

A meromorphic function on an algebraic curve is a mapping from this curve into the projective line. However, it is natural to consider also mappings into other complex curves, first of all, one-to-one mappings from a complex curve to itself, i.e., automorphisms of a curve. All automorphisms of a given curve form a group. For a curve of genus 0 (projective line), this group is three-dimensional. For any curve of genus 1 (elliptic curve), it is one-dimensional. For curves of higher genus it is finite, and for curves of genus g > 2 it usually consists only of the identity mapping. Curves with a large symmetry group are of special interest: like any symmetric object, they can be very beautiful.

6.1 Automorphisms of the Riemann Sphere A biholomorphic mapping from a curve into itself is called an automorphism of the curve. All automorphisms of a given curve form a group. For any curve of genus g > 1, this group is finite. The automorphism groups of the rational curve and any elliptic curve are infinite. The following theorem describes the automorphism group of the rational curve. Theorem 6.1 Let z be an arbitrary coordinate on the Riemann sphere CP1 . Every automorphism of CP1 is a nondegenerate linear fractional transformation z → az+b cz+d , a, b, c, d ∈ C, ad − bc = 0. Proof A mapping from CP1 into itself is a meromorphic function on CP1 , and, by Theorem 3.3, every such function is a rational function f (z) = P (z)/Q(z). The degree of f is equal to the maximum of the degrees of the polynomials P and Q. Indeed, let m be this maximum. Then for almost all c, the polynomial equation P (z) = cQ(z) has degree m. Thus for almost all c, the set f −1 (c) consists of m points, and hence the degree of f equals m. An automorphism of © Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_6

71

72

6 Mappings of Curves

a curve is necessarily a mapping of degree 1, hence every automorphism f is a linear fractional transformation. It must be nondegenerate, since a degenerate linear fractional transformation is a constant. Corollary 6.1 The automorphism group of the rational curve is the projective group PSL(2, C) of complex 2 × 2 matrices with determinant 1. The group PSL(2, C) is the quotient of the group SL(2, C) of complex 2 × 2 matrices with determinant 1 by the normal subgroup consisting of the matrices I and −I , where I is the identity matrix. In particular, the automorphism group of the rational curve is a three-dimensional complex Lie group. Proof Indeed, let us associate with each linear fractional transformation z → az+b cz+d   a b the matrix . The composition of linear fractional transformations corresponds cd to the product of matrices, as can be checked by a straightforward calculation. Two nondegenerate linear fractional transformations determine the same automorphism of CP1 if and only if the numerator and denominator of the second transformation are obtained from those of the first one by multiplication by the same constant. Multiplying the numerator and denominator of a given linear fractional transformation by an appropriate number, we can ensure that the determinant of the corresponding matrix is equal to 1. But simultaneous multiplication of the numerator and denominator by −1 does not affect the transformation, which makes it necessary to quotient the group of matrices with determinant 1 by the subgroup of scalar matrices. The corollary is proved.

6.2 Mappings of Elliptic Curves Let L and M be two lattices in C, and let X = C/L and Y = C/M be the corresponding elliptic curves. We will determine the structure of an arbitrary (nonconstant) holomorphic mapping f : X → Y . A mapping z → z + a induces an automorphism of the curve Y ; composing with such an automorphism, we may assume that f (0) = 0. The Riemann–Hurwitz formula implies that a holomorphic mapping between two tori has no ramification points, hence f : X → Y is a covering. The mapping p

f

C − → X − → Y , where p : C → X is the natural projection, is also a covering. The space C is simply connected, hence the obtained covering is isomorphic to the covering q : C → Y (both of them are isomorphic to the universal covering over Y ). Thus we obtain the commutative diagram

6.2 Mappings of Elliptic Curves

73

The mapping F : C → C is holomorphic, and the equality f (0) = 0 implies that F (0) is a point of the lattice L. We may assume that F (0) = 0, since the translation p

f

→X− → Y. by an element of L does not affect the mapping C − Under the mapping F , the set p−1 (0) goes to q −1 (0), whence F (L) ⊂ M. If l ∈ L and z is an arbitrary complex number, then F (z + l) ≡ F (z) (mod M), i.e., F (z + l) − F (z) = wl (z) ∈ M. The set M is discrete, while the set C is connected, hence the function wl is constant, i.e., wl (z) depends only (z+l) on l. Therefore, dF dz − dFdz(z) = 0, i.e., the function dFdz(z) is invariant under translations by elements of L. Thus the set of all values of the function dFdz(z) coincides with the set of its values on a fundamental parallelogram of the lattice L. Therefore, this function is constant, because every bounded holomorphic function is constant. Since F (0) = 0, we see that F (z) = cz. The constant c must satisfy the condition cL ⊂ M. As a result, we see that every holomorphic mapping between elliptic curves is induced by a mapping of the form F (z) = cz + a, where cL ⊂ M. Using this property, we can describe the group of automorphisms of an elliptic curve leaving the point 0 fixed. Theorem 6.2 The group of automorphisms (leaving the point 0 fixed) of an elliptic curve X = C/L is the cyclic group of order 4 for a square lattice, of order 6 for a triangular lattice, and of order 2 in all the other cases. Proof First of all, note that to an automorphism of an elliptic curve there corresponds a mapping F (z) = cz such that cL = L. Indeed, if cL = L, then c−1 L = L, hence the mapping z → c−1 z induces the inverse mapping. Conversely, if a mapping z → c1 c2 z induces the identity mapping of the torus, then c1 c2 = 1 (to prove this, it suffices to consider small values of z). Hence the inclusions c1 L ⊂ L and c2 L ⊂ L imply that L ⊂ c2−1 L = c1 L ⊂ L. Under the mapping z → cz, the area of a fundamental parallelogram of the lattice gets multiplied by |c|2 , hence |c| = 1. Moreover, c is a root of unity. Indeed, otherwise the endpoints of the vectors 1, c, c2 , c3 , . . . would form a dense subset of the circle, while all of them are lattice vectors. Every elliptic curve has automorphisms corresponding to c = ±1. Let us find out which elliptic curves have other automorphisms. First of all, note that c is a root of a quadratic equation with integer coefficients. Indeed, if the lattice is generated by vectors ω1 and ω2 , then cω1 = pω1 + qω2 and cω2 = rω1 + sω2 , hence c is a root of the quadratic equation p − x q  n    r s − x  = 0. The polynomial x − 1 factors into a product of irreducible cyclotomic polynomials Φd (x), where d ranges over all divisors of n; the degree of Φd (x) is equal to ϕ(d), where ϕ is Euler’s totient function, i.e., ϕ(d) is the number of positive integers smaller than and relatively prime with d (the proof can be found, e.g., in [18]). The quadratic irreducible cyclotomic polynomials are x 2 + x + 1 and x 2 + 1. In the first case, we obtain a lattice consisting of regular triangles, since x 3 = 1; in the second case, we obtain a square lattice.

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Corollary 6.2 There exist nonisomorphic elliptic curves. Indeed, the automorphism groups of isomorphic curves are isomorphic, but we have given examples of elliptic curves with a marked point whose automorphism groups have different orders. On every elliptic curve C there exists a meromorphic function f : C → CP1 of degree 2. It has 4 critical values, i.e., 4 ramification points. We can choose such a function so that one of the critical points coincides with the marked point on C. Automorphisms of a curve C are closely related to automorphisms of the quadruple of critical values of the corresponding function. The most symmetrical arrangement of four points on a sphere is the one in which they occupy the vertices of a regular tetrahedron. An elliptic curve that is a two-sheeted covering of CP1 with such a collection of ramification points corresponds to a regular triangular lattice, and the automorphism group of this curve is isomorphic to Z6 . If (for an appropriate choice of a projective coordinate) the ramification points occupy the vertices of a square on an equator of the sphere, then the covering curve corresponds to a square lattice, and its automorphism group is isomorphic to Z4 . All other curves have no symmetries different from the transposition of the sheets of the covering. Exercise 6.1 Show that the quotient of an elliptic curve by the group of automorphisms leaving the point 0 fixed is the Riemann sphere.

6.3 Moduli of Elliptic Curves The properties of holomorphic mappings of elliptic curves proved in the previous section allow one to understand the structure of the moduli space of elliptic curves. First of all, note that every elliptic curve is isomorphic to that obtained as the quotient of C by a lattice Lτ generated by 1 and a number τ with Im τ > 0. Indeed, if a lattice L is generated by numbers ω1 and ω2 , then the mapping z → cz, where c = ±1/ω1 , sends L to the lattice generated by 1 and ω2 /ω1 . For τ one can take whichever of the numbers ±ω2 /ω1 has a positive imaginary part. Now let us find out when elliptic curves corresponding to parameters τ and τ  are isomorphic. For this, there should exist a number c such that cLτ = Lτ  , i.e., c and cτ generate the lattice Lτ  . Therefore, c = p + qτ  and cτ = r + sτ  , where r+sτ  p, q, r, s are integers. Thus τ = p+qτ  . So far, we have used only the fact that the numbers c and cτ belong to the lattice Lτ  . They generate this lattice if and only if qr − ps = ±1. Since Im τ > 0 and Im τ  > 0, we obtain that qr − ps = 1. Thus the elliptic curves corresponding to τ and τ  are isomorphic if and only if r+sτ  these parameters are related by a linear fractional transformation τ = p+qτ  where   r s is an integer matrix with determinant 1. pq The group of 2 × 2 matrices with integer entries and determinant 1 is denoted by SL(2, Z). Any two proportional matrices (and only they) give rise to the same

6.3 Moduli of Elliptic Curves

75

Fig. 6.1 A fundamental domain of the modular group

linear fractional transformation. Hence the group of transformations under study is isomorphic to the quotient SL(2, Z)/±I = PSL(2, Z); here I is the identity matrix. Exercise 6.2 Check that the subgroup ±I in SL(2, Z) is normal. The group PSL(2, Z) is called the modular group.   Theorem 6.3 The triangle D with angles 0, π3 , π3 shown in Fig. 6.1 is a fundamental domain of the modular group. Proof Consider the elements S(z) = −1/z and T (z) = z + 1 in G = PSL(2, Z). They generate a subgroup G ⊂ G. First we will prove that D is a fundamental domain of G , and then that G = G, i.e., that the elements S and T generate the whole group G. To begin with, we will show that the triangles g  D (with g  ∈ G ) cover the whole upper half-plane, i.e., if Im(z) > 0, then g  z ∈ D for some g  ∈ G . Lemma 6.1 If Im(z) > 0, then Im(gz) with g ∈ G takes only finitely many values greater than Im(z). Proof Clearly, Im(gz) = Im

 az + b  cz + d

= Im

adz + bc¯z Im(z) = . |cz + d|2 |cz + d|2

Hence Im(gz)  Im(z) only if |cz + d|  1. The latter inequality holds only for finitely many pairs of integers (c, d), and for every such pair the value Im(gz) is uniquely determined. The group G is contained in G, hence for every point z in the upper half-plane one can choose an element g  ∈ G with the largest possible value of Im(g  z). The transformation T (z) = z + 1 does not change the imaginary part of z, hence for

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6 Mappings of Curves

some element w = T k g  z, with k ∈ Z, we have | Re(w)|  1/2, while the value Im(w) is still the largest possible one. In particular,  1  Im(w) Im(w) ≥ Im − . = w |w|2 Thus |w| ≥ 1, whence w ∈ D. Lemma 6.2 If z is an interior point of the domain D and gz ∈ D for g ∈ G, then g is the identity transformation. Proof Let g(z) = az+b cz+d . First consider the case c = 0. Then ad = 1, i.e., g(z) = z ± b. If b = 0, then the image of D has a nonempty intersection with D only for the transformations g(z) = z ± 1. But this intersection belongs to the set |Re(z)| = 1/2, which contains no interior points of D. Now assume that c = 0. Then g(z) =

1 a − , c c(cz + d)

whence  d  a   1  g(z) −  · z +  = 2 . c c c

(6.1)

Since a/c and d/c are real, the imaginary parts of g(z) − a/c and z + d/c are equal to the imaginary parts of g(z) and z. But the imaginary part of every point of D √ 3/2, hence the absolute is at least √ √ values of g(z) − a/c and z + d/c are also at least 3/2. Therefore, |c|  2/ 3, where c is a nonzero integer. Thus c = ±1, hence (6.1) can be written in the form |g(z) ∓ a| · |z ± d| = 1. But if g(z) ∈ D and z is an interior point of D, then |g(z) ∓ a| ≥ 1 and |z ± d| > 1 for any integers a and d. Lemma 6.2 implies, in particular, that for distinct elements g1 and g2 of the group G the sets g1 D and g2 D have no common interior points. Thus D is a fundamental domain of G . Now one can easily prove that G = G . Indeed, let g be an arbitrary element of G. Take an arbitrary interior point z of the domain D. The point gz lies in the upper half-plane, hence there exists an element g  ∈ G such that g  (gz) ∈ D. The transformation g  g ∈ G sends the interior point z of D to a point of D. Then it follows from Lemma 6.2 that g  g is the identity transformation, i.e., g = (g  )−1 ∈ G . The proof of the theorem is completed.

6.4 Lattices and Cubic Curves

77

6.4 Lattices and Cubic Curves A smooth plane algebraic curve of degree 3 (cubic) has genus 1, hence it is natural to expect that every curve of the form C/Λ, where Λ is a lattice, is isomorphic to a cubic. In Sect. 9.3 we will show that every smooth curve of genus 1 is given by an equation of the form y 2 = P3 (x), i.e., is an elliptic curve. Here we will explicitly describe an isomorphism of the curve C/Λ onto a cubic, i.e., associate a third-order equation with every lattice. As a preliminary step, we will find a standard form for the equation of a cubic. Every cubic has an inflection point (in fact, it has nine inflection points, see Sect. 3.3). Choose coordinates in the plane so that one of the inflection points is (0 : 1 : 0) and the equation ofthe tangent at this point is z = 0. Now, if the cubic is given by an equation aij x i y j z3−i−j = 0, then x = 0 is a root 3 2 of the polynomial a30 x + a21 x y + a12xy 2 + a03 y 3 of multiplicity 3. Therefore, a21 = a12 = a03 = 0 and a30 = 0. The tangent at (0 : 1 : 0) is given by the equation Fx x + Fy y + Fz z = 0, where the derivatives are evaluated at (0 : 1 : 0); therefore, Fx x = 0, Fy y = 0, and Fz z = 0. We may assume that Fz z(0, 1, 0) = 1. Thus in affine coordinates the curve is given by the equation y 2 − 2(ax + b)y + P3 (x) = 0, where P3 is a polynomial of degree 3. By the change of variables y1 = y − ax − b we can reduce it to the form y 2 = Q3 (x), where Q3 is a polynomial of degree 3 without multiple roots (otherwise the curve would have a singular point). So, we may assume that the curve is given by an equation y 2 = x(x − 1)(x − λ), where λ is a number different from 0 and 1. Making the change of variables x = x1 + 1+λ 3 , we may assume that the equation has the form y 2 = x 3 + ax + b, where a =

−λ2 +λ−1 3

+3λ−2 and b = −2λ +3λ . Here 4a 3 + 27b 2 = −λ2 (1 − λ)2 = 0. 27 For every lattice Λ = {mω1 + nω2 | m, n ∈ Z}, ω1 , ω2 ∈ C, Im ω1 /ω2 > 0, we can consider the function 3

2

℘Λ (z) =

  1 1  1 , + − z2 (z − ω)2 ω2

(6.2)

where the prime means that the sum is taken over all nonzero elements ω ∈ Λ. The grouping of terms  in the square brackets is indispensable, since each of the series  (z − ω)−2 and  ω−2 taken separately is divergent. First we will prove that the series (6.2) determines a meromorphic function on the complex line C. On every compact set K that contains no points of the lattice, the series converges uniformly and absolutely. Indeed, 1 2zω − z2 ω 2z − z2 ω−1 1 − 2 = 2 = 4 . 2 2 (z − ω) ω ω (z − ω) ω (zω−1 − 1)2

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6 Mappings of Curves 2 −1

2z−z ω  If |ω| is sufficiently large, then (zω −1 −1)2 ≈ 2z. Hence for all ω ∈ Λ with sufficiently large |ω| and for all z ∈ K there is a constant C such that

  

1 C 1  − . < (z − ω)2 ω2 |ω|3

Besides, |z − ω| >  for all z ∈ K and ω ∈ Λ , hence  such a constant exists also for all ω ∈ Λ . One can easily check that the series  |ω|−3 converges. Indeed, 

|ω|−3 =

∞ 



|pω1 + qω2 |−3 ≤

n=1 max(|p|,|q|)=n

∞ 

8n (nh)−3 ,

n=1

where h is the smaller of the heights of the fundamental parallelogram. Thus ℘Λ (z) is a meromorphic function with poles at the nodes of Λ. It is called the Weierstrass function. Now we proceed to the proof that the function ℘Λ (z) is periodic. For this, consider its derivative   (z) = −2 (z − ω)−3 ℘Λ (this time, the sum is over all nodes of the lattice). Obviously, ω1 and ω2 are periods of ℘  (z). Hence the functions ℘Λ (z + ωi ) and ℘Λ (z) can differ only by a constant c. Substituting the value z = −ωi /2 into the equality ℘Λ (z + ωi ) = ℘Λ (z) + c, we see that ℘Λ (ωi /2) = ℘Λ (−ωi /2) + c. But it follows from (6.2) that the function ℘Λ (z) is even. Hence c = 0, i.e., ω1 and ω2 are periods of ℘Λ (z). The function ℘Λ has double poles at the nodes of L and no other singular points. Inside the fundamental parallelogram there is exactly one node of L. Hence the sum of the poles of ℘Λ lying inside the fundamental parallelogram is congruent to zero modulo Λ. Therefore, inside the fundamental parallelogram there are two zeros u and v of ℘Λ with u + v ≡ 0 (mod Λ). For every constant c, the poles of the function ℘Λ (z) − c coincide with the poles of ℘Λ (z), hence inside the fundamental parallelogram there are exactly two points u and v such that ℘Λ (u) = ℘Λ (v) = c, and they satisfy the relation u+v ≡ 0 (mod Λ). In the case where u ≡ −u (mod Λ), these two points coincide, and the corresponding value is taken by ℘Λ with multiplicity 2. At points where two zeros of the function ℘Λ (z) − c merge, the  (z) vanishes. One can choose a fundamental parallelogram so that its derivative ℘Λ interior contains exactly four points for which u ≡ −u (mod Λ), namely, 0,

ω1 ω2 ω1 + ω2 , , and . 2 2 2

6.4 Lattices and Cubic Curves

79

The first of them is a pole of the function ℘Λ , and the other three are zeros of its  . So, the values derivative ℘Λ e1 = ℘ Λ

ω  1 , 2

e2 = ℘ Λ

ω + ω  1 2 , 2

and e3 = ℘Λ

ω  2 , 2

and only they, are taken by ℘Λ with multiplicity 2. Values of multiplicity 2  (z) = 0 if and only if correspond to zeros of the derivative, hence ℘Λ z≡

ω1 ω2 ω1 + ω2 , , (mod Λ). 2 2 2

Note that the values e1 , e2 , and e3 are pairwise distinct. Indeed, assume, for instance, that e1 = e3 . Then the function ℘Λ (z) − e1 has double zeros at ω1 /2 and ω2 /2, i.e., inside the fundamental parallelogram there are at least four zeros of this function, which is impossible. Let us derive a differential equation for the function ℘Λ (z). If the coefficients of  (z))2 and all nonpositive powers of z in the Laurent expansions of the functions (℘Λ 3 2 a℘Λ (z)+b℘Λ (z)+c℘Λ (z)+d coincide, then these functions are equal. Indeed, their difference is an elliptic function without poles that vanishes at the origin. Therefore, this difference is a constant, the constant being zero. Since 

1 2 d  1  = 1 + 2x + 3x 2 + . . . , = 1−x dx 1 − x

we have   1 1  1 + − z2 (z − ω)2 ω2      1  z z 2 1  1 1 + 2 + 3 + . . . − = 2+ z ω2 ω ω ω2

℘Λ (z) =

= where Gk =



1 + 3G4 z2 + 5G6 z4 + . . . , z2

ω−k (for odd k, this sum vanishes by symmetry). Hence ℘Λ (z) = z−2 + . . . , 2 ℘Λ (z) = z−4 + 6G4 + . . . , 3 ℘Λ (z) = z−6 + 9G4 z−2 + 15G6 + . . . ,  (℘Λ (z))2 = 4z−6 − 24G4 z−2 − 80G6 + . . .

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6 Mappings of Curves

(we have written only those terms of the Laurent expansion that are relevant for our purposes). Thus 3 2 a℘Λ (z) + b℘Λ (z) + c℘Λ (z) + d

= az−6 + bz−4 + (9aG4 + c) z−2 + (15aG6 + 6bG4 + d) + . . . 3 + b℘ 2 + c℘ + d = (℘  )2 if Therefore, a℘Λ Λ Λ Λ

⎧ ⎪ ⎪ a = 4, ⎪ ⎪ ⎪ ⎨ b = 0, ⎪ 9aG4 + c = −24G4 , ⎪ ⎪ ⎪ ⎪ ⎩ 15aG6 + 6bG4 + d = −80G6. Obviously, the obtained system of equations has the solution ⎧ a = 4, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ b = 0, ⎪ c = −60G4 , ⎪ ⎪ ⎪ ⎪ ⎩ d = −140G6. To simplify the formulas, one usually sets 

ω−4 ,  g3 = 140G6 = 140 ω−6 . g2 = 60G4 = 60

Then  3 (℘Λ (z))2 = 4℘Λ (z) − g2 ℘Λ (z) − g3 .

One can use the function ℘Λ to parametrize the cubic curve y 2 = 4x 3 − g2 x − g3 by setting x = ℘Λ (z),

 y = ℘Λ (z).

(6.3)

6.5 The j -Invariant Revisited

81

Changing to homogeneous coordinates in CP2 , we define a mapping f : C/Λ→ CP2 as follows:  z → (℘Λ (z) : ℘Λ (z) : 1) for z = 0,

z → (0 : 1 : 0)

for z = 0.

Obviously, f is analytic at all points different from the nodes of the lattice. Writing it in the form   1 ℘Λ (z) z →  (z) : 1 : ℘  (z) , ℘Λ Λ one can check that f is also analytic around the nodes of the lattice. It is a one-to-one mapping from the torus C/Λ onto the cubic curve y 2 z = 4x 3 − g2 xz2 − g3 z3 in CP2 . Indeed, the line at infinity z = 0 contains only the point (0 : 1 : 0) of this curve. It is the image of all nodes of the lattice (which all correspond to the same point of the torus). For the other points, we can consider the affine curve y 2 = 4x 3 − g2 x − g3  (z)). The equation ℘ (z) = c can have and the mapping z → (℘Λ (z), ℘Λ Λ  (z) = 0, and one or two solutions. The latter takes place in the case where ℘Λ the two solutions have the form ±z. The images of these two points under the  (z)) do not coincide, since the nonzero numbers ℘  (z) mapping z → (℘Λ (z), ℘Λ Λ   and ℘Λ (−z) = −℘Λ (z) have different signs.

6.5 The j -Invariant Revisited Recall from p. 40 that for a cubic C given by an equation y 2 = x(x − 1)(x − λ) we have defined, from geometric considerations, the j -invariant j (C) = j (λ) = 28

(1 − λ + λ2 )3 . λ2 (1 − λ)2

Using the correspondence between plane cubic curves and lattices, we can assign a j -invariant to every lattice. Let us show that the j -invariant can be expressed in terms of the functions g2 and g3 introduced in the previous section. For this, we first make the change of 2 variables x = x  + λ+1 3 , which sends the curve y = x(x − 1)(x − λ) to the curve

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6 Mappings of Curves

+3λ−2 y 2 = x 3 + ax  + b, where a = −λ +λ−1 and b = −2λ +3λ , and then make 3 27 √ 3   2 3 − g x  − g , the change of variables x = 4x , resulting in the curve y = 4x 2 3 √ where g2 = − 3 4a and g3 = −b. Thus 2

j (C) = j (λ) = 28

3

2

1728g23 1728 · 4a 3 (1 − λ + λ2 )3 = = , λ2 (1 − λ2 ) 4a 3 + 27b 2 g23 − 27g32

and we have expressed the j -invariant in terms of g2 and g3 . The functions g2 and g3 are so-called modular functions, and all the other modular functions can be algebraically expressed in terms of g2 and g3 . Before giving a general definition of a modular function, we recall that g2 and g3 are the following  functions of a lattice Λ: g2 = 60G4 and g3 = 140G6 , where G2k (Λ) = ω∈Λ ω−2k . We will say that F is a function of weight 2k (defined on lattices) if F (μΛ) = μ−2k F (Λ) for every lattice Λ and every complex number μ = 0. In particular, G2k is a function of weight 2k. Let ω1 , ω2 ∈ C be a positively oriented basis of Λ. A function F (Λ) can also be regarded as a function F (ω1 , ω2 ). Then the condition that F is a function of weight 2k takes the form F (μω1 , μω2 ) = μ−2k F (ω1 , ω2 ). For such a function, ω22k F (ω1 , ω2 ) = F

ω

1

ω2

,

ω  ω  ω2  1 1 =F ,1 = f ω2 ω2 ω2

for some function f on the upper half-plane H . The function f depends only on the lattice and does not depend on the choice of a basis, hence for every matrix   ab ∈ SL2 (Z) cd we have ω22k f

ω  1

ω2 

= (cω1 + dω2 )2k f

 aω + bω  1 2 , cω1 + dω2

 ω1 i.e., f (z) = (cz + d)2k f az+b cz+d , where z = ω2 . Such a meromorphic function f on the upper half-plane is said to be weakly modular. A weakly modular function is modular if it additionally satisfies the following condition of being meromorphic at infinity. A function f of a variable z ∈ H can be regarded as a function f˜ of the variable q = e2πiz defined everywhere except the origin. If it can be defined at the origin so as to obtain a function meromorphic at the origin, we will say that the original function f is meromorphic at infinity. Denote by vp (f ) the order of f at a point p (if p is a pole of f , then vp (f ) is negative; if p is a zero of f , then vp (f ) is positive; and if f (p) = 0, then

6.5 The j -Invariant Revisited

83

vp (f ) = 0). We also define the order of f at infinity as follows. Let f˜(q) = ∞  cn q n with c−N = 0. Then v∞ (f ) = −N. n=−N

Theorem 6.4 For every modular function f of weight 2k,  1 1 k vp (f ) = , v∞ (f ) + vρ (f ) + vi (f ) + 3 2 6 p √

where ρ = 1+i2 3 and the sum is over all points p of the fundamental domain, with equivalent points on the boundary identified. Proof Consider the closed contour C shown in Fig. 6.2. If f has zeros or poles at ρ or i, then we turn around these points along circular arcs; if there is a zero or a pole at the vertical boundary, then we cut out a semicircle around it and add the corresponding semicircle to the parallel boundary (and do the same for zeros and poles lying at the arc of the unit circle).  df (z) 1 Let us compute the integral 2πi f (z) in two ways. On the one hand, C

1 2πi

 C

 f  (z) df (z)  = Res = vp (f ). f (z) f (z) p =i,ρ

On the other hand, the integral can be computed directly, and we proceed to this computation. If we replace z by e2πiz , then the upper line segment transforms into a circle centered at the origin and traversed clockwise, hence the integral along this segment is equal to −v∞ (f ). The integrals along the vertical sides cancel, since f (z + 1) = f (z). Let f (z − i) = c(z − i)m + . . ., where m = vi (f ). Then m f  (z − i) = + ..., f (z − i) z−i Fig. 6.2 The contour C

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6 Mappings of Curves

hence the integral of f  /f along a small circle centered at i traversed clockwise is equal to −vi (f ). Therefore, the integral along half of this circle is equal to − 12 vi (f ). √ In a similar way, the integrals along the circular arcs centered at (±1 + i 3)/2 sum to − 13 vρ (f ), since each of them tends to 1/6 of the circle as the radius decreases. Finally, the most difficult part is the computation of the integral along the arc of the unit circle. We will use the fact that the transformation z → 1/z sends a half of the arc to the other half changing the orientation. Clearly,  1 = d(z2k f (z)) = z2k df (z) + 2kz2k−1f (z) dz. df − z Hence  1 df − dz f (z)dz df (z) z2k df (z) z + 2k . + 2kz2k−1 2k =  1  = 2k f (z) z z f (z) z f (z) f − z  dz 1 Thus the integrals of df/f cancel, and we are left with the integral 2πi 2k z along 1/12 of the circle traversed counterclockwise. In total, we obtain 

1 1 k vp (f ) = −v∞ (f ) − vρ (f ) − vi (f ) + , 3 2 6

as desired. The functions g2 (z) = 60



1 , (nz + m)4

g4 (z) =



1 (nz + m)6

are modular functions of weights 4 and 6, respectively. They are holomorphic not only in the whole upper half-plane, but also at infinity. Indeed, the terms with n = 0 vanish as z → ∞, hence g2 (∞) = 60

∞   1 1 4 = 120 = π4 4 4 3 m m m=1

and g3 (∞) = 280

∞  1 8 6 π . = 6 m 27

m=1

6.6 Automorphisms of Elliptic Curves and Poncelet’s Closure Theorem

85

Theorem 6.4 allows one to find the zeros of the functions g2 and g3 . For g2 we obtain the expansion 26 = n + n21 + n32 , where n, n1 , and n2 are nonnegative integers. Thus n = n1 = 0 and n2 = 1, i.e., g2 has a simple zero at ρ and no other zeros. For g3 we obtain the expansion 36 = n + n21 + n32 . Hence n = n2 = 0 and n1 = 1, i.e., g3 has a simple zero at i and no other zeros. Theorem 6.5 The moduli space of elliptic curves is isomorphic to the Riemann sphere, the isomorphism being given by the j -invariant. Proof The function j =

1728g23 g23 −27g32

is a modular function of weight 0, since both

numerator and denominator are modular functions of weight 12. The denominator 8 6 Δ = g23 − 27g32 vanishes at infinity, since g2 (∞) = 43 π 4 and g3 (∞) = 27 π . Clearly, Δ does not vanish at i and ρ. The function Δ is holomorphic in the upper half-plane and v∞ (Δ) ≥ 1, hence Theorem 6.4 implies that v∞ (Δ) = 1 and vp (Δ) = 0 for p = ∞, i.e., Δ does not vanish in the whole upper half-plane. Taking into account that g2 (∞) = 0, we see that j has a simple pole at infinity. Now let us prove that for every λ the equation j (z) = λ has a unique solution. This equation is equivalent to the equation 1728g32 −λΔ = 0. Applying Theorem 6.4 to the holomorphic modular function 1728g32 − λΔ of weight 12, we obtain the relation 1 = n + n21 + n32 . Thus (n, n1 , n2 ) is one of the triples (1, 0, 0), (0, 2, 0), (0, 0, 3). In each of these cases, the function under consideration has exactly one zero. For the curve C given by the equation y 2 = x 3 + ax + b, where a = and b =

−2λ3 +3λ2 +3λ−2 , 27

−λ2 +λ−1 3

we obtain

j (C) = j (λ) = 28

1728(4a 3) (1 − λ + λ2 )3 = 3 . 2 2 λ (1 − λ) 4a + 27b 2

If a = 0, i.e., λ2 − λ + 1 = 0, we obtain j (C) = 0. If λ = −1, we obtain b = 0 and j (C) = 1728. Easy calculations show how to find the equation of a plane cubic with j -invariant equal to a given value j : if j is different from 0 and 1728, then the desired curve is y2 = x3 −

j 27 j 27 x− . 4 j − 1728 4 j − 1728

6.6 Automorphisms of Elliptic Curves and Poncelet’s Closure Theorem Using our knowledge of the structure of automorphisms of elliptic curves, we will prove the following Poncelet’s closure theorem.

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x2

Fig. 6.3 Closed 4-segment polygonal lines inscribed in a conic and touching an inner conic

l1

x1

x

l2

Consider two plane conics C1 and C2 . Choose a point x1 ∈ C1 and construct a sequence of points x1 , x2 , x3 , . . . of C1 as follows: the line x1 x2 touches C2 , the line x2 x3 touches C2 and x3 = x1 , and so on (Fig. 6.3). Poncelet’s theorem says that if xn+1 = x1 (i.e., the segments of tangents form a closed n-segment polygonal line) for some initial point on C1 , then every other choice of a point x1 ∈ C1 also results in a closed n-segment polygonal line. In the real case, we cannot draw a tangent to a conic from any point, hence we will assume that the conics C1 and C2 are in CP2 . We will also assume that these conics are in general position, i.e., that they meet in four distinct points. Let C2∨ be the dual conic of C2 lying in the dual projective plane (CP2 )∨ . To prove Poncelet’s theorem, consider the subset E in C1 × C2∨ ⊂ CP2 ×(CP2 )∨ consisting of the pairs (x, l) such that x ∈ l, i.e., x is a point of C1 lying at a tangent l to C2 . To begin with, note that E is an elliptic curve. Indeed, the natural projection E → C1 (as well as the projection E → C2∨ ) is a ramified 2-sheeted covering with 4 ramification points (which are the intersection points of C1 and C2 , since two tangents to C2 from a point x ∈ C1 merge if x lies on C2 ). Hence it follows from the Riemann–Hurwitz formula that χ(E) = 0. On the elliptic curve E there are two involutions: (x1 , l) ↔ (x2 , l) and (x, l1 ) ↔ (x, l2 ) (the notation is the same as in Fig. 6.3). Denote them by σ1 and σ2 . The description of automorphisms of an elliptic curve implies that σ has the form σ (x) = c − x, where c is a constant. Thus σ1 (x) = c1 − x and σ2 (x) = c2 − x. Consider the composition τ of the involutions σ1 and σ2 . On the one hand, τ (x) = (c1 − c2 ) + x, i.e., τ is a translation. On the other hand, τ is the mapping (x1 , l1 ) → (x2 , l1 ) → (x2 , l2 ). Hence τ n (x1 , l1 ) = (x1 , l1 ), i.e., the mapping τ n has a fixed point. But τ n is also a translation, and a translation having a fixed point is the identity mapping. Remark 6.1 In the proof of Poncelet’s theorem we have used the fact (proved in Sect. 9.3) that every smooth curve of genus 1 is an elliptic curve.

6.7 Automorphisms of Curves of Higher Genus:Hurwitz’s Theorem

87

6.7 Automorphisms of Curves of Higher Genus: Hurwitz’s Theorem Let C be an algebraic curve and G be a finite subgroup in Aut(C). Taking the quotient of C by the action of G, we obtain a two-dimensional surface C  = C/G. The natural projection p : C → C  is a ramified covering. For almost all points x ∈ C, the orbit {gx | g ∈ G} consists of |G| points; the ramification index of such a point is equal to 1. For finitely many points x ∈ C, the group Gx = {g | gx = x} (the stabilizer of x) is nontrivial. The orbit of every such point x consists of |G|/|Gx | points. The ramification index of each point of this orbit equals |Gx |. Hence, applying the Riemann–Hurwitz formula, we obtain   |G|  1  1− ; χ(C) = |G|χ(C  ) − (|Gx | − 1) = |G| χ(C  ) − |Gx | |Gx | here the sum is over all points x of the curve C  (and the contribution of each point except ramification points is 0). By Gx we have denoted the stabilizer of the preimage of such a point. Using this formula, we will prove Theorems 6.6 and 6.7. Theorem 6.6 (Hurwitz) The order of a finite subgroup G of the automorphism group of a curve of genus g ≥ 2 does not exceed 84(g − 1). Proof Let g  be the genus of the topological surface C  = C/G. We may assume that |G| > 1, hence g  < g. We now consider three cases. 1. g  ≥ 2. Then 2g − 2 ≥ 2|G|, i.e., |G| ≤ g − 1. Thus, if the quotient of a curve of genus g by the action of a finite group of automorphisms has genus g  ≥ 2, then this group contains at most g − 1 elements.    2. g = 1. Then 2g −2 = |G| 1− |G1x | . If there are no ramification points, then 2g − 2 = 0, i.e., g = 1, but g > 1 by assumption. If there is a ramification point, then the right-hand side contains the sum of the terms 1 − |G1x | , each of which is

at least 1/2, since |Gx | ≥ 2. Therefore, 2g − 2 ≥ |G| 2 , i.e., |G| ≤ 4(g − 1). In other words, if the quotient of a curve of genus g ≥ 2 by the action of a group of automorphisms is a torus, then most 4(g − 1) elements.  at   this  group has 1  3. g = 0. Then 2g − 2 = |G| 1 − |Gx | − 2 . The numbers 2g − 2 and |G| are positive, and each of the terms 1 − at least 3 such terms.

1 |Gx |

is strictly less than 1, hence there are  1−



1 5 |Gx | is at least 2 , since each of the summands is at least 12 . Therefore, 2(g − 1) ≥ |G| 2 , i.e., |G| ≤ 4(g − 1). If there are exactly 4 terms, then at least one of the numbers |Gx | is greater than 2,   1 1 − |Gx | = 2. Therefore, since otherwise

If there are more than 4 of them, then the sum

2(g − 1) ≥ |G|

3 2

+

 2 1 −2 = |G|, 3 12

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i.e., |G| ≤ 24(g − 1). Thus, if the quotient of a curve of genus g ≥ 2 by the action of a group of automorphisms is a sphere, and the number of ramification points of the covering is 4, then this group has at most 24(g − 1) elements. It remains to consider the case where there are 3 terms. Denote the three  numbers |Gx | by a ≤ b ≤ c. The sum 1 − a1 + 1 − b1 + 1 − 1c must be greater than 2, hence c > 3; moreover, b ≥ 3. If c ≥ 7, then |G| ≤ 84(g − 1). If c = 6 and a = 2, then b ≥ 4, whence |G| ≤ 24(g − 1). If c = 6 and a ≥ 3, then |G| ≤ 12(g − 1). If c = 5 and a = 2, then b ≥ 4, whence |G| ≤ 40(g − 1). If c = 5 and a ≥ 3, then |G| ≤ 15(g − 1). If c = 4 and a ≥ 3, then |G| ≤ 24(g − 1). The value |G| = 84(g − 1) is reached only for a = 2, b = 3, c = 7. Thus a curve of genus g ≥ 2 can have a group of automorphisms with 84(g − 1) elements only if there is a meromorphic function on this curve with exactly three critical values and the preimages of each of these critical values are critical points of multiplicity 2, 3, and 7, respectively. All cases are covered, so the proof is completed. Remark 6.2 Actually, the automorphism group of a curve of genus g ≥ 2 is finite, hence its order does not exceed 84(g − 1). But the finiteness of the automorphism group will be proved later, in Sect. 11.4. Let N > 1 be a positive integer. Consider the homomorphism of the group PSL(2, Z) to the group PSL(2, ZN ) that sends each matrix entry to its residue modulo N. This is obviously an epimorphism. Its kernel is called the principal congruence subgroup of level N and denoted by Γ (N). This subgroup consists of all 2 × 2 matrices (regarded up to multiplication by −1) with determinant 1 whose diagonal entries are congruent to 1 modulo N and off-diagonal entries are divisible by N. Exercise 6.3 Show that the group PGL(2, ZN ) has N(N − 1)(N + 1) elements and the number of elements in PSL(2, ZN ) is twice less. In particular, the number of elements in PSL(2, Z7 ) is equal to 7·6·8 = 168. 2 Exercise 6.4 The Klein quartic is the curve given by the equation xy 3 + yz3 + zx 3 = 0 in the plane. a) Show that this curve is smooth and has genus 3. b) Show that the quotient of the upper half-plane by Γ (7), the principal congruence subgroup of level 7, is biholomorphic to the Klein quartic punctured at 24 points.

6.7 Automorphisms of Curves of Higher Genus:Hurwitz’s Theorem

89

c) Find the 168 automorphisms of the Klein quartic. In particular, show that it has an automorphism of order 7 given by the formula (x : y : z) → (x, ζ 4 y, ζ 5 z), where ζ is a primitive 7th root of unity, ζ 7 = 1. The automorphism group of the Klein quartic is isomorphic to PSL(2, Z7 ) = PSL(3, Z2 ). Its order is equal to 168 = 84 · (3 − 1). The Klein quartic is the first among the Hurwitz curves, i.e., curves of genus g ≥ 2 whose automorphism group has 84(g − 1) elements. Such curves exist not for all values of g; the next such value is g = 7 (the corresponding curve is called the Macbeath surface). However, there exist arbitrarily large g for which there are Hurwitz curves of genus g. The question of what is the maximal order of the automorphism group of a curve of a given genus g is not studied completely. Exercise 6.5 Show that there are no Hurwitz curves of genus g = 2 and genus g = 4. Exercise 6.6 Find the order of the automorphism group of the Bolza surface, the most symmetric curve of genus g = 2; this is the hyperelliptic curve given by the equation y 2 = x 5 − x. Exercise 6.7 Find the automorphism group of the Fermat curve x n + y n + zn = 0. Assume that an action of a finite group G of automorphisms of a curve C has k orbits with nontrivial groups Gx , whose orders are r1 , . . . , rk . We will say that this action is of type {r1 , . . . , rk }. Theorem 6.7 The action of a nontrivial finite subgroup G of automorphisms of the Riemann sphere has one of the following types: {r, r}, {2, 2, r} (where r ≥ 2 is an arbitrary number), {2, 3, 3}, {2, 3, 4}, {2, 3, 5}. Proof The Riemann sphere C = CP1 has genus 0, hence the quotient curve C/G is a topologicalsphere and  its Euler characteristic is 2. Therefore, 2 = |G|(2 − R),  where R = 1 − |G1x | . Hence for a nontrivial group G we have 0 < R < 2; in particular, there exist ramification points, because R = 0. Let k be the number of orbits with nontrivial groups Gx . If k = 1, then 0 < R < 1, hence 2 > 2 − R > 2 1 and the number |G| = 2−R cannot be an integer. Therefore, k ≥ 2. On the other hand, each of the summands in R is greater than 1/2, hence the number of summands does not exceed 3, i.e., k ≤ 3. For k = 2, we obtain two points x1 and x2 with ramification indices r1 and r2 . Let y1 and y2 be the images of these points in C/G. On a sphere with punctures at the points y1 and y2 , a circle around y1 is homotopic to a circle around y2 . Therefore, r1 = r2 = r.

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6 Mappings of Curves

      Now let k = 3 and R = 1 − a1 + 1 − b1 + 1 − 1c . Then |G| =

2 2−R ,

where

2 − R = + + − 1. For definiteness, let a ≤ b ≤ c. If a ≥ 3, then 2 − R ≤ 0. Consequently, a = 2 and 2 − R = b1 + 1c − 12 . Hence b is 2 or 3. For b = 2 we obtain |G| = 2c, and for b = 3 we obtain |G| = 1 2 1 . Therefore, c is equal to 3, 4, 1 a

or 5.

1 b

1 c

c−6

Remark 6.3 Actions of all types mentioned above can indeed be constructed. An action of type {r, r} is given by the rotations around some axis by multiples of 2π/r. An action of type {2, 2, r} is provided by the group generated by the rotation by π around some axis and the rotations by multiples of π/r around an axis perpendicular to the first one (one can imagine it as the group of rigid motions of a sphere that preserve a regular 2r-gon inscribed in an equatorial circle). Actions of type {2, 3, 3}, {2, 3, 4}, and {2, 3, 5} are provided by the groups of rigid motions of a sphere that preserve a regular tetrahedron, cube (or octahedron), and dodecahedron (or icosahedron), respectively. Exercise 6.8 Find the orders of subgroups G of automorphisms of the types mentioned in Theorem 6.7. Check that a group of automorphisms of type {2, 3, 5} is isomorphic to PSL(2, Z5 ).

Chapter 7

Differential 1-Forms on Curves

It is convenient to express the basic properties of curves and, more generally, arbitrary complex manifolds, in terms of various related objects. We mean primarily spaces of meromorphic functions, vector fields, and differential forms. These spaces are endowed with natural algebraic structures, which allows one to express properties of curves in algebraic terms.

7.1 Tangent and Cotangent Bundles A Riemann surface is a one-dimensional complex manifold. Over every manifold, whether real or complex, there are natural vector bundles. For definiteness, in what follows we will speak about complex manifolds, which are our prime interest. First of all, given a manifold M, we can consider the product vector bundle M × C with the natural projection to the first factor. Of course, we could consider the product of M with a vector space of higher dimension, but we do not yet need this construction. The other two natural vector bundles defined over every manifold are the tangent and cotangent bundles. Let us recall their definitions. A tangent vector at a point t of a manifold M can be defined in the following ways. First, one may think of it as an equivalence class of holomorphic mappings D → M from the unit disk D ⊂ C to M sending 0 to t. Here two mappings γ1 , γ2 are equivalent if the distance ρ(γ1 (x), γ2 (x)) tends to zero faster than x as x → 0; this means that lim ρ(γ1 (x), γ2 (x))/|x| = 0,

x→0

x ∈ D.

In this definition, ρ is any metric that induces the topology of M.

© Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_7

91

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7 Differential 1-Forms on Curves

The other definition is more algebraic, but easier to work with. It says that a tangent vector at a point t is a derivation from the ring of germs at t to the ring of complex numbers. A derivation is a linear mapping δ between rings satisfying the Leibniz rule δ(fg) = δ(f )g(t) + f (t)δ(g). For instance, it is clear from this definition that all tangent vectors at a point form a vector space, with dimension equal to the dimension of the manifold. In particular, the tangent space of a curve is one-dimensional. The dual of the tangent space is called the cotangent space. It consists of all functionals on the tangent space, which are called cotangent vectors. d In a local coordinate z on a curve, a tangent vector can be written as a dz , and ∂ in local coordinates z1 , . . . , zn on a manifold of dimension n, as a1 ∂z1 + . . . + an ∂z∂ n . Accordingly, the coordinate representation of a cotangent vector is bdz or, in the general case, b1 dz1 + . . . + bn dzn . A cotangent vector bdz acts on a tangent d vector a dz as bdz : a

d → ab. dz

The collection of all tangent (respectively, cotangent) spaces at all points of a manifold M is called the tangent (respectively, cotangent) bundle of M. Definition 7.1 A line bundle over a manifold is a vector bundle of rank 1, i.e., a bundle with one-dimensional fibers. The tangent and cotangent bundles of a complex curve are line bundles, since their fibers are one-dimensional. A vector field on a manifold M is an assignment of a tangent vector to each point t of M. The vector must depend holomorphically on t. In other words, a vector field is a holomorphic section of the tangent bundle. Accordingly, a holomorphic section of the cotangent bundle is called a holomorphic differential 1-form. In a similar way, a holomorphic function on M is a holomorphic section of the trivial line bundle, i.e., of the direct product M × C regarded as a bundle over M. In a local coordinate z, a holomorphic vector field and a holomorphic 1-form can be written as a(z)

d dz

and b(z)dz,

respectively, where a, b are holomorphic functions.

7.1 Tangent and Cotangent Bundles

93

Note that the coordinate representations of a vector field and a 1-form behave differently under changes of coordinates. In a new coordinate z = g(z1 ), a 1-form b(z)dz can be written as b(z)dz = b(g(z1 ))dg(z1 ) = b(g(z1 ))g  (z1 )dz1 . To describe the corresponding transformation of a vector field is a more difficult d acts on the function z1 : task. Let us see how a vector field a(z) dz a(z)

dz1 dg −1 (z) a(g(z1 )) = a(z) =  . dz dz g (z1 )

Hence the change of coordinate z = g(z1 ) has the following effect on the vector field: a(z)

a(g(z1 )) d d =  . dz g (z1 ) dz1

For any coordinate z, every holomorphic function in a neighborhood of the given point z = 0 can be represented by a power series in positive powers of z: a(z) = a0 + a1 z + a2 z2 + . . . . However, holomorphic vector fields and differential forms on compact complex curves, as well as holomorphic functions on these curves, are extremely rare. So, we will also consider meromorphic vector fields and differential 1-forms, i.e., meromorphic sections of tangent and cotangent bundles. Each point of such a section, except finitely many, has a neighborhood in which the coefficient of d/dz (respectively, dz) in any local coordinate z is a holomorphic function. At the special points, this coefficient is allowed to have poles. The order of such a pole is called the order of the corresponding pole of the meromorphic vector field or meromorphic 1-form. It does not depend on the choice of a local coordinate. Every meromorphic function, in any coordinate z in a punctured neighborhood of the pole z = 0 of order k > 0, can be represented by a power series in powers of z of order ≥ −k: a(z) =

a−k a−k+1 + k−1 + . . . + a0 + a1 z + a2 z2 + . . . . zk z

All meromorphic functions on a given curve C form a field: they can be added, multiplied, and divided. Each of the C-spaces of meromorphic vector fields and meromorphic 1-forms on C is also a vector space over the field of meromorphic functions on C.

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7 Differential 1-Forms on Curves

Exercise 7.1 Show that for an appropriate choice of a local coordinate z, in a neighborhood of the pole z = 0 of order k > 0 a meromorphic function can be written as a(z) = z−k . Exercise 7.2 Is it true that for an appropriate choice of a local coordinate z, in a neighborhood of the pole z = 0 of order k > 0 a given meromorphic 1-form can be written as z−k dz? A meromorphic function is a meromorphic section of the trivial line bundle. A meromorphic vector field can also be defined as a derivation from the ring of meromorphic functions to itself.

7.2 How to Define Vector Fields and Differential Forms First consider the case of the rational curve. In any coordinate z on CP1 , a meromorphic differential form can be written as a(z)dz, where a is a meromorphic (i.e., rational) function. The change of coordinate z = 1/z1 shows that the coefficient of dz1 is also meromorphic at z = ∞. In a similar way, every meromorphic vector field can be written as a(z)d/dz. Let us see how holomorphic differential 1-forms and vector fields look like. Consider the differential 1-form dz on CP1 . The substitution z = 1/z1 shows that it has a pole of order 2 at z = ∞: dz = −

1 dz1 . z12

But this means that every nonzero meromorphic 1-form also has poles! Indeed, let ω be an arbitrary meromorphic 1-form. The ratio ω/dz is a nonzero meromorphic function; denote it by f . Then ω = f dz, and ω has a pole at every point where f does. If f has no poles other than z = ∞, then ω necessarily has a pole at z = ∞. Thus we have proved that on CP1 there are no nonzero holomorphic 1-forms, or, in other words, that the dimension of the space of holomorphic 1-forms on CP1 is equal to zero. Exercise 7.3 Find the dimension of the space of holomorphic vector fields on CP1 . The claim that a nonzero meromorphic 1-form on CP1 has poles can be refined. Namely, consider the orders of all zeros and all poles of an arbitrary meromorphic 1-form ω. Then the difference between the sum of the orders of all zeros and the sum of the orders of all poles is equal to −2. Indeed, this holds for the form dz, and multiplying it by a nonzero meromorphic function increases the sum of the orders of all zeros by the same amount as the sum of the orders of all poles. Exercise 7.4 What is the difference between the sum of the orders of all zeros and the sum of the orders of all poles of a meromorphic vector field on CP1 ?

7.2 How to Define Vector Fields and Differential Forms

95

Later we will see that it is much easier to work with differential forms than with vector fields; so, our focus will be primarily on 1-forms on curves. With every meromorphic function f on a curve C we can associate a meromorphic 1-form on C, namely, its differential df . For instance, the 1-form dz is the differential of the meromorphic function z. Exercise 7.5 Check that the meromorphic 1-form dz/z cannot be represented as the differential of a meromorphic function. Thus the differential defines a linear mapping from the space of meromorphic functions to the space of meromorphic 1-forms. Elements of the image of this mapping are called exact differential 1-forms. As the previous exercise shows, not every meromorphic 1-form on a curve is exact. Exercise 7.6 State a simple necessary and sufficient condition for a meromorphic 1-form on CP1 to be exact. Under a mapping between manifolds, differential forms (like functions, but unlike vector fields) get pulled back in the opposite direction as compared with the mapping itself. This allows us to define them by the same methods as were used to define functions. If a curve C is embedded into a projective space, then a meromorphic 1-form can be defined as the restriction to C of a meromorphic 1-form on the ambient space. Indeed, a tangent to C at a point can be naturally embedded into the tangent space to the ambient space at the same point. A differential 1-form is a family of linear functionals on tangent spaces chosen at each point of the ambient space. Restricting it to tangent lines to C at various points, we obtain a family of linear functionals on tangent lines to C, i.e., a differential 1-form on C. Exercise 7.7 Given a curve C embedded into a projective space, the restriction of a vector field on this space to C is not, in general, a vector field on C. Why? Exercise 7.8 Given a curve C embedded into a projective space, check that if a meromorphic 1-form on the ambient space is the differential of a meromorphic function on this space, then its restriction to C is the differential of the restriction of the function to C. Exercise 7.9 Find the orders of zeros and poles of the restriction to the curve xn + yn = 1 of the differential 1-forms a) x dx, b) x dy on the plane. In a similar way, a holomorphic mapping g : C1 → C2 between complex curves allows one to construct, given a meromorphic 1-form ω on C2 , a meromorphic 1-form on C1 , which will be denoted by g ∗ ω and called the lifting of ω to C1 via g. By definition, the value of g ∗ ω on a tangent vector τ is equal to ω(dg(τ )), where dg(τ ) is a tangent vector in C2 , the image of τ under the tangent mapping dg. This definition fully coincides with that of the restriction of a 1-form to a curve in

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7 Differential 1-Forms on Curves

a projective space, but instead of an embedding of a curve into a projective space we have a mapping from a curve to another curve. Exercise 7.10 Consider the projection of the curve x 2 + y 2 = 1 to the x axis. Is it true that every meromorphic 1-form on this curve is the lifting of a 1-form from the x axis via this mapping? On the other hand, if (as is typically the case) the automorphism group of a curve C acts discretely on this curve, then every meromorphic 1-form on C invariant under this group can be descended to a meromorphic 1-form on the quotient curve. The same is also true for vector fields. Example 7.1 On the complex line C with coordinate z, consider the differential 1-form dz. It is invariant under all translations z → z + z0 of C and, consequently, can be descended to any elliptic curve. This form has no poles on C, hence its image on any elliptic curve C is a holomorphic 1-form on C. Thus we have constructed a holomorphic 1-form on every elliptic curve. Exercise 7.11 Show that every holomorphic 1-form on an elliptic curve is proportional to the 1-form constructed above with a constant coefficient. This means that the space of holomorphic 1-forms on an elliptic curve is one-dimensional over C. The constructed holomorphic 1-form on an elliptic curve has no poles and no zeros. This means that for every meromorphic 1-form on an elliptic curve, the difference between the sum of the orders of all zeros and the sum of the orders of all poles is zero.

7.3 The Dimension of the Space of Holomorphic 1-Forms on a Plane Curve The above calculations show that the dimension of the space of holomorphic 1-forms is 0 for CP1 and 1 for an elliptic curve. This observation makes one suspect that the dimension of this space is closely related to the genus of the curve. Now we will give another extremely important argument in favor of the conjecture that this dimension is just equal to the genus. We will prove that this is the case for smooth plane curves. (Recall that not every curve can be embedded into the plane; in particular, the embeddability of a curve imposes a strong restriction on its genus.) Theorem 7.1 The dimension of the space of holomorphic 1-forms on a smooth plane curve is equal to the genus of this curve. The proof will proceed in two steps. First we will prove that the dimension of the space of holomorphic 1-forms is not less than the genus of the curve. The reverse inequality will be proved in the next section. Let d be the degree of a plane curve C. As we know, then the genus of C is equal to g = (d − 1)(d − 2)/2. Curves of degree 1 and 2 are rational, and for them the

7.3 The Dimension of the Space of Holomorphic 1-Forms on a Plane Curve

97

theorem is true. In what follows, we assume that d ≥ 3. Consider a line in CP2 that intersects C in d different points, and choose a coordinate system (x : y : z) so that this line has the equation z = 0. Now consider the space S d−3 of homogeneous polynomials of degree d − 3 in variables x, y, z. This is a vector space over C. Exercise 7.12 Show that the dimension of this space is equal to (d − 1)(d − 2)/2. Given such a polynomial, we will construct a holomorphic 1-form on C so that the resulting mapping from S d−3 to the space of holomorphic 1-forms is linear and injective. Let C be a curve given by a homogeneous equation F (x, y, z) = 0 of degree d that has the form f (x, y) = 0 in the chart z = 1. With an arbitrary homogeneous polynomial G(x, y, z) we associate the polynomial g(x, y) = G(x, y, 1) and the differential 1-form g dx . ∂f/∂y This 1-form is defined at those points of the plane at which the partial derivative of f with respect to y does not vanish, and its restriction to C at these points is a holomorphic 1-form. What happens at points where ∂f/∂y = 0? Since the differential of f vanishes on C, i.e., ∂f ∂f dx + dy = 0, ∂x ∂y we conclude that dx dy =− ∂f/∂y ∂f/∂x on C. Thus the above differential 1-form on C can be rewritten as −

g dy . ∂f/∂x

Since the curve C is smooth, its partial derivatives with respect to x and y cannot vanish simultaneously at any point of the curve, hence the constructed 1-form is holomorphic on the affine part of the curve. Now let us check that if G has degree d − 3, then the constructed 1-form has a holomorphic extension to the points of intersection of C with the line z = 0. Change the coordinates from (x : y : 1) to (1 : u : v), i.e., set x = 1/v;

y = u/v.

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In the coordinates u, v, our 1-form can be written as  1 u  dv , 2 − v v v , ∂f 1 u , ∂y v v g

or, multiplying the numerator and denominator by v d−1 , 1 u , dv v v . −  ∂f 1 u  , v d−1 ∂y v v v d−3 g

The numerator of the latter expression is a polynomial, since g is a polynomial of degree d − 3, and the denominator is just the derivative of F (1, u, v) with respect to u. If it happens to be zero, then we can pass to the derivative with respect to v, in the same way as we earlier replaced y with x if the derivative with respect to y was zero. Thus the above fraction is the ratio of a polynomial and a function that does not vanish at the chosen point, hence it cannot have a pole at this point. Therefore, all 1-forms constructed from polynomials of degree d − 3 are holomorphic. g dx The mapping g → ∂f/∂y , which associates a 1-form with a polynomial, is a linear mapping of vector spaces. Its restriction to the space of polynomials of degree d − 3 is injective, i.e., has no kernel. Indeed, this mapping sends a polynomial to zero only if it is identically zero on the curve C. This cannot happen for polynomials of degree d −3, since the degree of C is d > d −3 and the smoothness of C implies that it is irreducible. Hence the number of linearly independent 1-forms on a plane curve of degree d is not less than the dimension of the space of homogeneous polynomials of degree d − 3 in three variables. Remark 7.1 The choice of a mapping that associates a 1-form with a polynomial is not fortuitous. In fact, in CP2 there is a distinguished 2-form ω, which can be written as dx ∧ dy in the chart z = 1. Given a polynomial f , let us associate with it the ω 1-form df . The above considerations show that its restriction to the curve f = 0 is a well-defined holomorphic 1-form. Such a 1-form (as well as its multidimensional generalizations) is called a Gelfand–Leray form. If g is a polynomial of degree at most d −3, then the 1-form gω/df is also a holomorphic 1-form on the curve, which allows one to construct a space of holomorphic 1-forms with dimension equal to the dimension of the space of polynomials of degree at most d − 3.

7.4 Integrating 1-Forms

99

7.4 Integrating 1-Forms It remains to prove that the dimension of the space of holomorphic 1-forms does not exceed the genus of the curve. A crucial fact here is that a 1-form can be integrated along a real curve on the surface. A meromorphic 1-form can be regarded as a complex-valued 1-form on a real two-dimensional surface, a(z)dz = a(u + iv)(du + i dv),

z = u + iv.

Fix a point x0 on a curve C and consider a smooth curve γ : [0, 1] → C, γ (0) = x0 . Then the integral 

1 ω=

γ

γ ∗ω

0

is defined for every holomorphic 1-form ω on C. Moreover, by Stokes’ theorem, the integral depends only on the homotopy class of γ in the space of curves with fixed endpoints. If we slightly change the endpoint γ (1) of the curve preserving its homotopy type, then the integral becomes a function of the second endpoint. If ω is exact, i.e., ω = df for some function f , then, by the fundamental theorem of calculus, the integral of ω along a path γ that starts at x0 and ends at x1 is equal to x1

 ω= γ

df = f (x1 ) − f (x0 ). x0

In particular, the integral of an exact 1-form along any closed contour (i.e., a path whose beginning and end coincide) vanishes. Now consider a collection of closed paths γ1 , . . . , γ2g that start and end at x0 whose classes constitute a basis in the first homology group H1 (C, Z).  Lemma 7.1 Let ω be a real 1-form on C. If the integrals ω of ω along all cycles γi

γi , i = 1, . . . , 2g, vanish, then ω is exact, ω = df . The function f is defined uniquely up to an additive constant. If the form ω is holomorphic, then the function f is also holomorphic. Proof The proof is standard: given a 1-form ω, we construct a function f according to the rule x f (x) =

ω. x0

By the condition on integrals along cycles, this integral does not depend on the choice of a path connecting x0 and x.

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Now we know that the dimension of the space of holomorphic 1-forms on the curve C does not exceed 2g. Indeed, with such a 1-form we can associate the collection of its integrals along the cycles γi . If for two holomorphic 1-forms all the integrals coincide, then their difference has zero integrals, and hence is the differential of a holomorphic function. But on a compact curve there are no nonconstant holomorphic functions, so this difference is zero. Hence the space of holomorphic 1-forms is at most 2g-dimensional. And we want to prove that it is at most g-dimensional. For this, along with holomorphic forms ω consider also complex conjugate antiholomorphic forms ω. ¯ We want to prove that if holomorphic 1-forms ω1 , . . . , ωk are linearly independent, then the homology classes of the 1-forms ω1 , . . . , ωk , ω¯ 1 , . . . , ω¯ k are linearly independent. This will imply the desired inequality. The required assertion is a consequence of the following lemma. Lemma 7.2 Given a pair of holomorphic 1-forms ω1 , ω2 on C, if the sum ω1 + ω¯ 2 is the differential of a smooth function f , then ω1 = ω2 = 0. Proof Let ω1 = g1 (z)dz, ω2 = g2 (z)dz in a local coordinate z on C. If z = u + iv and z¯ = u − iv, then dz ∧ d z¯ = −2i du ∧ dv. Hence i i ω2 ∧ ω¯ 2 = |g2 (z)|2 dz ∧ d z¯ = |g2 (z)|2 du ∧ dv. 2 2 Assume that ω2 = 0. Then  C

i ω2 ∧ ω¯ 2 = 2

 |g2 (z)|2 du ∧ dv > 0. C

On the other hand, ω2 ∧ ω¯ 2 = ω2 ∧ ω1 + ω2 ∧ ω¯ 2 = ω2 ∧ (ω1 + ω¯ 2 ) = ω2 ∧ df, since  ω1 ∧ ω2 = 0. Therefore, to obtain a contradiction, it suffices to prove that ω2 ∧ df = 0. C

To this end, observe that the function g2 is holomorphic, hence dω2 = d(g2 dz) = dg2 ∧ dz =

 ∂g

2

∂z

dz +

d(f ω2 ) = df ∧ ω2 + f dω2 = df ∧ ω2 .  C

= 0, and thus

∂g2  d z¯ ∧ dz = 0. ∂ z¯

Therefore,

Thus the form df ∧ ω2 is exact, whence

∂g2 ∂ z¯

df ∧ ω2 = 0.

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7.5 The Dimension of the Space of Holomorphic 1-Forms on a Curve with Singularities For plane curves that do have singularities, but only of the simplest possible kind (double points, or nodes), Theorem 7.1 remains valid if we define the genus of a singular curve as the genus of its normalization. This means that the dimension of the space of holomorphic 1-forms is equal to g for every smooth curve of genus g. Theorem 7.2 The dimension of the space of holomorphic 1-forms on a plane curve with double points is equal to the genus of this curve. Proof The proof that the dimension of the space of holomorphic 1-forms does not exceed the genus of the curve requires no changes to be made. The proof of the reverse inequality should be modified as follows. For a curve C with double points, instead of the whole space of homogeneous polynomials of degree d − 3 in variables x, y, z one should consider only polynomials G(x, y, z) that vanish at all double points of C. Then the 1-form g(x, y) dy g(x, y) dx =− , ∂f/∂y ∂f/∂x where g(x, y) = G(x, y, 1), has no poles at double points of C. Indeed, choose local coordinates so that in a neighborhood of a double point the curve is given by the equation f (x, y) = 0, where f (x, y) = ax 2 +2bxy +cy 2 +. . . with b2 −ac = 0. If a point (x, y) = (0, 0) lies on the branch touching the line fx = 0, then y ≈ −ax b ,

g(x,y) dx whence fy (x, y) ≈ 2bx + 2cy ≈ 2 b −ac has no b x. In this case, the 1-form fy pole at the origin, since g(0, 0) = 0. The dimension of the space of polynomials that vanish at all double points of C is equal to (d−1)(d−2) − δ. As we already know (see Sect. 4.2), this number is equal 2 to the genus of the normalized curve, and the theorem follows. 2

7.6 Residues and Integrals of Meromorphic 1-Forms A holomorphic 1-form ω on a complex curve is always closed, dω = 0. Hence the integral of a holomorphic 1-form along a contractible loop on a complex curve vanishes. For meromorphic 1-forms, this is no longer true. More exactly, this is true if we regard a meromorphic 1-form ω as defined on the complex curve punctured at the poles of ω. On such a surface, a small circle centered at a puncture x0 is usually no longer contractible (the exception being the rational curve with one puncture). The value  1 ω = Resx0 ω, 2πi

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where the integral is taken along such a circle traversed in the positive direction (“counterclockwise”), is called the residue of the 1-form ω at the point x0 . Introducing a local coordinate z in a neighborhood of x0 , a 1-form ω that has a pole of order k at x0 admits a local expansion   ω = a−k z−k + a−k+1 z−k+1 + . . . + a−1 z−1 + a0 + . . . dz,

a−k = 0,

and its residue coincides with the coefficient a−1 of z−1 . Exercise 7.13 Check directly that the coefficient of z−1 in the Laurent series expansion of a meromorphic 1-form in a neighborhood of any point of the curve does not depend on the choice of a local coordinate z in this neighborhood. The residue of an exact meromorphic 1-form at every its pole is zero.  Theorem 7.3 For a meromorphic 1-form ω on a smooth curve, Resp (ω) = 0, where the sum is over all points at which the residue is not zero. We will give two proofs of this theorem. First Proof Indeed, let us glue our surface out of one polygon so that none of the poles of ω lies on the boundary of the polygon. Then the sum of the residues of ω is equal, up to the factor 1/2πi, to the integral of ω along the cycle that traverses the boundary of the polygon in the positive direction. In the surface glued out of this polygon, the cycle is homologous to zero, since it traverses each segment twice in opposite directions. Hence the integral of ω along this cycle vanishes, as required. Second Proof Consider the domain Ω obtained by removing small neighborhoods  of the poles of ω from the curve C. The form ω is holomorphic on Ω, hence ω = ∂Ω   0. On the other hand, ω = − Resp (ω). ∂Ω

Now we can give another proof of a fact we already know. Theorem 7.4 The sum of the orders of all zeros and poles of every meromorphic function f on a smooth curve vanishes. Proof Apply Theorem 7.3 to the meromorphic 1-form ω = df f . If, in a neighborn hood of a point p ∈ C, we have f (z) = az + . . ., where n is an integer and the dz dots stand for terms of higher order, then ω = df f = n z in this neighborhood. Therefore, the residue of ω at p is equal to n. Thus the sum of the orders of all zeros and poles of every meromorphic function f is equal to the sum of the residues of the 1-form ω, i.e., zero.

Chapter 8

Line Bundles, Linear Systems, and Divisors

We have already come across line bundles over curves: trivial, tangent, and cotangent bundles and their tensor powers. Thus, a natural question arises: given a curve, are there other line bundles over it? For instance, in the case of elliptic curves, all line bundles mentioned above are trivial, so this question means, in particular, whether there are nontrivial line bundles over a given elliptic curve. In this chapter we discuss the relation between line bundles over a curve and collections of points on this curve, called divisors. The language of divisors allows one to effectively describe and work with line bundles. We also discuss how a line bundle over a curve can be used to construct mappings from this curve to various projective spaces.

8.1 The Divisor of a Meromorphic Section of a Line Bundle There are two collections of points naturally associated with every meromorphic section of a line bundle over a curve C: the zeros and the poles of this section. Each point of these collections is assigned an integer multiplicity. In a neighborhood of every point, a section of a line bundle is described by a meromorphic function, and the multiplicity of a point x is the order of the zero or pole of this function at x (with the orders of zeros assumed to be positive and those of poles assumed to be negative). The formal sum of the zeros of a section σ , counted with multiplicity, is called the divisor of zeros of σ and denoted by (σ )0 . The formal sum of the poles of σ , counted with multiplicity, is called the divisor of poles and denoted by (σ )∞ . The formal sum of the zeros and poles of a section σ , counted with multiplicity, is called the divisor of σ and denoted by (σ ) = (σ )0 + (σ )∞ . Example 8.1 Let C = CP1 be the projective line with coordinate z. The divisor of the constant zero function on C is zero: (const) = 0. The divisor of the function z is (z) = 1 · 0 − 1 · ∞, since it has a simple zero at 0 and a simple pole at ∞. The © Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_8

103

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divisor of the differential 1-form ω = dz is (dz) = −2 · ∞, and the divisor of the dual vector field ∂/∂z is (∂/∂z) = 2 · ∞. Exercise 8.1 Let p, q be two distinct points of an elliptic curve C. Can a meromorphic function on C have a divisor of the form p − q? Exercise 8.2 Find the divisors of zeros and poles of the following objects: a) the polynomial zn on CP1 ; b) the 1-form (z − 1)dz on CP1 ; c) the vector field z2 d/dz on CP1 ; d) the Weierstrass function on an elliptic curve; e) the function on the Fermat curve x n + y n = 1 determined by its projection to the x axis; f) the function on the Klein quartic xy 3 + yz3 + zx 3 = 0 determined by its projection to the x axis. Let us give a general definition of a divisor on a curve. Definition 8.1 A divisor on a curve C is a formal finite linear combination of points of C.



ai xi

Another, often more convenient, notation for divisors is to write a divisor as a sum ax x over all points of C with only finitely many nonzero coefficients. In what x∈C

follows, we will consider only divisors with integer coefficients, ai ∈ Z. Divisors can be added (coefficientwise), and they form a commutative group under addition. The zero of this group is the zero divisor. Now let σ1 : C → E and σ2 : C → E be two nonzero meromorphic sections of the same line bundle E over a curve C. Since E is a line bundle (i.e., its fibers are one-dimensional), each of these sections is proportional to the other, i.e., can be obtained by multiplying the other by a meromorphic function: σ2 = f σ1 for some function f . Writing the sections σ1 , σ2 and the function f in a local coordinate, we see that (σ2 ) = (σ1 )+(f ), i.e., the divisor of the section σ2 is the sum of the divisors of the section σ1 and the function f . Conversely, if σ : C → E is a section of a line bundle and f is an arbitrary meromorphic function on C, then f σ is also a section of E, with (f σ ) = (f )+(σ ). Thus the divisors of any two sections of a given line bundle differ by a divisor of a meromorphic function, and adding a divisor of a meromorphic function to a divisor of a section yields again a divisor of a section of the same bundle. So, it is natural to introduce the following equivalence relation on the group of divisors: divisors D1 , D2 are said to be linearly equivalent if there is a meromorphic function f on the curve C with the divisor D1 − D2 . Exercise 8.3 Check that the addition of divisors induces a group structure on the linear equivalence classes of divisors. The quotient of the group of divisors of a curve C by the linear equivalence relation is called the Picard group of C and denoted by Pic(C). Every line bundle over C gives rise to an element of the Picard group Pic(C), namely, the linear equivalence class of divisors of holomorphic sections of this bundle.

8.2 The Degree of a Divisor and the Degree of a Bundle

105

8.2 The Degree of a Divisor and the Degree of a Bundle A divisor has an characteristic, the degree. The degree of  important quantitative  a divisor D = ai pi is the sum ai of its coefficients. This is an integer, which is usually denoted by deg D. The degree defines an epimorphism from the group of classes of divisors to the additive group of integers. It follows, in particular, that the classes of divisors of degree 0 form a subgroup in the former group. For a given curve C, this subgroup is denoted by Pic0 (C); more generally, the collection of classes of divisors of a given degree d, d ∈ Z, is denoted by Picd (C), and the whole Picard group can be written as the disjoint union Pic(C) = . . .  Pic−2 (C)  Pic−1 (C)  Pic0 (C)  Pic1 (C)  Pic2 (C)  . . . . Every subset Picd (C) in Pic(C) can be represented as Picd (C) = D + Pic0 (C), where D ∈ Picd (C) is an arbitrary class of divisors. Hence the structure of every subset Picd (C) essentially coincides with the structure of the group Pic0 (C), although the identification is not canonical, but depends on the choice of an element D ∈ Picd (C). Consequently, knowing the group Pic0 (C), one obtains a description of the whole group Pic(C). The degree of the divisor of any meromorphic function is 0, since the number of preimages of zero and the number of preimages of infinity (counting multiplicities) coincide and are equal to the degree of the function. Therefore, linearly equivalent divisors have the same degree, and hence all sections of a given line bundle also have the same degree. This common degree is called the degree of the bundle. Divisors of meromorphic functions are called principal divisors. They constitute the linear equivalence class of the zero divisor. Note that for g > 0, the fact that a divisor has zero degree does not in general imply that it is a principal divisor. Exercise 8.4 Show that any two divisors of the same degree on the projective line are linearly equivalent. In other words, the group Pic0 (CP1 ) is trivial and the group Pic(CP1 ) is isomorphic to Z. Example 8.2 Given a curve C embedded into a projective space, one can define the class of hyperplane divisors on C. A representative of this class is the divisor of the intersection of C with a hyperplane that does not contain C. Any two such divisors are linearly equivalent. Indeed, let two hyperplanes be given by homogeneous linear equations F1 = 0 and F2 = 0. Then the restriction of the ratio F1 /F2 to the curve C is a meromorphic function on C which establishes an equivalence of the two divisors. Analogously, for a projective curve we have the class of divisors corresponding to intersections of this curve with hypersurfaces of a given degree k. Exercise 8.5 Express the degree of the class of hyperplane divisors of a projective curve C in terms of the degree of C itself. The degree of the tangent bundle of a curve C of genus g is equal to the Euler characteristic of C, i.e., deg(T C) = χ(C) = 2−2g. Indeed, the Euler characteristic

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of a manifold can be defined as the sum of the indices of the singular points of a generic vector field on this manifold. In turn, the singular points of a meromorphic vector field on a curve are exactly its zeros and poles, with indices equal to the orders of zeros and poles, respectively. Exercise 8.6 Let a meromorphic vector field on a curve in a neighborhood of a singular point have the form zk d/dz. Show that the index of the field at this point is equal to k. Taking the tensor product of line bundles corresponds to taking the sum of the corresponding classes of divisors in the Picard group. Indeed, if E → C and F → C are holomorphic line bundles and σ : C → E and τ : C → F are meromorphic sections of these bundles, then the tensor product E ⊗ F → C has the meromorphic section σ ⊗ τ : C → E ⊗ F . The divisor of σ ⊗ τ is the sum (σ ⊗ τ ) = (σ ) + (τ ), which implies the desired conclusion. Of course, the degree of the tensor product of line bundles is also equal to the sum of the degrees of the factors. Since the cotangent bundle of a curve is dual to the tangent bundle, their tensor product is the trivial line bundle. This means, in particular, that its degree is 0. Hence the degree of the cotangent bundle is equal to −χ(C) = 2g(C) − 2; it is positive for g(C) ≥ 2. The class of divisors of the cotangent bundle is called the canonical class, and its elements are called canonical divisors.

8.3 The Tautological Line Bundle Over the Projective Line The so-called tautological line bundle O(−1) is defined over a projective space of any dimension in the following way. The projective space CPn is the space of lines in the vector space Cn+1 passing through the origin. Then O(−1) is the line bundle over CPn whose fiber over a given point of CPn is exactly the line in Cn+1 that determines this point. The total space of O(−1) can be imagined as the submanifold in CPn ×Cn+1 consisting of the pairs (a point in CPn , a point of the corresponding line in Cn+1 ). For n = 1, this construction yields a line bundle O(−1) over the projective line CP1 . One can construct a section of this bundle taking an arbitrary line  in C2 that does not pass through the origin. This line has one intersection point with every line passing through the origin except the unique line  parallel to . Exercise 8.7 Show that the intersection points of  with lines passing through the origin determine a meromorphic section CP1 → O(−1) of the line bundle O(−1). Check that this section has no zeros and a unique pole of order 1 at the point  ∈ CP1 . This exercise shows that the line bundle O(−1) has degree −1. Thus it cannot coincide with the trivial, tangent, or cotangent bundle. The dual O(1) of the tautological bundle has degree 1. Tensor powers of these bundles are denoted by O(−1)⊗d = O(−d) and O(1)⊗d = O(d), respectively, where d is a nonnegative

8.4 Recovering a Line Bundle from a Class of Divisors

107

integer. The degree of O(d) is equal to d. Below we will see that O(2) is isomorphic to the tangent bundle of the projective line CP1 , and O(−2) is isomorphic to the cotangent bundle of CP1 .

8.4 Recovering a Line Bundle from a Class of Divisors It turns out that the correspondence associating with a line bundle the class of divisors of its sections is invertible: for every linear equivalence class of divisors there is a unique holomorphic line bundle for which the class of divisors of sections coincides with the given one. To describe this correspondence, we will finally give a rigorous definition of a line bundle. Definition 8.2 Let C be a smooth complex curve. A holomorphic mapping π : E → C from a two-dimensional complex surface to C is called a line bundle if every point in C has an open neighborhood U ⊂ C such that the preimage π −1 (U ) admits a trivialization, i.e., there exists a biholomorphic mapping ϕ : U × C → π −1 (U ) and for every x ∈ U the following properties hold: • π ◦ ϕ(x, c) = x for all c ∈ C; • the mapping c → ϕ(x, c) is an isomorphism of the line C onto the fiber π −1 (x). Two line bundles π1 : E1 → C and π2 : E2 → C over a curve C are said to be isomorphic if there exists a biholomorphic mapping h : E1 → E2 such that π2 = h ◦ π1 . Let π : E → C be a holomorphic line bundle, U, V be two open subsets in C, and ϕU : U × C → π −1 (U ), ϕV : V × C → π −1 (V ) be trivializations of E over U and V , respectively. Then we obtain the mapping ϕV−1 ◦ ϕU (x, c) = (x, fU V (x) · c) over the intersection U ∩ V . Here fU V (x) is a holomorphic function. The functions fU V for a given collection of trivializations are called transition functions. If the curve C is covered by open sets U1 , . . . , Um , then the collection of transition functions between trivializations over these sets uniquely determines the line bundle. Exercise 8.8 Show that over the intersection of three open sets U, V , W , the transition functions for trivializations of a given line bundle satisfy the identity fU V (x)fV W (x)fW U (x) = 1. Below we assume without proof that every holomorphic line bundle has at least one meromorphic section (one can also think that we restrict ourselves to considering only such bundles).

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Theorem 8.1 For every linear equivalence class of divisors D ∈ Pic(C) on a given smooth compact complex curve C there is a unique line bundle over C whose class of divisors of sections coincides with D. The line bundle from the statement of the theorem is denoted by O(D). Since on CP1 all divisors of the same degree are linearly equivalent, this notation agrees with the notation O(d). Proof We will explicitly construct the desired line bundle O(D). Let x1 , . . . , xm be the points of C that occur in D with nonzero coefficients. Consider an open cover U1 , . . . , Um , W of C consisting of m small disks Ui centered at xi and the curve C punctured at these points, W = C \ {x1 , . . . , xm }. Here Ui are chosen so small that they are pairwise distinct. In each disk Ui , choose an arbitrary holomorphic local coordinate zi . Over each punctured disk Ui ∩ W , glue the trivial line bundles over Ui and W via the −ai transition function fUi W (z i ) = zi , where ai is the multiplicity of the point xi in the divisor D, i.e., D = ai xi , ai ∈ Z. Then the identity section of the trivial line bundle over W can be extended to a meromorphic section of the constructed line bundle over C whose divisor coincides with D. We will denote this bundle by O(D). Now observe that a line bundle for which the class of divisors of sections is zero is necessarily trivial. Indeed, let σ be a meromorphic section of this line bundle; its divisor (σ ) coincides with the divisor of a meromorphic function f , i.e., (σ ) = (f ). Hence the section f1 σ has neither zeros no poles and, consequently, trivializes our bundle over the whole curve. Now let D ∈ Pic(C) be the class of divisors of sections of a given bundle E → C. Then the class of divisors of sections of the line bundle E ⊗ O(−D) is zero, i.e., this bundle is trivial. Therefore, E = O(D), and hence we have described all line bundles over C.

8.5 Mappings from Curves to Projective Spaces Associated with Line Bundles Let L be a line bundle over a curve C, and let V be a vector space of sections of L. Then we have a mapping from C to the projective space P V ∨ , the projectivization of the dual vector space of V . It has the following structure. For x ∈ C and σ ∈ V , the map x : σ → σ (x) is not a linear functional on V , since σ (x) is not an element of C, but an element of the fiber Lx of L over x. However, the ratio σ1 (x) : σ2 (x) for σ1 , σ2 ∈ V is already a well-defined complex number (if none of the two sections vanishes at x). Thus the mapping x : σ → σ (x) determines a linear functional on V defined up to multiplication by a constant, i.e., a point in P V ∨ , as desired.

8.6 Linear Systems and Mappings Between Curves

109

More exactly, the mapping is well defined at all points of C at which at least one of the sections σ ∈ V is not zero or infinity. However, it is not difficult to prove that it has a unique holomorphic extension to these points. Exercise 8.9 Check that if dim V > 0, then the above mapping from C to P V ∨ , defined originally on the complement to the set of common zeros and poles of all sections from V , can be extended to a holomorphic mapping of the whole curve. The space of all holomorphic sections of a given line bundle L is denoted by H 0 (L) (this notation stems from sheaf cohomology, a tool that we do not consider in this book). Taking for V the space H 0 (L) of all holomorphic sections of a line bundle L over a curve C, we denote the corresponding mapping by ϕL : C → P (H 0 (L))∨ . If L is the line bundle O(D) associated with a divisor D (more exactly, with the linear equivalence class of this divisor), then we also denote ϕL by ϕD . For two linearly equivalent divisors D and D  , the mappings ϕD and ϕD  coincide. The mapping under consideration is especially appealing and natural in the case where L = T ∨ (C) is the cotangent bundle and V is the space of holomorphic 1-forms H 0 (T ∨ (C)) = Ω 1 (C). Due to uniqueness, the image of the curve under this mapping is called the canonical curve, and the mapping itself is called the canonical map. As we will see later, the canonical map is almost always an embedding. Exercise 8.10 Why the canonical map is constructed from the cotangent bundle rather than the trivial or tangent bundle? Similar mappings can also be constructed from tensor powers of the cotangent bundle. They are called pluricanonical maps. For sufficiently large k (say, k = 3), the pluricanonical map constructed from the kth tensor power is an embedding on every curve (for g ≥ 2).

8.6 Linear Systems and Mappings Between Curves The space of holomorphic differentials is not the only finite-dimensional space associated with a complex curve. Weakening the naturalness requirements, we may consider many finite-dimensional spaces of sections of line bundles. Such spaces are associated with divisors  on the curve. Let us say that a divisor ai xi is effective if all its coefficients ai are positive, ai > 0 (or nonnegative if the sum is taken over all points of the curve). The zero divisor is also considered effective. If a divisor D is effective, we will write D ≥ 0. The effective divisors form a cone: they can be added and multiplied by positive numbers, but the difference of two effective divisors may not be effective. For an arbitrary line bundle over a curve C and a fixed divisor D, we can consider the set of

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sections σ of this bundle such that (σ ) + D ≥ 0. This is a finite-dimensional vector space with respect to natural operations on sections. In this way we significantly extend the collection of vector spaces associated with a given curve. Let D be a divisor on a curve C. Consider all rational functions f for which the divisor (f )+D is effective. They constitute a finite-dimensional vector space, which will be denoted by L(D). It can be identified with the space of holomorphic sections of the line bundle O(D). The dimension of the space L(D) is denoted by l(D). Example 8.3 Let D be a divisor of nonnegative degreeon the Riemann sphere. Write it in the form D = ai pi + a∞ ∞. Set fD (z) = (z − ai )−pi . Then L(D) is the space of functions of the form P (z)fD (z), where P (z) is a polynomial of degree at most deg D. Proof Let of degree n. Then (P ) ≥ −n∞. It is also P (z) be a polynomial  clear that ) = (−a p ) + ( a )∞. Hence (Pf ) + D ≥ (−a p ) + ( ai )∞ + (f D i i i D i i    (−ai pi ) + ( ai )∞ − n∞ ≥ ( ai + a∞ − n)∞ = (deg D − n)∞. Thus if n ≤ deg D, then (PfD ) + D ≥ 0, i.e., PfD ∈ L(D). Conversely, let h ∈ L(D) be a nonzero function. Consider  the function P (z) = h(z)/fD (z). Then (P ) = (h) − (fD ) ≥ −D − (fD ) = (− ai − a∞ )∞ = − deg D∞. This means that P (z) has no finite poles, and at infinity it can have a pole of order at most deg D. Thus P (z) is a polynomial of degree at most deg D. In the case of the trivial bundle, the projective space corresponding to a given divisor D admits the following realization. Consider the set |D| of all effective divisors linearly equivalent to D. Clearly, it is empty whenever the degree of D is negative. The set |D| can be naturally identified with the projectivization of the vector space L(D). Indeed, |D| = {(f ) + D|f ∈ L(D) \ {0}} and (f ) + D = (g) + D ⇔ g = λf, where λ is a nonzero complex number. The dimension of the projective space |D| is equal to l(D) − 1. Definition 8.3 A linear system (or linear series) on a complex curve C is a projective subspace in |D|. A linear system coinciding with |D| is said to be complete. Example 8.4 The hyperplane divisors on a curve in a projective space form a linear system. Example 8.5 A special case of the previous construction is a meromorphic function on a curve (i.e., a mapping from a curve to CP1 ). The preimage of every point under such a mapping is an effective divisor, and all such divisors are linearly equivalent. They form a linear system of divisors on the curve.

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111

Example 8.6 The canonical divisors form a complete linear system. All divisors belonging to the same linear system have the same degree. This common degree is called the degree of the linear system. A linear system of degree d for which the projective space has dimension n is a gdn linear system. Example 8.7 Every meromorphic function of degree d on a curve determines a linear system gd1 on this curve. In particular, a curve is hyperelliptic if and only if it admits a linear system g21 .

Chapter 9

Riemann–Roch Formula and Its Applications

The Riemann–Roch theorem establishes a relationship between two numbers: the dimension l(D) of the vector space L(D) of meromorphic functions with divisor ≥ −D and the dimension i(D) of the space Ω 1 (D) of meromorphic 1-forms with divisor ≥ D. It says that for a divisor D of degree d on a curve of genus g, l(D) = d − g + 1 + i(D). The dimension l(D) of the space of meromorphic functions with divisor ≥ −D does not change if we replace D with a linearly equivalent divisor D  . Indeed, let f be a function with divisor (f ) = D − D  (such a function is unique up to multiplication by a nonzero constant). Then the multiplication by f determines an isomorphism between the spaces L(D) and L(D  ). Thus the Riemann–Roch theorem deals not with individual divisors, but with linear equivalence classes of divisors. In particular, we can apply it to a canonical divisor K, and the dimension i(D) coincides with l(K − D). The Riemann–Roch formula and its various generalizations have numerous applications and are a tool widely used in modern algebraic geometry. In this chapter we discuss the simplest, yet very important, applications of this formula. Its proof is postponed to the next chapter.

9.1 Mittag-Leffler’s Problem Mittag-Leffler’s problem was the starting point that gave rise to the theory of sheaves. It is formulated as follows. Given a collection of principal parts of local functions at poles on a curve, determine whether they are the principal parts of a meromorphic function that is defined on the whole curve and has no other poles. © Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_9

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9 Riemann–Roch Formula and Its Applications

The statement of the problem should be made more precise, and we proceed to necessary definitions. For simplicity, we begin by considering the case of simple poles. So, let C be a complex curve and x ∈ C; we consider meromorphic functions defined in a neighborhood of x and having a simple pole or no pole at x. Two functions f1 , f2 are said to have the same principal part at x if their difference has no pole at x. Clearly, “having the same principal part” is an equivalence relation on the set of such functions. The principal part of a function f at x is the equivalence class containing f . If we introduce in a neighborhood of x a local coordinate z centered at x, then every function f with a simple pole or no pole at x can be written as a Laurent series f (z) =

a−1 + a0 + a1 z + . . . . z

Two functions f1 , f2 have the same principal part if and only if the coefficients of z−1 in their Laurent expansions coincide. Indeed, it is in this case that the difference f1 − f2 has no pole at x. Remark 9.1 A function f with a simple pole or no pole at a point x ∈ C determines a linear functional on the space of holomorphic local differential 1-forms in a neighborhood of x, associating with every such form ω the residue Resx f ω. If f has no pole at x, then the functional thus defined is zero. If f does have a pole at x, then the value of the functional on a 1-form ω vanishes if and only if ω(x) = 0. Thus the value of the functional on a 1-form depends only on the cotangent vector determined by this form. Hence f gives rise to a linear functional on the cotangent line at x and, consequently, a tangent vector at this point. On the other hand, this tangent vector is uniquely determined by the principal part of f at x. Thus a principal part at a simple pole can be interpreted as a tangent vector at this point. Now let f : C → CP1 be a meromorphic function on a curve C with simple poles at points x1 , . . . , xd and no other poles. Then for every holomorphic 1-form ω on C, d 

Resxi f ω = 0,

i=1

i.e., the sum of the residues of the 1-form f ω at all poles vanishes. This equality can also be understood as follows: if we associate with f the collection f1 , f2 , . . . , fd of its principal parts at the poles x1 , . . . , xd , then d  i=1

Resxi fi ω = 0.

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115

(Note that some of the points x1 , . . . , xd may not be poles of f ω. This happens if ω vanishes at some xi .) The following theorem says that the converse is also true and thus gives a complete solution to Mittag-Leffler’s problem for the case of functions with simple poles. Theorem 9.1 (Riemann) Given a collection f1 , f2 , . . . , fd of principal parts of functions with simple poles at points x1 , x2 , . . . , xd of a curve C (i.e., a collection of tangent vectors at these points), a function f with this collection of principal parts exists if and only if d 

Resxi fi ω = 0

(9.1)

i=1

for every holomorphic 1-form ω on C. Since the space of holomorphic 1-forms on a curve of genus g is finite-dimensional and has dimension g, it suffices to choose a basis ω1 , . . . , ωg in this space and check (9.1) for these forms. Proof It suffices to prove the “if” part of the theorem: if (9.1) holds for all holomorphic 1-forms, then a function with these principal parts does exist. Consider the effective divisor D = x1 + . . . + xd . By the Riemann–Roch theorem, the dimension of the space of functions with poles of order at most one at the points x1 , . . . , xd and no other poles is equal to l(D) = i(D) + d + 1 − g. Here i(D) is the dimension of the space of holomorphic 1-forms vanishing at all points xi of the divisor D. These 1-forms impose no restrictions on principal parts of functions at these points. Thus the dimension of the space of linearly independent linear restrictions on principal parts of functions at the points x1 , . . . , xd is equal to g − i(D). But the dimension of the space of these principal parts is equal to d. Thus the Riemann–Roch formula shows that there are no other restrictions: the dimension of the whole space of functions, including the subspace of constants, is equal to (d + 1) − (g − i(D)) and, therefore, coincides with the dimension of the space of principal parts annihilating the holomorphic 1-forms. Remark 9.2 If two meromorphic functions f, g on a curve C have the same collection of simple poles and the same principal parts at these poles, then their difference f − g is a meromorphic function without poles and, consequently, a constant. Therefore, the collection of principal parts of a meromorphic function with simple poles determines it uniquely up to an additive constant. Exercise 9.1 Deduce from the Riemann–Roch theorem the following solution to the general Mittag-Leffler’s problem: given a collection of principal parts f1 , . . . , fd at points x1 , . . . , xd of a curve C, a meromorphic function that has these principal

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parts at these points and no other poles exists if and only if Resx1 f1 ω + . . . + Resxd fd ω = 0 for every holomorphic 1-form ω on C. By the principal part of a function at a pole we mean the sum of the “negative order terms” of its expansion in a neighborhood of this pole: given a function f that in a neighborhood of a pole of order k in an arbitrary coordinate z has an expansion f (z) =

a−k+1 a−1 a−k + a0 + a1 z + . . . , + k−1 + . . . + zk z z

its principal part is equal to a−k a−k+1 a−1 . + k−1 + . . . + k z z z More exactly, two functions f1 , f2 have the same principal part at a common pole if their difference f1 −f2 has no pole at this point, and a principal part is an equivalence class of functions with respect to this equivalence relation. Since the residue of the 1-form f ω at a pole of f is determined by the principal part of this function, the residue of the product of a principal part and a holomorphic 1-form is also well defined. As in the case of simple poles, the collection of principal parts of a function at its poles of arbitrary order determines it uniquely up to an additive constant.

9.2 The Rational Curve We have often mentioned that the projective line CP1 is the unique complex curve of genus 0. The Riemann–Roch theorem makes it possible to prove this. Theorem 9.2 Every smooth compact complex curve of genus 0 is isomorphic to CP1 . Proof Given a complex curve C of genus 0, take the divisor D on C consisting of a single point x taken with multiplicity 1. The dimension i(D) of the space of meromorphic 1-forms with zero at x is equal to zero, since the degree of a canonical divisor on a curve of genus g = 0 is equal to 2g − 2 = −2. As to the dimension of the space L(D) of meromorphic functions on such a curve with a pole of order at most one at x, the Riemann–Roch theorem says that it is equal to l(D) = d − g + i(D) + 1 = 1 − 0 + 0 + 1 = 2. Choosing in the two-dimensional space L(D) a basis f0 , f1 where f0 is a constant, f0 = 1, we obtain a meromorphic function f1 on C. The degree of f1 is equal to 1,

9.3 Elliptic Curves

117

since it has one simple pole; hence it determines a biholomorphic mapping from C onto the projective line, as required.

9.3 Elliptic Curves In Sect. 6.4 we proved that the curve C/Λ, where Λ is a lattice, is given by an equation of the form y 2 = P3 (x), i.e., is an elliptic curve. The proof used an explicit formula for the Weierstrass function as a series over the lattice. In this section we will use the Riemann–Roch formula to prove that every smooth curve of genus 1, i.e., a smooth curve homeomorphic to a torus, is given by an equation of the form y 2 = P3 (x), and will give an alternative proof of the existence of the Weierstrass function. Theorem 9.3 Every smooth curve of genus 1 can be embedded into the projective plane. Now it follows from the formula for the genus of a plane curve that the degree of an embedded elliptic curve is equal to 3. Proof As we know, the dimension of the space of holomorphic 1-forms on a curve of genus 1 is equal to 1. Moreover, a generator of this space has no zeros, hence on an elliptic curve there are no nontrivial holomorphic 1-forms that have zeros but do not have poles. Let us compute the dimension of the space of meromorphic functions with a fixed pole of order at most one and no other poles. By the Riemann–Roch theorem, it is equal to d − g + i(D) + 1 = 1 − 1 + 0 + 1 = 1, i.e., all such functions are constants. However, functions with one pole of order two do exist. Indeed, for the divisor D = 2x, we have l(D) = d − g + i(D) + 1 = 2 − 1 + 1 = 2. Hence on an elliptic curve C there is a meromorphic function with a pole of order two at a given point and no other poles. The lifting of such a function to the complex line C via the quotient map C → C, like the original function itself, is defined uniquely up to an additive constant, a multiplicative constant, and a translation. One of these functions is called the Weierstrass function, see Sect. 6.4. This is a simplest elliptic function. The Weierstrass function ℘ (z) distinguishes itself by its expansion at z = 0. At this point, it has a pole of order two, and the coefficient of 1/z in this expansion vanishes, since otherwise the 1-form ℘ (z)dz would have a nonzero residue at 0. It follows that the Weierstrass function is even: otherwise ℘ (z) − ℘ (−z) would be a nonconstant holomorphic function on the curve. Adding to ℘ (z) a constant if

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necessary, we may assume that its expansion at 0 has the form ℘ (z) =

1 + a2 z 2 + a4 z 4 + . . . . z2

Since the Weierstrass function is defined on C, its derivative ℘  (z) is also defined and can be descended to the corresponding elliptic curve. The derivative has the expansion ℘  (z) = −

2 + 2a2 z + 4a4 z3 + . . . . z3

Now consider the function 4℘ 3 (z) − (℘  (z))2 . It is even, and its expansion at z = 0 has a pole of order two. Therefore, it can be written as a linear combination of the Weierstrass function and the constant function 1. Thus we have the identity (℘  (z))2 = 4℘ 3 (z) + a℘ (z) + b for some constants a and b. In other words, the triple of functions (℘ (z) : ℘  (z) : 1), which is a basis in the space L(3 · 0), determines a mapping from C to a plane cubic curve. Since the first of these functions is even and the second one is odd, the mapping is one-to-one onto its image. The complex numbers a and b in this representation are parameters of the curve; they depend on the chosen lattice (i.e., on a vector τ in the upper half-plane). In the affine chart z = 1, the image of this mapping is given by the equation y 2 = 4x 3 + ax + b. The projection of this cubic curve to the x axis coincides with the Weierstrass function on the elliptic curve, and the projection to the y axis coincides with its derivative. Exercise 9.2 Check that for any two points of an elliptic curve there is a meromorphic function of degree 2 with poles of order 2 at these points. Express this function in terms of the Weierstrass function. Exercise 9.3 Is it true that every divisor of zero degree on an elliptic curve is a) principal (i.e., coincides with the divisor of some meromorphic function); b) canonical (i.e., coincides with the divisor of some meromorphic 1-form)?

9.4 Hyperelliptic Curves and Curves of Genus 2

119

9.4 Hyperelliptic Curves and Curves of Genus 2 Recall that for a curve C of genus ≥ 2, the canonical map is the map ϕ : x → (ω1 (x) : . . . : ωg (x)) to the projective space CPg−1 determined by a choice of a basis (ω1 , . . . , ωg ) in the space of holomorphic 1-forms on C (see Sect. 8.5). Taking a more invariant point of view, consider the dual projective space P Λ∨ of the projectivized space P Λ of holomorphic 1-forms on C. Then almost every point x ∈ C determines a point in P Λ∨ : for any two nonzero holomorphic 1-forms ω1 , ω2 on C, the ratio ω1 (x) : ω2 (x) is a well-defined number at almost every point x. Now let us prove that there is no point of C at which all holomorphic 1-forms vanish simultaneously. Indeed, if such a point x existed, then, by the Riemann– Roch theorem, there would exist a meromorphic function on C with a simple pole at x (and no other poles): l(x) = 1 − g + i(x) + 1 = 2,

since i(x) = g.

This would mean that our curve is rational. Theorem 9.4 Let ϕ : C → CPg−1 be the canonical map for a curve C of genus g. If ϕ(x1 ) = ϕ(x2 ) for two distinct points x1 , x2 of C, then this curve is hyperelliptic, i.e., there is a meromorphic function of degree 2 on C. Proof Since the values of holomorphic 1-forms at x2 are proportional to their values at x1 with coefficient common for all 1-forms, the dimension of the space of holomorphic 1-forms with a zero at x1 coincides with the dimension of the space of holomorphic 1-forms with zeros at x1 , x2 . Hence the Riemann–Roch theorem says that the dimension of the space of meromorphic functions with poles of order at most one at the points x1 , x2 is equal to 2: l(x1 + x2 ) = 2 − g + (g − 1) + 1 = 2. Thus there exists a nonconstant meromorphic function with this property, and it has degree 2. Exercise 9.4 Is it true that the canonical map on a hyperelliptic curve has degree 2? Exercise 9.5 Show that if for two points x1 , x2 of a curve (which is necessarily hyperelliptic) there exists a function with simple poles at these points and no other poles, then the canonical map sends x1 , x2 to the same point. Exercise 9.6 Show that the image of a hyperelliptic curve of any genus under the canonical map is the rational curve.

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9 Riemann–Roch Formula and Its Applications

Theorem 9.5 Every complex curve of genus 2 is hyperelliptic. Proof The canonical map of a curve of genus 2 is a ramified covering of the projective line. The degree of this covering is 2. Indeed, the degree of a canonical divisor on a curve of genus 2 is equal to 2, hence every nonzero holomorphic 1-form has two zeros (counting multiplicities), and two linearly independent 1-forms cannot have common zeros. Hence the point (0 : 1) has exactly two preimages under the canonical map x → (ω1 (x) : ω2 (x)). This means that the degree of the canonical map is 2, and it is a hyperelliptic covering of the projective line. Exercise 9.7 Every meromorphic function of degree 2 on a complex curve determines an involution on this curve—an automorphism of order 2 permuting the preimages of every point from CP1 . The fixed points of this involution are the critical points of the function. Find the dimension l(D) of the space L(D) of meromorphic functions on a hyperelliptic curve of genus g ≥ 2 for the following cases: a) b) c) d)

D D D D

= 2x where x is a fixed point of the involution; = 2x where x is not a fixed point of the involution; = x1 + x2 where points x1 , x2 constitute an orbit of the involution; = x1 + x2 where points x1 , x2 do not constitute an orbit of the involution.

9.5 Riemann’s Calculation The Riemann–Roch theorem allows one to compute, albeit perhaps without sufficient justification, the dimension of the space of curves of a given genus. We know the dimension of the space of curves of genus 0 (it is equal to 0) and the dimension of the space of curves of genus 1 (it is equal to 1). The corresponding calculation for curves of genus g ≥ 2 was carried out by Riemann. Consider the space of meromorphic functions of degree d on curves of genus g. For a generic function from this space, all ramification points are simple. By the Riemann–Hurwitz formula, the number of these simple ramification points is equal to 2d +2g −2. This means that the dimension of the space of meromorphic functions of degree d on curves of genus g is equal to 2d +2g −2. Indeed, Riemann’s theorem shows that the number of meromorphic functions with fixed ramification points and given ramification indices at these points is finite, hence the space of such functions can be locally parametrized by their ramification points. Therefore, its dimension is equal to the dimension of the space of such points. Thus we know the dimension of the space of meromorphic functions with d poles of order at most one on curves of an arbitrary genus g. On the other hand, we can use the Riemann–Roch theorem to compute the dimension of the space of such functions on every individual curve of genus g. For d ≥ 2g, this dimension is equal to 2d − g + 1. Indeed, a holomorphic differential on a curve of genus g has 2g − 2 zeros, hence i(D) = 0 for every divisor D of degree at least 2g − 1.

9.6 Curves of Genus 3, 4, and 5

121

The automorphism group of a curve of genus g (and, therefore, of the space of meromorphic functions on such a curve) is finite. Hence the dimension dim Mg of the space Mg of curves of genus g ≥ 2 is equal to dim Mg = (2d + 2g − 2) − (2d − g + 1) = 3g − 3. A natural method to make such a formula “general,” i.e., to extend it also to curves of genus 0 and 1, is to make these curves “rigid” by “killing” their continuous automorphisms. For this, one should consider curves with marked points. Let us mark n points x1 , . . . , xn on a curve and say that two curves with marked points are isomorphic if there is a biholomorphic mapping between them that sends the marked points to the marked points. Remark 9.3 There are two versions of this construction. In the first one marked points are distinguishable, while in the second one they are indistinguishable. Accordingly, an isomorphism in the first version is a biholomorphism that sends every marked point to the point with the same mark, while in the second version it may permute marked points. In what follows, we will consider only the first version. On curves of genus g ≥ 2, adding yet another marked point increases the dimension of the space of such curves by one: the dimension of the space of curves of genus g with n marked points is equal to 3g − 3 + n. This is also true for curves of smaller genus, but for a sufficiently large number of points added. Elliptic curves (g = 1) already acquire rigidity for one marked point, and the dimension of such curves with n marked points is equal to 3g − 3 + n = n for n ≥ 1. To make the rational curve CP1 rigid, one should mark three points: for any two ordered triples of points on CP1 there exists a unique automorphism of CP1 that sends the first triple to the second one preserving the order of points. For n ≥ 3, the dimension of the space of rational curves with n marked points is also equal to 3g − 3 + n = n − 3. One can choose local coordinates on this space to be the coordinates of the points x4 , . . . , xn in the coordinate system in which x1 = ∞, x2 = 0, x3 = 1.

9.6 Curves of Genus 3, 4, and 5 We already know that every curve of genus 0 is biholomorphic to the projective line CP1 , every curve of genus 1 is biholomorphic to a plane cubic, and every curve of genus 2 is hyperelliptic (the hyperelliptic covering coinciding with the canonical map C → P (Ω 1 (C))∨ ≡ CP1 ). In this section we give an analogous description for curves of genus 3, 4, and 5. It uses the construction of the canonical curve and is based on the following observation. Proposition 9.1 Every smooth curve of genus g and degree 2g − 2 in CPg−1 that is not contained in any hyperplane is canonical.

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Proof Let C ⊂ CPg−1 be a smooth curve of genus g and degree 2g − 2. Let D be the divisor of a hyperplane section on C and K be a canonical divisor. Then the degree of the divisor K − D is 0, whence l(K − D) = 1 if D is linearly equivalent to K, and l(K − D) = 0 otherwise. In the second case, by the Riemann–Roch theorem, l(D) = g − 1; hence if C is not canonical, then it must be contained in a hyperplane. Theorem 9.6 Every smooth plane quartic (a curve of degree 4) is a canonical nonhyperelliptic curve of genus 3. Proof Indeed, the canonical map is a one-to-one map from a nonhyperelliptic curve of genus g = 3 onto a smooth plane curve of degree 2g − 2 = 4. On the other hand, we have just proved that every smooth plane curve of degree 4 is canonical. Theorem 9.7 Every nonhyperelliptic curve of genus 4 is biholomorphic to the intersection of a quadric and a cubic in CP3 . Proof As we have already seen (see Exercise 4.6), the transversal intersection of a quadric and a cubic in CP3 has indeed genus 4. Now consider a curve C embedded into CP3 via the canonical map. Every homogeneous polynomial of degree 2 in four variables (coordinates in CP3 ) determines on C a section of the square of the cotangent bundle corresponding to the class of divisors 2K. The space Sym2 C4 of 5 such polynomials has dimension 3 = 10. On the other hand, by the Riemann– Roch formula, l(2K) = 2(2g − 2) − g + 1 = 3g − 3, which is equal to 9 for g = 4. Hence the kernel of the natural mapping Sym2 C4 → L(2K) is at least one-dimensional. A nonzero element of this kernel determines a quadric Q2 containing C. In a similar way,   6 = 20, dim Sym C = 3 3

4

l(3K) = 3(2g − 2) − g + 1 = 5g − 5 = 15.

Hence the dimension of the space of cubic forms that vanish on C is at least 20 − 15 = 5. However, the space of cubic forms divisible by Q2 is fourdimensional. Thus the curve C lies on some cubic Q3 that does not contain Q2 . Therefore, C lies on the intersection of the surfaces Q2 and Q3 , and hence coincides with it. Exercise 9.8 Show that every nonhyperelliptic curve of genus 5 is biholomorphic to the intersection of three quadrics in CP4 . The result of calculating the dimensions via the analysis of canonical maps of curves of small genus coincides with the result of Riemann’s calculation, as the following table shows:

9.6 Curves of Genus 3, 4, and 5 g

d

Canonical curve

3

4

Q4 ⊂ CP2

4

6

Q2 ∩ Q3 ⊂ CP3

8

Q2 ∩ Q2

5

∩ Q2

⊂ CP

4

123 Number of parameters  6 2 − 1 = 14  5  6  4 3 − 1 + 3 − 3 − 1 = 24    3 64 − 3 = 36

dim PGL(g)

dim Mg

32 − 1 = 8

14 − 8 = 6

42 − 1 = 15

24 − 15 = 9

52

− 1 = 24

36 − 24 = 12

Now let us find the dimension of the space of hyperelliptic curves. On every hyperelliptic curve there is a meromorphic function of degree 2. By the Riemann– Hurwitz formula, such a function on a curve of genus g has 2g + 2 simple critical values. The space of critical values is acted upon by the group of linear fractional transformations of the image, whose dimension is equal to 3. Functions corresponding to each other under such a transformation are defined on the same hyperelliptic curve. Hence the space of hyperelliptic curves of genus g has dimension 2g − 1. For g = 2, the dimension 2g − 1 = 3 of the space of hyperelliptic curves coincides with the dimension 3g − 3 = 3 of the space of all curves, which agrees with the result proved earlier that every curve of genus 2 is hyperelliptic. For g = 3, the dimension of the space of all curves is 3g −3 = 6, while the hyperelliptic curves constitute only a subvariety of dimension 2g − 1 = 5, i.e., a hypersurface, in this space. Exercise 9.9 Find the dimension of the space of plane curves of degree 4 with one ordinary double point. Exercise 9.10 Show that every curve of genus 2 can be embedded into the projective plane as a curve of degree 4 with one ordinary double point. Exercise 9.11 Show, by counting dimensions, that not every complex curve of genus 10 can be realized as a plane curve. Exercise 9.12 Estimate the degree of plane curves (with double points) needed to realize every curve of a given genus g.

Chapter 10

Proof of the Riemann–Roch Formula

In the first section of this chapter, we give a proof of the Riemann–Roch formula l(D) − l(K − D) = d − g + 1. In the second section, we present a geometric interpretation of the quantities occurring in the Riemann–Roch formula in terms of canonical curves.

10.1 Proof When comparing the numbers l(D) and i(D) = l(K − D), it is useful to know that with every divisor D one can associate a divisor E such that l(D) = i(E) and l(E) = i(D). This correspondence is provided by the Brill–Noether duality. Theorem 10.1 (Brill–Noether Duality) Let ω be an arbitrary nonzero holomorphic 1-form on C and (ω) = D + E. Then L(D) ∼ = Ω 1 (E) and L(E) ∼ = Ω 1 (D). Proof By definition, L(D) is the space of meromorphic functions f such that (f ) ≥ −D. Hence (f ω) = (f ) + (ω) ≥ −D + D + E = E. Thus the formula f → f ω defines a mapping L(D) → Ω 1 (E). It is an isomorphism of vector spaces; the inverse mapping is given by the formula η → ωη . The isomorphism L(E) ∼ = Ω 1 (D) follows from the fact that D and E play the same role. By the Riemann–Roch theorem, the difference l(D) − (d − g + 1) = i(D) is nonnegative, i.e., l(D) ≥ d − g + 1. It is this inequality that was obtained by Riemann, and later Roch gave an interpretation of the difference l(D) − (d − g + 1). We begin the proof of the Riemann–Roch theorem by proving Riemann’s inequality. Theorem 10.2 (Riemann’s Inequality) For a divisor D of degree d on a smooth curve C of genus g, the inequality l(D) ≥ d − g + 1 holds. © Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_10

125

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10 Proof of the Riemann–Roch Formula

Proof We may assume that C is a normalization of a plane curve C  whose all singularities are double points. Let C  be given by an equation F = 0 of degree m and have δ double points. Further, let {pi } be the double points C  and {pi1 }, {pi2 } of δ be the corresponding points of C. Consider the divisor Δ = i=1 (pi1 + pi2 ) on C. Changing coordinates, we may assume that neither points of Δ, nor points of D correspond to points at infinity. Let us write the divisor D in the form D = D  − D  where D  and D  are nonnegative divisors; let d  and d  be the degrees of these divisors. Let S n be the space of homogeneous polynomials of degree n in three variables; the dimension of this vector space is equal to (n+1)(n+2) . Let us try to choose 2 a polynomial G ∈ S n so that the following two conditions are satisfied: 1) the polynomial F does not divide G; 2) G·C ≥ D  +Δ, where G·C is the divisor on C corresponding to the intersection points of the curves G = 0 and C  . Let us show that the second condition means that the coefficients of G satisfy d  + δ linear equations. Let D =

δ 

(ni1 pi1 + ni2 pi2 ) +



mj q j ,

i=1

where ni1 , ni2 , mj are positive numbers. Then D + Δ =

δ 

((ni1 + 1)pi1 + (ni2 + 1)pi2 ) +



mj q j .

i=1

For every i, the inequality G · C ≥ (ni1 + 1)pi1 is equivalent to (ni1 + 1) linear equations on the coefficients of G. Indeed, choose an affine coordinate system with the origin at pi1 . Then, in a small neighborhood of the origin, the projection of the branch containing pi1 is given by the equation y = a1 x + a2 x 2 + . . .. Under the above change of coordinates, the polynomial G(x, y, 1) goes to a polyno˜ ˜ mial G(x, y). Substituting into G(x, y) the expression y = a1 x+a2x 2 +. . . obtained above, we get a polynomial b0 + b1 x + b2 x 2 + . . . whose coefficients can be linearly expressed in terms of the coefficients of G. The inequality G · C ≥ (ni1 + 1)pi1 is equivalent to b0 = b1 = · · · = bni1 = 0. Analogously, the inequality G · C ≥ (ni2 + 1)pi2 is equivalent to c0 = c1 = · · · = cni2 = 0, where c0 , c1 , . . . can be linearly expressed in terms of the coefficients of G and can be obtained by the procedure described above. Now observe that the equations b0 = 0 and c0 = 0 are equivalent. Indeed, each of them is equivalent to the equation G(pi ) = 0. Thus the inequality G · C ≥ (ni1 + 1)pi1 + (ni2 + 1)pi2

10.1 Proof

127

is equivalent to ni1 + ni2 + 1 linear equations. To qj there corresponds a nonsingular point of C, hence the inequality G · C ≥ mj qj is equivalent to mj linear equations on the coefficients of G. In total, we obtain δ δ     (ni1 + ni2 + 1) + mj = (ni1 + ni2 ) + mj + δ = d  + δ i=1

i=1

linear equations on the coefficients of G. Thus the dimension of the space of polynomials for which the second condition is satisfied is at least 12 (n + 1)(n + 2) − d  − δ. On the other hand, since the degree of F is m, the polynomials divisible by F constitute a space S n−m of dimension 1 2 (n − m + 1)(n − m + 2). Hence if n is sufficiently large, then the dimension of the space of polynomials for which the second condition is satisfied is greater than the dimension of the space of polynomials for which the first condition is not satisfied. Fix a polynomial G satisfying both conditions and consider the divisor E = G · C − D  − Δ. By Bézout’s theorem, deg(G · C) = mn, whence deg E = mn − d  − 2δ. By the choice of G, the divisor E is effective. Let us find the dimension of the space S = {H ∈ S n | H · C ≥ Δ + E + D  }. As shown above, the condition H · C ≥ Δ + E + D  is equivalent to a system of δ +deg E +deg D  = δ +(mn−d  −2δ)+d  = mn−δ −d  +d  = mn−δ −d linear equations. Hence dim S ≥

1 (n + 1)(n + 2) + δ + d − mn. 2

With each polynomial H ∈ S we associate a function fH on the curve C as follows: the value of this function at a point x ∈ C is   to the value taken by  equal  H /G at the corresponding point of C  , i.e., fH = π ∗ H G C  , where π : C → C is the normalization mapping. Then (fH ) + D = (H · C) − (G · C) + D ≥ (Δ + E + D  ) − (E + D  + Δ) + D = 0. Thus f ∈ L(D), and we obtain a mapping α : S → L(D), which is given by the formula α(H ) = fH . It is clear that α is a linear mapping of vector spaces. It is also clear that fH = 0 if and only if H vanishes on C  , i.e., H = F · Q, where Q is a polynomial of degree n − m. Therefore, Ker α = F S n−m , and we obtain an

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10 Proof of the Riemann–Roch Formula

injective linear mapping α¯ : S/F S n−m → L(D). Thus l(D) ≥ dim S − dim S n−m 1 1 (n + 1)(n + 2) + δ + d − mn − (n − m + 1)(n − m + 2) 2 2 1 = d − (m2 − 3m) + δ = d − g + 1. 2 ≥

In the last equality, we have used the formula for the genus of a curve with singular points (see Sect. 4.2). Corollary 10.1 If d ≥ g, then l(D) = 0. Using the Brill–Noether duality (Theorem 10.1), one can deduce from Riemann’s inequality the following result. Theorem 10.3 For a divisor D of degree d on a smooth curve C of genus g, the inequality i(D) ≥ g − d − 1 holds. Proof If ω is an arbitrary holomorphic 1-form on C, then, by the Brill–Noether duality, i(d) = l((ω) − D) ≥ deg((ω) − D) − g + 1 = 2g − 2 − d − g + 1 = g − d − 1. Corollary 10.2 If d ≤ g − 2, then i(D) = 0. Now we deduce the Riemann–Roch theorem from Riemann’s inequality. Let D be an effective divisor, D ≥ 0. Lemma 10.1 For an effective divisor D, l(D) − l(K − D) ≤ d − g + 1.  Proof We argue as in Sect. 9.1. Let D = di pi , di > 0. The number l(D) is the dimension of the space of meromorphic functions having a pole of order at most di at each of the points pi ∈ C and no other poles. For each of the points pi , consider the space of principal parts of functions at this point (see Sect. 9.1). It has dimension di , and the sum of the dimensions of all these spaces is equal to d. The residues of these principal parts paired with an arbitrary holomorphic 1-form sum to zero. Since the space of holomorphic 1-forms on the curve C is g-dimensional, we have g linear conditions imposed on principal parts at the points of the divisor D. However, among them there are only g − l(K − D) linearly independent conditions, since l(K − D) is the dimension of the space of holomorphic 1-forms on C having at each pi a zero of multiplicity at least di . Taking into account the fact that the space of functions L(D) has a one-dimensional subspace of constants, we obtain the desired inequality.

10.2 Divisors on the Canonical Curve

129

As we know from Sect. 9.1, the Riemann–Roch theorem implies that the nonstrict inequality we have proved is in fact an equality. Now let us prove that for an arbitrary divisor D, l(D) − l(K − D) ≤ d − g + 1.

(10.1)

This suffices to prove the Riemann–Roch theorem, since the reverse inequality follows from (10.1) by replacing D with K − D. Indeed, deg(K − D) = 2g − 2 − d, and hence (10.1) means that l(K − D) − l(D) ≤ (2g − 2 − d) − g + 1 = g − d − 1, i.e., l(D) − l(K − D) ≥ d − g + 1. First let l(D) = 0. In this case, it suffices to prove that l(K − D) ≤ g − d − 1 = (2g − 2 − d) − g + 1 = deg(K − D) − g + 1. But this is exactly Riemann’s inequality for the divisor K − D (Theorem 10.3). If l(D) > 0, then there exists a function f such that (f ) + D ≥ 0. The divisor (f ) + D is linearly equivalent to D. It is positive, so we may apply the lemma, and the statement of the Riemann–Roch theorem does not change if we replace D with a linearly equivalent divisor. Thus the proof of the Riemann–Roch theorem is completed.

10.2 Divisors on the Canonical Curve In this section we discuss a geometric interpretation of the numbers i(D) and l(D). As above, let C be a smooth curve of genus g ≥ 3, and assume that its canonical map to CPg−1 is an embedding (i.e., C is not hyperelliptic). Fix a basis ω1 , . . . , ωg in the space of holomorphic 1-forms on C. Consider an effective divisor D = x1 + · · · + xd of degree d on C with pairwise distinct points x1 , . . . , xd , and let yj = (ω1 (xj ) : · · · : ωg (xj )) be the image of xj on the canonical curve in CPg−1 . The condition  (ω) ≥ D means, in particular, that the 1-form ω is holomorphic, hence ω = λk ωk for some constants λ1 , . . . , λg , and we can associate with ω the  hyperplane Πω in CPg−1 given by the equation λk ωk = 0. Then the condition  (ω) ≥ D is equivalent to the fact that λk ωk (xj ) = 0, i.e., the hyperplane Πω passes through the points y1 , . . . , yd . Thus i(D) − 1 is the dimension of the space of hyperplanes in CPg−1 passing through the points y1 , . . . , yd of the canonical curve. Hence, denoting by s the dimension of the space spanned by the points y1 , . . . , yd , we see that i(D) − 1 = (g − 1) − r − 1, i.e., i(D) = g − 1 − s. Substituting this

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10 Proof of the Riemann–Roch Formula

expression into the Riemann–Roch relation l(D) = d − g + 1 + i(D), we obtain that l(D) = d − s, i.e., l(D) − 1 = d − s − 1. The number l(D) − 1 is equal to the dimension r of the complete linear system |D|, whence s = d − r − 1. Thus the Riemann–Roch theorem for a divisor D is equivalent to the following statement. Theorem 10.4 The dimension r of the complete linear system |D| is equal to the total number of linearly independent relations between the points y1 , . . . , yd of the canonical curve; in other words, the points y1 , . . . , yd span a (d − r − 1)-plane in CPg−1 . This theorem can be directly extended to the case where the multiplicity of points comprising an effective divisor D can be greater than one. If a point xi occurs in D with multiplicity ai , then a hyperplane passing through yi must intersect the canonical curve in this point with multiplicity at least ai . For instance, if ai = 2, then the plane simply touches the curve; if ai = 3, then it touches the curve with multiplicity at least 3, etc. The dimension of the space of such hyperplanes is exactly i(D) − 1 for a general effective divisor on a nonhyperelliptic curve. For a generic collection of points yi on the canonical curve C, the dimension of the corresponding space of hyperplanes is the minimum possible. As the points xi of the divisor move along C, additional linear dependencies can arise between their images (counting with multiplicities) under the canonical map yi , which causes the dimension of the corresponding space of hyperplanes to increase.

Chapter 11

Weierstrass Points

On curves of genus g ≥ 2, points differ from each other. For example, every nontrivial automorphism of such a curve has finitely many fixed points, and the set of points that are fixed by some nontrivial automorphism is also finite. In this chapter, we discuss another method of singling out points with special properties. This method, due to Weierstrass, also singles out a finite set of special points on every curve. This set is closely related to the set of fixed points of nontrivial automorphisms, but does not in general coincide with it. In particular, Weierstrass points on a curve do exist even if it has no nontrivial automorphisms. As we have seen, on every hyperelliptic curve there is a finite set of points that can be poles of order 2 of functions having no other poles. The definition of Weierstrass points relies upon a generalization of this observation.

11.1 Definition of Weierstrass Points As we will see below, the automorphism group of a curve of genus g ≥ 2 is finite, hence, in contrast to the case of smaller genus, two randomly chosen points of such a curve are, in general, not equivalent: no automorphism of the curve sends the first point to the second one. Let us discuss what numerical characteristics can be assigned to a point x of a complex curve C. Given a positive integer k, consider the dimension l(k · x) of the space of meromorphic functions with a pole of order at most k at x and no other poles. This dimension is related to the dimension i(k · x) of the space of holomorphic 1-forms with a zero of order at least k at x by the Riemann–Roch formula l(k · x) = k − g + i(k · x) + 1.

© Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_11

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For k ≥ 2g − 1, the value i(k · x) vanishes, since a nonzero holomorphic 1form cannot have zeros of order greater than 2g − 2. Hence for such k we have l(k · x) = k − g + 1 for every point x. Thus, points of the curve do not differ by the value l(k · x) for k ≥ 2g − 1. However, the situation is drastically different if we consider the function l(k · x) for k ≤ 2g − 2. As we know, l(x) = 1 for every point x of a curve C of positive genus. On the other hand, on a curve of genus g ≥ 2 there is a point x with l(2 · x) = 2 if and only if the curve is hyperelliptic. Exercise 11.1 Show that on a hyperelliptic curve of genus g ≥ 2 there are at least 2g + 2 points x with l(2 · x) = 2. Exercise 11.2 Show that on a hyperelliptic curve of genus g ≥ 2 points x with l(2 · x) = 2 are fixed points of the hyperelliptic involution. With every point x of a complex curve C we associate the nondecreasing sequence of positive integers l(x), l(2 · x), l(3 · x), . . . , l(k · x), . . .. For g > 0, this sequence begins with 1; for k ≥ 2g − 1, it coincides with the linear sequence k − g + 1. The value l(k·x) either coincides with l((k−1)·x), or is greater than l((k − 1) · x) by 1. The second case occurs if there is a meromorphic function on C with a pole of order exactly k at x and no other poles. Since the first element of the sequence is 1 and the (2g − 1)th element is g, among the first 2g − 1 terms of the sequence there are exactly g terms without jumps. Consider the sequence a1 , . . . , ag of these g numbers, 1 = a1 < a2 < . . . < ag ≤ 2g − 1. They are called the gaps of x. Example 11.1 Let us find the gap sequence for a point x on a hyperelliptic curve C such that l(2 · x) = 2. Since on C there is a meromorphic function with a pole of order 2 at x, there are also functions with poles of order 4, 6, 8, . . .: it suffices to take the square, cube, fourth power, etc. of the function with a pole of order 2. Thus all even values of k are not gaps. This means that the values 1, 3, 5, . . . , 2g − 1 are the gaps: otherwise, there would be less than g gaps. Hence the whole set of values that are not gaps is the union of the set of even numbers from 2 to 2g − 2 and the set of all positive integers starting from 2g. Exercise 11.3 Show that the set of non-gaps at every point x (i.e., the set of positive integers k for which there exists a meromorphic function with a pole of order exactly k at x) is a semigroup under addition: along with any two elements k1 , k2 , it contains their sum k1 + k2 . Hurwitz’s problem of describing all semigroups that can arise as semigroups of non-gaps of Weierstrass points is not yet completely solved. Exercise 11.4 Use the result of the previous exercise to deduce Clifford’s theorem: if a curve C is not hyperelliptic, then l(kx) < k2 + 1 for every point x ∈ C and k = 1, . . . , 2g − 1. For a generic point of a curve C, we have ai = i for i = 1, . . . , g, i.e., jumps start to appear from the (g + 1)th element. If this is not the case, i.e., l(k · x) = 2

11.3 Weights of Weierstrass Points

133

for some k ≤ g, then x is called a Weierstrass point. An equivalent definition is as follows: x is a Weierstrass point if l(g · x) ≥ 2, i.e., there exists a function on C with a unique pole at x of order at most g. A point at which the gap sequence has the form 1, 2, 3, . . . , g − 1, g + 1 is called a normal Weierstrass point. Below we will show that Weierstrass points exist on every curve of genus g ≥ 2.

11.2 Weierstrass Points on Curves of Genus 3 and Inflection Points of Plane Quartics Let C be a smooth plane quartic, i.e., a curve of degree 4 in the projective plane. As we know, the genus of a quartic is 3 · 2/2 = 3. By the Riemann–Roch theorem, for every point x of C there is a function with a pole of order 4 at x. Now let A ∈ C be an ordinary inflection point of C (see Sect. 3.3). The tangent to C at A meets C transversally in another point B. Hence the projection of C from the point B is a ramified covering of degree 3 of the projective line, and one of the points of the image (namely, the point corresponding to the line AB) has a unique preimage. Choosing a coordinate in the projective line formed by the lines passing through B so that the line AB has coordinate ∞, we obtain a meromorphic function of degree 3 on C with a pole of order 3 at A. This means that A is a Weierstrass point of C. A generic smooth quartic has 3 · 4 · 2 = 24 ordinary inflection points, and each of them is a Weierstrass point. Inflection points of a nongeneric quartic are not necessarily ordinary. Smooth plane quartics are canonical curves of genus 3. Weierstrass points of curves of higher genus are generalizations of inflection points of canonical curves. Exercise 11.5 Find the gap sequence for a simple inflection point of a plane quartic. Exercise 11.6 Find the inflection points of the Klein quartic x 3 y + y 3 z + z3 x = 0 and describe the action of the automorphism group of this curve on the set of inflection points.

11.3 Weights of Weierstrass Points Given a curve C, fix a basis of holomorphic 1-forms on C and a coordinate z in a neighborhood of a point x. In this coordinate, the basis 1-forms can be written as ϕ1 (z)dz, . . . , ϕg (z)dz. We define the weight of the point x as the order of vanishing

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11 Weierstrass Points

at x of the function   ϕ (z) ϕ  (z) . . . ϕ (g−1)(z)  1  1 1   W (z) =  ........... ,   (g−1) ϕg (z) ϕg (z) . . . ϕg (z) the determinant of the Wronski matrix. This function is called the Wronskian of the system of functions ϕ1 (z), . . . , ϕg (z). Since every basis in the space of holomorphic differentials on a curve can be linearly expressed in terms of any other basis, and this expression extends to derivatives of arbitrary order of the functions ϕi (z), the order of vanishing of the Wronskian at any point does not depend on the choice of a basis in this space. Proposition 11.1 A point x of a curve C of genus g ≥ 2 is a Weierstrass point if and only if its weight is positive, i.e., the Wronskian of a basis of holomorphic 1-forms vanishes at x. Proof For convenience, we modify the chosen basis in the space of holomorphic 1-forms as follows (elements of the new basis will be denoted by the same letters). As the first element of the new basis, we take a holomorphic 1-form ω1 that does not vanish at x (as we know, all holomorphic 1-forms cannot vanish simultaneously, hence such a 1-form does exist). The space Ω 1 (C) of holomorphic 1-forms on C decomposes into the direct sum of the set of 1-forms proportional to ω1 and the (g − 1)-dimensional subspace of 1-forms vanishing at x. Now let b2 be the smallest order of vanishing at x of 1-forms from this subspace. Choose a 1-form ω2 with the order of vanishing exactly b2 . Represent the space Ω 1 (C) as the direct sum of the plane spanned by the 1-forms ω1 and ω2 and the (g − 2)-dimensional subspace of holomorphic 1-forms with the order of vanishing strictly greater than b2 . Take ω3 to be an arbitrary 1-form from this subspace with the smallest possible order of vanishing b3 . All subsequent orders of vanishing bi and 1-forms ωi are defined in a similar way. Thus we obtain a sequence of positive integers bg > bg−1 > . . . > b2 > b1 = 0 and a basis ω1 , . . . , ωg in the space of holomorphic 1-forms such that the order of vanishing of ωi at x is bi , i = 1, 2, . . . , g. In this basis, the Wronski matrix at x is upper triangular, since bi ≥ i − 1 for all i. Moreover, this is a matrix in a row echelon form, with the height of every nonzero step equal to 1. But the Wronskian is not zero if and only if bi = i − 1 for all i = 2, . . . , g. Indeed, only in this case all diagonal elements of the upper triangular Wronski matrix are nonzero. On the other hand, there exists a meromorphic function on C with a unique pole of order at most g at the point x if and only if for some nonzero principal part f (z) =

c−g c−g+1 c−1 + g−1 + . . . + g z z z

the condition Resz=0 f ωi = 0 holds for all i = 1, 2, . . . , g. If bg = g − 1, i.e., bi = i − 1 for all i, then Resz=0 f ωg = 0 is a linear condition on the coefficient c−g ,

11.3 Weights of Weierstrass Points

135

which implies that c−g is zero. Then the condition Resz=0 f ωg−1 = 0 implies that c−g+1 is also zero. Arguing inductively, we arrive at the conclusion that all coefficients of the principal part f are zero, and hence the principal part itself is zero. If bg > g − 1, then the equation Resz=0 f ωg = 0 is automatically satisfied, and the system of equations Resz=0 f ωi = 0, i = 1, . . . , g − 1, is a system of g − 1 homogeneous linear equations on g unknown coefficients c−1 , . . . , c−g ; thus it has a nontrivial solution. The vanishing of the Wronskian at a point x ∈ C means that the canonical curve becomes “infinitesimally flat” at x, much in the same way as it happens at inflection points of a plane quartic. Corollary 11.1 The weights of all Weierstrass points on a curve of genus g sum to (g − 1)g(g + 1). Proof The order of vanishing of the determinant in the definition of the weight of a Weierstrass point does not depend on the choice of a local coordinate in a neighborhood of this point. Hence, in order to find the total weight of all Weierstrass points, it suffices to compute the sum of the orders of all zeros of the Wronskian. Every 1-form ω is a section of the cotangent bundle. Its derivative is nothing else than a section of the tensor square of this bundle. If in a local coordinate z a 1-form ω is written as ω = ϕ(z)dz, then its derivative is ϕ  (z)(dz)2 . Analogously, the ith derivative of ω is ϕ (i) (z)(dz)i+1 . Thus, the zeros of the Wronskian are the zeros of a section of the tensor product of the first, second, . . . , gth tensor powers of the cotangent bundle. The degree of the divisor of zeros of such a section is equal to (2g − 2) + 2 · (2g − 2) + 3 · (2g − 2) + . . . + g · (2g − 2) = (2g − 2)(1 + 2 + . . . + g) = (2g − 2)

g(g + 1) = (g − 1)g(g + 1), 2

as required. Exercise 11.7 Show that on every curve there are finitely many Weierstrass points. Example 11.2 On a generic plane quartic, we have found 24 Weierstrass points. On the other hand, the total weight of all Weierstrass points on a curve of genus 3 is equal to 2 · 3 · 4 = 24. Hence the weight of each inflection point is 1, and a quartic has no other Weierstrass points. In the proof of Proposition 11.1, with each point x of a curve we have associated an increasing sequence of nonnegative integers 0 = b1 < b2 < . . . < bg , which are the smallest possible orders of vanishing at x of the basis 1-forms. This sequence and the gap sequence 1 = a1 < a2 < . . . < ag uniquely determine each other. In particular, if x is not a Weierstrass point, then the sequence of bi is 0, 1, . . . , g −

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11 Weierstrass Points

1, and the gap sequence is 1, 2, . . . , g. In the general case, this relation looks as follows. Proposition 11.2 The gap sequence a1 , . . . , ag and the sequence b1 , . . . , bg of orders of vanishing of the basis 1-forms are related by the following formulas: ag = g +

g  (bi − i + 1); i=1

ag−1 = (g − 1) +

g−1  (bi − i + 1); i=1

................................ a2 = 2 + (b2 − 1 + 1); a1 = 1. Proof Follows by duality: if bi+1 − bi > 1, then the condition Resz=0 f ω = 0 imposes no additional restrictions on the coefficients of the principal part f , and the dimension of the space of solutions increases by 1. For instance, a2 = 2 if b2 = 1, i.e., if there exists a holomorphic 1-form vanishing at x whose derivative at x does not vanish, and a2 = 3 if b2 = 2. The subsequent values of gaps can be found in a similar way. Corollary 11.2 The weight of a Weierstrass point x is equal to

g 

(ai − i), where

i=1

a1 , . . . , ag is the gap sequence of x. In particular, the weight of a normal Weierstrass point, i.e., a point with the sequence of gaps 1, 2, . . . , g − 1, g + 1, is equal to (1 − 1) + (2 − 2) + . . . + ((g − 1) − (g − 1)) + ((g + 1) − g) = 1. Thus, if all Weierstrass points on a curve of genus g are normal, then there are (g − 1)g(g + 1) of them. Exercise 11.8 Find the weight of all Weierstrass points x on a hyperelliptic curve for which l(2 · x) = 2. Use the obtained result to show that hyperelliptic curves have no other Weierstrass points. Show that these points have the greatest possible weight for curves of a given genus. Exercise 11.9 Check that the Fermat curve x 4 + y 4 = 1 has 12 points of “double” inflection, each being a Weierstrass point of weight 2. Exercise 11.10 Show that on a nonhyperelliptic curve of genus g ≥ 2 there are at least 2g + 6 Weierstrass points.

11.4 Weierstrass Points and the Finiteness of the Automorphism Group

137

The above discussion allows us to draw some conclusions about points of a curve that can be mapped to each other by its automorphisms: • no automorphism of a curve maps a non-Weierstrass point to a Weierstrass point and vice versa; • if an automorphism maps a Weierstrass point to another Weierstrass point, then these points have the same sequence of gaps. Of course, the fact that two Weierstrass points have the same gap sequence does not necessarily imply that they can be mapped to each other by an automorphism of the curve. Exercise 11.11 Give an example of a curve and two Weierstrass points on this curve that have the same gap sequence but cannot be mapped to each other by an automorphism of the curve.

11.4 Weierstrass Points and the Finiteness of the Automorphism Group Every automorphism of a curve of genus g ≥ 2 must preserve the set of Weierstrass points. Using this fact, we can now prove that the group of automorphisms of a curve is finite. For this, it suffices to show that the group of automorphisms that leave all Weierstrass points fixed is finite. Lemma 11.1 A nontrivial automorphism of a curve of genus g ≥ 2 has at most 2g + 2 fixed points. Proof Let η be an automorphism with s fixed points. Let D be a divisor consisting of g + 1 distinct points none of which is a fixed point of η. Applying the Riemann– Roch theorem l(D) − l(K − D) = (g + 1) − g + 1 = 2 to this divisor shows that there exists a nonconstant function f with simple poles at (possibly, not all) points of D. The function f − f ◦ η is not constant and has at most 2g + 2 simple poles and at least s zeros (every fixed point of η is a zero of this function). Hence s ≤ 2(g + 1). Corollary 11.3 The group of automorphisms of a curve of genus g ≥ 2 is finite. Proof The weight of a Weierstrass point x for which l(2 · x) = 2 is equal to g(g−1) 2 , and this is the greatest possible weight of a Weierstrass point. Hence on a curve C of genus g ≥ 2 there are at least 2(g + 1) Weierstrass points, and this value is attained only in the case of a hyperelliptic curve. If C is not hyperelliptic, then the number of Weierstrass points on C is greater than 2(g + 1), thus the group of automorphisms

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11 Weierstrass Points

preserving these points is trivial, and hence the group of all automorphisms of C is finite. If C is hyperelliptic, then the only nontrivial automorphism of C preserving all Weierstrass points is the hyperelliptic involution. Indeed, let f : C → CP1 be the hyperelliptic covering and η : C → C be an automorphism of C preserving the Weierstrass points, i.e., the ramification points of f . Then the function f ◦ η : C → CP1 has the same critical values as f , and hence coincides either with f , or with the image of f under the hyperelliptic involution. The following exercise shows that Weierstrass points and fixed points of automorphisms are closely related in the general case too. Exercise 11.12 Prove Schoeneberg’s lemma: if the number of fixed points of a nontrivial automorphism of a curve of genus g ≥ 2 is greater than 4, then these fixed points are Weierstrass points.

Chapter 12

Abel’s Theorem

The line bundles over a given complex curve are in a one-to-one correspondence with the linear equivalence classes of divisors on this curve. Such a class has an integer-valued characteristic, the degree. Since divisors consist of points of the curve, it is natural to expect that the set of classes of divisors of the same degree is endowed with additional structures. It must be a topological space and, moreover, a complex variety. Abel’s theorem identifies the space of classes of divisors of zero degree on a curve of genus g with a g-dimensional complex torus, the Jacobian of the curve. In particular, Abel’s theorem answers the question: Which divisors D can be represented in the form D = (f ) where f is a meromorphic function? Such divisors, obviously, satisfy the condition deg D = 0, since the orders of all zeros and poles of every meromorphic function sum to zero. The space of classes of divisors of nonzero degree has the same form as the space of classes of divisors of degree 0; however, the isomorphism of this space with the Jacobian is not canonical. Besides providing a description of the space of line bundles, the geometry of the Jacobian contains a great deal of other information on the structure of the curve.

12.1 Jacobian Let γ be a path on a curve C. The integral along this path determines a linear functional γ : Ω 1 (C) → C on the space of holomorphic 1-forms on C, given by the formula γ : ω → ω. By Stokes’ theorem, this functional is determined γ

by the homotopy equivalence class of γ among all paths with given endpoints. If γ is closed, then this functional is determined by the corresponding element [γ ] ∈ H1 (C, Z) of the first homology group (if two paths represent the same

© Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_12

139

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12 Abel’s Theorem

homology class, then the corresponding functionals coincide even if the paths are not homotopically equivalent). For a curve C of genus g, all functionals corresponding to closed paths form a discrete subgroup in the g-dimensional vector space (Ω 1 (C))∨ dual to the space of holomorphic 1-forms on C. Moreover, as we know (see Lemma 7.2), this subgroup is a full-dimensional lattice, its rank is equal to 2g. For generators of this lattice, one can take the collection of functionals constructed from an arbitrary basis in H1 (C, Z). The obtained lattice is called the period lattice of the curve C. The quotient of the space (Ω 1 (C))∨ by the lattice of periods is a g-dimensional complex torus. It is called the Jacobian of the curve C and denoted by J (C). Example 12.1 Jacobians of elliptic curves are one-dimensional, i.e., are themselves elliptic curves. Moreover, the Jacobian of an elliptic curve C is isomorphic to C. Indeed, let C be the quotient of the complex line C with coordinate z by the lattice spanned by 1 and a vector τ . Choose a basis in H1 (C, Z) consisting of the parallel and meridian of the torus, i.e., of the classes of curves obtained as the quotients of the intervals [0, 1] and [0, τ ] in C, respectively. The first curve determines a linear functional on Ω 1 (C) whose value at the basis holomorphic 1-form ω = dz is equal to 1 ω = 1; 0

the second curve determines a functional with τ ω = τ, 0

and this implies that J (C) = C. This example already shows that the Jacobians of different curves of the same genus are, in general, not biholomorphic to each other. Remark 12.1 In contrast to the original elliptic curve, the Jacobian always contains a distinguished point, namely, the identically zero functional, i.e., the result of projecting the defining lattice of the Jacobian. The zero functional determines a distinguished point on the Jacobian of an arbitrary curve. Let q ∈ C be a fixed point and ω be a holomorphic form on C. For every p point p ∈ C, consider the integral ω along an arbitrary path from q to p. If one q

chooses another path, the integral changes by an integer linear combination of the integrals of ω along the basis paths, i.e., by the value at ω of an element of the lattice.

12.1 Jacobian

141

Thus, with every pair of points q, p on C we have associated a unique point

p

of

q

the Jacobian J (C). Hence, for every point q we have the well-defined Abel–Jacobi map uq : C → J (C) given by the formula p uq : p →

, q

which can be extended to a map uq : Div(C) → J (C) on the space of divisors: uq : D =

k 

ni pi →

i=1

k  i=1

pi ni

. q

Clearly, this is a group homomorphism. In particular, for zero degree divisors k k   D= pi − qi ∈ Div0 (C) we obtain i=1

i=1 k  

pi

u(D) =

i=1 q

k  

qi

−

i=1 q

k  

pi

=

.

i=1 qi

This mapping no longer depends on the choice of an initial point q ∈ C. Abel’s theorem says that a divisor D ∈ Div0 (C) can be represented in the form D = (f ) where f is a meromorphic function if and only if u(D) = 0. Moreover, the following more precise result holds. Theorem 12.1 (Abel–Jacobi) Let L∗ (C) be the group of (not identically zero) meromorphic functions on C under multiplication, and let () be the mapping that sends a function f to its divisor (f ). Then the sequence of homomorphisms ()

u

L∗ (C) − → Div0 (C) − → J (C) → 0 is exact, i.e., Ker u = Im(), and the mapping u is surjective. The Abel–Jacobi theorem can be reformulated as follows. The quotient group Div0 (C)/ Im() = Pic(C) is called the Picard variety of C. The Abel–Jacobi theorem says that the mapping Pic(C) → J (C) induced by the Abel–Jacobi map u is an isomorphism. The claim that the mapping u : Div0 → J (C) is surjective is often called Jacobi’s inversion theorem .

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12 Abel’s Theorem

12.2 Proof of the Necessity Part Let us prove that the condition u(D) = 0 is necessary for a divisor D to be representable in the form D = (f ). Let f be a meromorphic function of degree d on C. For t ∈ CP1 = C ∪ {∞}, consider the effective divisor Dt = f −1 (t) ∈ Div(C). Clearly, D = (f ) = f −1 (0) − f −1 (∞) = D0 − D∞ , hence in order to prove that u(D) = 0, it suffices to show that u(Dt ) = const. Let t0 ∈ CP1 be a point over which f has no ramification points, i.e., a point such that the preimage f −1 (t0 ) = {p1 (t0 ), . . . , pd (t0 )} consists of exactly d points. Then, over a sufficiently small neighborhood U (t0 ) of t0 , the mapping f is an (unramified) covering, and the preimage f −1 (U (t0 )) of U (t0 ) is a disjoint union of d neighborhoods biholomorphic to U (t0 ). Denote them by U1 , . . . , Ud , and denote the preimage of a point t ∈ U (t0 ) lying in Uj by pj (t), j = 1, . . . , d. Set Dt = f −1 (t) = p1 (t) + . . . + pd (t). Then for every holomorphic 1-form ω and a chosen point q, p1 (t )

uq (Dt )(ω) =

pd (t )

ω +...+ q

ω. q

We want to show that for some, and hence any, point q, the value uq (Dt )(ω) does not depend on t for every 1-form ω. Let us find the derivative of each summand with respect to t at the point t0 . Since the restriction of f to each neighborhood Uj is a biholomorphic mapping, t is a local coordinate in each of these neighborhoods. Let ω = ϕ(t)dt in this coordinate, where ϕ(t) is a holomorphic function. Then d dt

pj (t )

 ω

t =t0

d = dt

q

t

 ϕ(t)dt t =t = ϕ(t0 ). 0

q

The result of this calculation can be written as d dt

pj (t )

q

 ω 1 Respj (t0 ) ωt =t = . 0 2πi f − t0

Indeed, we have verified the last equality in the local coordinate t in Uj ; hence, since the right-hand side is invariant, it also holds in any other coordinate. Besides, since the choice of a point t0 in the neighborhood U (t0 ) is arbitrary, we conclude that for every t, d dt

pj (t )

ω= q

ω 1 Respj (t ) . 2πi f −t

12.3 Proof of the Sufficiency Part: Beginning

143

Thus, d d uq (Dt )(ω) = dt dt

p1 (t )

d ω + ...+ dt

q

=

pn (t )

ω q

1 1 ω ω Resp1 (t ) +...+ Respn (t ) = 0, 2πi f −t 2πi f −t

since the residues of the meromorphic 1-form

ω f −t

on a compact curve sum to zero.

12.3 Proof of the Sufficiency Part: Beginning We start proving that the condition u(D) = 0 is sufficient for a divisor D to be representable in the form D = (f ). Assume that u(D) = 0 for a divisor D ∈ Div0 (C). We must find a meromorphic function f such that D = (f ). Let us make some preliminary remarks. Let f : C → CP1 be a meromorphic function. Consider its logarithmic 1 df differential, the meromorphic 1-form αf = 2πi f on C. It has the following properties: (1) the divisor of poles (αf )∞ consists of the zeros and poles of f , all of them being simple poles of αf ; nj (2) at every pole pj , the  residue of αf is equal to Respj αf = 2πi where nj is a nonzero integer and nj = 0; j

(3) the integral of αf along every closed path on C is an integer. Theorem 12.2 Let α be a meromorphic 1-form satisfying the above properties. Fix a point q ∈ C and set 

p  fq (p) = exp 2πi α , q

where the integral is taken along any path from q to p that does not pass through k  nj pj = D. the poles of α. Then fq is a meromorphic function on C with (fq ) = j =1

Note that if a 1-form α has the third property, then it necessarily has the second one: the residue of a 1-form at a pole is proportional to its integral along a small closed path centered at this pole.

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12 Abel’s Theorem

Proof Let us check that fq (p) does not depend on the choice of a path from q to p. p Let I and I  be two integrals α along two different paths from q to p. The third q

property guarantees that I − I  is an integer; therefore, exp(2πiI ) = exp(2πiI  ), and hence the function fq (p) is meromorphic. The fact that the 1-forms α and αfq have the same residues at all poles follows directly from the definition of fq .

12.4 Abelian Differentials of the First, Second, and Third Kind Meromorphic (in particular, holomorphic) differential 1-forms on a curve C are also called Abelian differentials. More exactly, a holomorphic 1-form is called an Abelian differential of the first kind; a meromorphic 1-form whose all residues are zero is called an Abelian differential of the second kind (an Abelian differential of the first kind is also an Abelian differential of the second kind), and an arbitrary meromorphic 1-form is called an Abelian differential of the third kind. By an elementary differential of the third kind we mean a meromorphic 1-form α 1 1 with two simple poles p and q such that Resp α = 2πi , Resq α = − 2πi . Theorem 12.3 For any two distinct points p and q of a curve C there exists an elementary differential of the third kind α with (α)∞ = p + q. Proof It suffices to prove that there exists a meromorphic 1-form α with (α)∞ = p + q. Indeed, in this case p is a simple pole of α, hence Resp α = 0. Besides, by the residue theorem, Resp α + Resq α = 0. Therefore, multiplying α by an appropriate constant, we obtain a desired 1-form. If p and q are distinct points of a curve C of genus g, the Riemann–Roch theorem applied to the divisor D = −(p + q) of degree −2 yields l(D) − i(D) = −2 + 1 − g. Since l(D) = 0 (there are no holomorphic functions that have zeros), we conclude that i(D) = g + 1. This means that the space of meromorphic 1-forms with poles of order at most one at p and q and no other poles has dimension g + 1, which is greater than the dimension of the space of holomorphic 1-forms. In particular, there exists a 1-form with poles of order exactly one at each of the points p and q. Example 12.2 In the case g = 0, one can easily produce such a 1-form. If p and q have coordinates 0 and ∞, respectively, then the desired 1-form is dz/z. In order to move the poles to another pair of points, it suffices to apply an appropriate fractional linear transformation.

12.5 Riemann’s Bilinear Relations

145

Exercise 12.1 Given a pair of points p, q on an elliptic curve, express a meromorphic 1-form with simple poles at p, q in terms of a holomorphic 1-form and the Weierstrass function of the curve. We edge towards the end of the proof of the sufficiency part of the theorem. Let k k   pj − qj ∈ Div0 (C), where pj (as well as qj ) are not necessarily D = j =1

j =1

1 distinct. Construct 1-forms α1 , . . . , αk such that (αj )∞ = pj + qj , Respj αj = 2πi , 1 and Resqj αj = − 2πi . Set α = α1 + . . . + αk . Then α is a differential of the third kind satisfying the first two required properties. For every holomorphic 1-form ω, the differential α + ω also satisfies the first two properties. Let us try to choose ω so that it satisfies the third property too. For this, we need Riemann’s bilinear relations, which will be discussed in a separate section.

12.5 Riemann’s Bilinear Relations Let C be a Riemann surface of genus g. Choose closed curves γ1 , . . . , γ2g on C whose homology classes generate the homology group H1 (C, Z) with intersection indices (γj , γj +g ) = −(γj +g , γj ) = 1

for j = 1, . . . , g

and all other intersection indices zero. For a holomorphic 1-form ω, consider the  periods πj (ω) = ω, j = 1, . . . , 2g. γj

Theorem 12.4 (Riemann’s Bilinear Relations) 1-forms ω and α, g 

For any nonzero holomorphic

(πj (ω)πj +g (α) − πj +g (ω)πj (α)) = 0,

j =1

i

g 

(πj (ω)πj +g (α) − πj +g (ω)πj (α)) > 0,

j =1

where the bar denotes complex conjugation. Proof The Riemann surface corresponding to the curve C can be glued out of a 4g-gon P so that the images of its sides under the gluing coincide with the curves γ1 , . . . , γ2g . Each of the curves γj is glued out of two sides of P , one having the same orientation along the boundary of P as γj (we denote it by cj ) and the other one having the opposite orientation (we denote it by cj−1 ). Fix a point q in P . Since

146

12 Abel’s Theorem

the polygon P is simply connected, the formula vq (p) =

p

ω defines a single-valued

q

function on P .   vq α = d(vq α) = 0. Now let us compute the By Stokes’ theorem, ∂P P  integral vq α in another way. Divide the sides of the polygon P into quadruples ∂P

corresponding to pairs of curves γj and γj +g for j = 1, . . . , g: g  

 vq α =

j =1

∂P

Further,

 vq α +

cj



vq α +

vq α =

cj−1

cj

 vq α .

 vq α +

cj−1

cg+j





 vq α +

−1 cg+j

((vq (p) − vq (p ))α,

γj

where p, p are the images of the endpoints of cj and cj−1 under the gluing. Here p



vq (p) − vq (p ) =

ω= p



q ω− p



q ω+

ω=−

γg+j

p

ω = −πg+j (ω).

γg+j

Thus, 

 vq α +

 vq α = −πg+j (ω)

cj−1

cj

α = −πg+j (ω)πj (α). γj

In a similar way, 

 vq α +

cj+1

vq α = πj (ω)πg+j (α),

−1 cj+1

and we get the desired equality. To prove the desired inequality, we should consider the differential form ivq ω. ¯ Since the form ω is holomorphic, d ω¯ = 0. Therefore, d(ivq ω) ¯ = iω ∧ ω¯ and 

 vq ω¯ = i

i ∂P

 d(vq ω) ¯ =i

P

The rest of the proof goes along the same lines.

ω ∧ ω¯ > 0. P

12.5 Riemann’s Bilinear Relations

147

Remark 12.2 Riemann’s bilinear relations can be rewritten as follows. Let Ig be the   0 Ig g × g identity matrix, Q = , and Π be the period matrix, that is, the −Ig 0 g × 2g-matrix with the entries γj ωi . Then ΠQΠ T = 0, and the matrix iΠQΠ T is Hermitian and positive definite. We have proved Riemann’s bilinear relations for a distinguished basis γ1 , . . . , γ2g of the one-dimensional homology group. Let us prove them for an  , i.e., a basis such that the intersection index of arbitrary canonical basis γ1 , . . . , γ2g   the curves γi and γi+g is equal to 1 for i = 1, . . . , g and the intersection indices of the other pairs of curves are equal to 0. Let Γ be the column consisting of the  . elements γ1 , . . . , γ2g , and Γ  be the column consisting of the elements γ1 , . . . , γ2g Then Γ  = AΓ for some square matrix A with integer entries. Since the bases are canonical, Γ Γ T = Q and Γ  (Γ  )T = Q; therefore, SΓ Γ T S T = Q and SQS T = Q. A matrix S satisfying the relation SQS T = Q is said to be symplectic. It is easy to check that the transpose of a symplectic matrix is itself symplectic. If Π is the period matrix for the basis γ1 , . . . , γ2g and Π  is the period matrix for  , then Π  = ΠS T . Hence the basis γ1 , . . . , γ2g Π  Q(Π  )T = ΠS T QSΠ T = ΠQΠ T = 0 and iΠ  QΠ T = iΠS T QS Π T = iΠS T QSΠ T = iΠQΠ T . The period matrix Π can be simplified by choosing an appropriate basis in the space of holomorphic forms Ω 1 (C). Let us write it in the form Π = (A, B) where A and B are square g × g matrices. In terms of these matrices, Riemann’s bilinear relations mean the following: 1) AB T = BAT ; 2) the matrix i(AB¯ T − B A¯ T ) is positive definite. The second property implies, in particular, that the matrix A is nondegenerate. Hence we may use the matrix (AT )−1 to transform the basis of the space Ω 1 (C). Then the transformed period matrix has the form Π  = (Ig , Z), where Ig is the g × g identity matrix. The matrix Z is called the normalized period matrix. In terms of the normalized period matrix, Riemann’s bilinear relations mean the following: (1) Z = Z T ; (2) the real matrix Im Z consisting of the imaginary parts of the elements of Z is positive definite.

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12 Abel’s Theorem

12.6 Completing the Proof of the Sufficiency Part Let D =

k  j =1

pj −

k  j =1

qj ∈ Div0 (C), where pj (and qj ) are not necessarily

distinct. Construct elementary differentials of the third kind α1 , . . . , αk such that 1 1 αj has simple poles at pj and qj with Respj αj = 2πi and Resqj αj = − 2πi . Set α = α1 + . . . + αk . Choose curves γ1 , . . . , γ2g representing a basis of the onedimensional homology group of C; we may assume that none of them passes through the poles of the 1-forms α1 , . . . , αk . Choose a basis ω1 , . . . , ωg of holomorphic 1-forms so that the period matrix has the form (Ig , Z). Let α = α −

g   s=1

 α ωs .

γs

The 1-form α  has the same poles and residues as α. Besides, for j = 1, . . . , g we obtain 

α =

γj

 α−

g   s=1

γj

γs



 ωs =

α γj

α− γj

g   s=1

   α πj (ωs ) = α − α = 0

γs

γj

γj

(we have used the fact that πj (ωs ) = δj s ). We will assume that the Riemann surface C is glued out of a polygon P in a standard way. Fix a point b ∈ P and, given a holomorphic 1-form ω, p set vb (p) = ω. Thus we have defined a holomorphic function on P . The b

meromorphic differential vb α  has the same poles as α. By the residue theorem, 2πi

 k  (Respj (vb α  ) − Resqj (vb α  )) = vb α  . j =1

∂P

Clearly,

2πi

k 

k k      (Respj (vb α ) − Resqj (vb α )) = (vb (pj ) − v(qj )) = ω. pj

j =1

j =1

j =1 qj

In exactly the same way as in the proof of Theorem 12.4, we get  ∂P

vb α  =

g  j =1

(πj (ω)πg+j (α  ) − πg+j (ω)πj (α  )) =

g  j =1

πj (ω)πg+j (α  ),

12.6 Completing the Proof of the Sufficiency Part

149

since πj (α  ) = 0. Setting ω = ωs for s = 1, . . . , g, we obtain k  

pj

g 

ω=

j =1 qj

πj (ωs )πg+j (α  ) = πg+s (α  ),

j =1

because πj (ωs ) = δj s . So far, we have not used the fact that D ∈ Ker(u : Div0 (C) → J (C)). Now we will assume that D has this property. This means that the sum of the integrals along the segments [qj , pj ] is an element of the period lattice, i.e., an integer linear combination of the integrals along the basis cycles: k  

pj

j =1 qj

=

2g 

 =

ml

l=1

   g   ml +mg+l , l=1

γl

γl

γg+l

where ml are integers. In particular, for every basis 1-form ωs we have k  

pj

ωs = ms +

j =1 qj

 mg+l

l=1

The property Z = Z T means that



ωs . γg+l



ωs =

γg+l k  

g 

ωl , hence

γg+s

pj

ωs = ms +

j =1 qj

g 

 mg+l

l=1

ωl . γg+s

Thus 

πg+s (α ) = ms +

g  l=1

 mg+l

ωl . γg+s

Let us show that the 1-form 



α =α −

g  l=1

mg+l ωl

150

12 Abel’s Theorem

has all the required properties. Indeed, since πs (ωl ) = δls and



α  = 0, we obtain

γs







α = γs



α −

g 

 ωl = −mg+s ∈ Z.

mg+l

l=1

γs

γs

And since 

πg+s (α ) = ms +

g 

 mg+l

l=1

ωl , γg+s

we obtain 





α = πg+s (α ) − γg+s

g  l=1

 ωl = ms ∈ Z.

mg+l γg+s

12.7 Proof of Jacobi’s Inversion Theorem We will prove a slightly more general result than the one stated at p. 141, see Theorem 12.8. Let us make some preliminary remarks. Given a smooth curve C, the symmetric power C (d) , d = 1, 2, . . ., of C is the set of effective divisors D = p1 + . . . + pd of degree d (repeated points are allowed among p1 , . . . , pd ). It is the same thing as the set of unordered collections {p1 , . . . , pd } of points of C. Theorem 12.5 The symmetric power C (d) has a structure of a complex variety of dimension d. Proof Consider the direct product C d = C × . . . × C; clearly, it has a structure of a complex variety. The topological space C (d) is the quotient of C d by the action of the group Sd of permutations of the set {1, 2, . . . , d} given by the formula ρ(p1 , . . . , pd ) = (pρ(1) , . . . , pρ(d) ),

ρ ∈ Sd .

Therefore, C (d) is compact and Hausdorff. Now we introduce local coordinates on C (d). Let D = k1 p1 + . . . + kl pl be an effective divisor of degree d; here p1 , . . . , pl are pairwise distinct points. In a neighborhood of pj choose a local coordinate zj . Then the collection (σ11 , . . . , σk1 1 , . . . , σ1l , . . . , σkl l ),

12.7 Proof of Jacobi’s Inversion Theorem (kj )

(1)

where σrj (zj , . . . , zj

151

) is the rth elementary symmetric function of the kj vari-

(k ) zj(1) , . . . , zj j

ables in a neighborhood of (pj , . . . , pj ), defines local holomorphic coordinates in a neighborhood of D ∈ C (d) . To prove this, it suffices to check that the mapping (z1 , . . . , zk ) → (σ1 , . . . , σk ), where σi = σi (z1 , . . . , zk ), defines local coordinates in a neighborhood of zero in C(k) . In fact, this mapping determines a homeomorphism C(k) → Ck . Indeed, the mapping is one-to-one, since it can be regarded as sending the collection of roots of a polynomial to the collection of its coefficients. To prove that the inverse mapping is continuous, it is convenient to extend the mapping C(k) → Ck to a mapping (CP1 )(k) → CPk . The latter is, obviously, continuous, and it is a one-to-one mapping of a compact space onto a Hausdorff space, which is always a homeomorphism. Let D ∈ Div(C) and deg D = d. Again consider the space L(D) of meromorphic functions f such that (f ) + D ≥ 0. The dimension of this space is l(D). With each function f ∈ L(D) we can associate the effective divisor E = (f ) + D of degree d, which is a point of C (d) . Functions f and λf where λ is a nonzero number give rise to the same divisor, hence we obtain a mapping αD : CPl(d)−1 → C (d) ,

[f ] → E = (f ) + D,

where [f ] is the equivalence class of f ∈ L(D) = Cl(d) in the projective space. Let us check that the mapping αD is monomorphic. Let αD [f ] = αD [g]. Then (f ) = (g), whence (f/g) = 0. Therefore, f/g is a nonzero constant, i.e., [f ] = [g]. Thus the projectivization of the space L(D) can be regarded as a subset in C (d). In fact, this subset is a subvariety. To check this, let us prove that the k  mj pj where pj are pairwise distinct mapping αD is holomorphic. Let D = j =1

points. Let l(d) = n + 1; fix a basis f0 , f1 , . . . , fn of the space L(D). Every n  λj fj = fλ . The collection (λ0 : function from L(D) can be written in the form j =0

. . . : λn ) is the homogeneous coordinates of the point [fλ ] in the projectivization of the space L(D). The mapping αD is given by the formula λ → Eλ = (fλ ) + D. Let us fix a point λ˜ = (λ˜ 0 : . . . : λ˜ n ) and check that the mapping αD is holomorphic near it. Let the divisor (fλ˜ ) contain, apart from the points p1 , . . . , pk , points pk+1 , . . . , pk+l . Choose pairwise disjoint neighborhoods of p1 , . . . , pk+l , and let W be the union of these neighborhoods. Construct a meromorphic function g on W such that g is identically equal to 1 in the neighborhoods of pk+1 , . . . , pk+l and the divisor (g) coincides with D in the neighborhoods of p1 , . . . , pk . Then for λ close to λ˜ , we have Eλ = (fλ g). We may assume that λ˜ 0 = 0. Then for points λ close to λ˜ , we obtain (λ0 : λ1 : . . . : λn ) = (1 : μ1 : . . . : μn ), where μj = λj /λ0 . Set f(μ) = f0 + μ1 f1 + . . . + μn fn . The product f(μ) g can be regarded as a function of μ and a

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˜  ˜ point of W . In the vicinity of μ˜ = λ˜ 1 : . . . : λ˜ n and p1 , . . . , pk+l , the function λ0 λ0 f(μ) g is holomorphic, since in the domain under consideration we have Eλ ≥ 0. Now the following lemma implies that the mapping αD is holomorphic. Lemma 12.1 Let μ = (μ1 , . . . , μn ) ∈ Cn and z ∈ C. Assume that a function h(μ, z) is holomorphic at (0, 0) and the function h(0, z) has a zero of order k at z = 0. Then one can choose ρ > 0 and ε > 0 so that the following properties are satisfied: 1) for every μ0 ∈ Uρ = {μ ∈ Cn | |μj | < ρ, j = 1, . . . , n}, the function h(μ0 , z) has exactly k roots z1 (μ0 ), . . . , zn (μ0 ) in the domain |z| < ε; 2) on the set Uρ , all k elementary symmetric functions in z1 (μ), . . . , zn (μ) are holomorphic functions of μ. Proof The function h(0, z) has an isolated zero at z = 0, hence we may choose ε > 0 such that |h(0, z)| ≥ δ for |z| = ε with some δ > 0. Then we may choose ρ > 0 such that |h(μ, z)| ≥ δ/2 for |z| = ε and μ ∈ Uρ .  (z) r = z−a + . . ., If f (z) = c0 (z − a)r + c1 (z − a)r+1 + . . . with c0 = 0, then ff (z) hence  1 f  (z) =r zm 2πi f (z) C

for every positively oriented contour C that contains a and does not contain any other zeros or poles of f . Thus, the integral 

1 2πi

f  (z) f (z)

C

for a holomorphic function f is equal to the number of zeros of f inside C. One can also easily see that the integral 1 2πi

 zm

f  (z) f (z)

C

is equal to z1m + . . . + zkm , where z1 , . . . , zk are the zeros of f lying inside C. Let  dz ∂h(μ, z) 1 · , zm sm (μ) = 2πi ∂z h(μ, z) |z|=ε

where m = 0, 1, . . . , k. The function sm is holomorphic on Uρ . The function s0 (μ) is equal to the number of zeros of the function h(μ, z) in the ball |z| < ε for a

12.7 Proof of Jacobi’s Inversion Theorem

153

fixed μ. It is continuous on Uρ and takes only integer values, hence σ0 (μ) = k for all μ ∈ Uρ . The first claim of the lemma is proved. To prove the second claim, it suffices to express the elementary symmetric functions of z1 , . . . , zk in terms of the power sums z1m + . . . + zkm . Denote by |D| the image of the projectivization of the space L(D) under the mapping αD . The restriction to C (d) of the Abel–Jacobi map u : Div(C) → J (C) is still denoted by u. In what follows, we assume that a point q ∈ C is fixed. A divisor D = p1 +. . .+pd where p1 , . . . , pd are pairwise distinct points is said to be generic. The mapping u : C (d) → J (C) is holomorphic in a neighborhood of a generic divisor D and locally bounded; hence, by the Riemann’s theorem on removable singularities, it is holomorphic on the whole variety C (d) . Theorem 12.6 Every fiber of the mapping u : C (d) → J (C) is a projective space, namely, u−1 (u(D)) = |D| for every divisor D ∈ C (d) . Proof Let us first check that u−1 (u(D)) ⊂ |D|. Let E ∈ C (d) be a divisor such that u(E) = u(D). Then u(E − D) = 0, hence, by Abel’s theorem, there exists a meromorphic function f with E − D = (f ), i.e., E = (f ) + D. Since E ∈ C (d) , it follows, in particular, that E ≥ 0, whence f ∈ L(D). Therefore, E = αD ([f ]) ∈ |D|. Now let us check that u−1 (u(D)) ⊃ |D|. Assume that E ∈ |D|, i.e., there exists a meromorphic function f such that E = (f ) + D. Then, by Abel’s theorem, u(E) = u((f )) + u(D) = 0 + u(D) = u(D), and hence E ∈ u−1 (u(D)). Let us investigate the structure of the differential of the mapping u : C (d) → J (C) for d ≤ g. Let D ∈ C (d) . Then the differential of u at the point D is a mapping of tangent spaces (u∗ )D : TD (C (d) ) → Tu(D) (J (C)). Let D = p1 + . . . + pd be a generic divisor and zj be a local coordinate in a neighborhood of pj . Then (z1 , . . . , zd ) are local coordinates in a neighborhood of D in C (d) . Let ω1 , . . . , ωg be a basis of the space of holomorphic 1-forms; in a neighborhood of pj , we have ωα = fαj (zj )dzj where fαj (zj ) is a holomorphic function. In a neighborhood of D, the Abel–Jacobi map can be written as u(z1 , . . . , zd ) = (u1 (z1 , . . . , zd ), . . . , ug (z1 , . . . , zd )) =

z  d j j =1 q

d  

zj

f1j (zj )dzj , . . . ,

j =1 q

 fgj (zj )dzj .

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Hence ⎛ ⎞ ∂u ⎞ . . . ∂z1g f11 (p1 ) . . . fg1 (p1 ) ⎜ ⎟ ⎠, (u∗ )D = ⎝ . . . . . ⎠ = ⎝ ......... ∂u ∂u1 g f1d (pd ) . . . fgd (pd ) ∂zd . . . ∂zd ⎛ ∂u

1 ∂z1

and the rank of the matrix (u∗ )D is equal to the rank of the so-called Brill–Noether ⎛ ⎞ ω1 (p1 ) . . . ωg (p1 ) ⎠. matrix ⎝ ........ ω1 (pd ) . . . ωg (pd ) Let C be a curve of genus g ≥ 2. Consider the canonical map ϕK : C → CPg−1 that sends a point p ∈ C to the point (ω1 (p) : . . . : ωg (p)) ∈ CPg−1 . Let D = p1 + . . . + pd be a generic divisor. The rank of the matrix (u∗ )D is equal to the rank of the Brill–Noether matrix, which, in turn, is equal to the dimension of the vector space spanned by ϕK (p1 ), . . . , ϕK (pd ). Hence, denoting by ϕK (D) the dimension of the projective space spanned by ϕK (p1 ), . . . , ϕK (pd ), we obtain rank(U∗ )D = dim ϕK (D) − 1. In algebraic geometry, the phrase “a generic object of a family parametrized by a complex variety has a given property” means that all objects that do not have this property can be parametrized by a subvariety of strictly smaller dimension. Theorem 12.7 Let C be a smooth curve of genus g ≥ 1. Then for a generic point D ∈ C (g) , we have rank(u∗ )D = g. Proof If g = 1, then (u∗ )p = ω(p) = 0 for every point p. In this case, rank u∗ = 1 for all points. Now let g ≥ 1. We must prove that divisors D = p1 + . . . + pg ∈ C (g) such that rank(u∗ )D ≤ g − 1 lie in a subvariety of C (g) of dimension less than dim C (g) = g. By assumption, dim ϕK (D) ≤ g − 2, i.e., there exists a (g − 2)-dimensional projective subspace H in CPg−1 containing all points ϕK (pj ). Thesubspace H cannot contain a component of the canonical curve ϕK (C). Indeed, if λs ωs (p) = 0 for all p, then the 1-forms ω1 , . . . , ωg would be linearly dependent. Hence the number of intersection points of the subspace H with the canonical curve ϕK (C) does not exceed the degree of ϕK (C), which is equal to 2g − 2 (the degree of the divisor of zeros of a holomorphic 1-form). Thus, the number of divisors D ∈ C (d) such that all points ϕK (pj ) lie in H does not exceed 2g−2. The variety of projective subspaces of dimension g − 2 in CPg−1 is isomorphic to CPg−1 , i.e., has dimension g − 1. To every such subspace there correspond at most 2g − 2 points D, hence all points D under consideration lie in a variety of dimension at most g − 1. Now we prove a strengthened version of Jacobi’s inversion theorem.

12.8 Theta Divisor and Theta Functions

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Theorem 12.8 The Abel–Jacobi map u : C (g) → J (C) is surjective; at a generic point, it is also injective. Proof We need the following property of holomorphic mappings between complex varieties of the same dimension. Lemma 12.2 Let X and Y be connected compact complex varieties of the same dimension. Assume that a holomorphic mapping f : X → Y has the following two properties: 1) the preimagef −1 (y) of every point y ∈ Y is closed; 2) the differential f∗ at a generic point is an isomorphism of tangent spaces. Then the mapping f is surjective; at a generic point, it is also injective. Proof The subset f (X) ⊂ Y is a subvariety. If the differential f∗ at a point x0 is an isomorphism, then f maps an open neighborhood of x0 onto an open neighborhood of f (x0 ). Hence f (X) is a subvariety in Y containing a subset open in Y . For a connected closed variety Y , this means that f (X) = Y . Thus f is surjective. By Sard’s theorem, for a generic point x ∈ X the differential f∗ is an isomorphism for all points x  ∈ f −1 (f (x)). Hence, by the inverse function theorem, for a generic point x ∈ X the set f −1 (f (x)) is discrete. But the set f −1 (f (x)) is connected for every point x ∈ X by assumption, hence it consists of a single point. It remains to verify that the Abel–Jacobi map has both properties stated in the lemma. But we know already that for every divisor D ∈ C (d) the set u−1 (u(D)) = |D| is the image of the projectivization of the space L(D) under the holomorphic mapping α. Besides, by Theorem 12.7, the differential of the Abel– Jacobi map at a generic point is an isomorphism. Jacobi’s inversion theorem can now be easily obtained from Theorem 12.8. Fix a point p ∈ C and denote by C (g) − g · p the set of divisors of the form D − g · p where D ∈ C (g) ; each of these divisors has degree 0. The restriction of the Abel– Jacobi map u to C (g) is surjective, hence the restriction of u to C (g) − g · p is also surjective. Therefore, the restriction of u to Div0 (C) is surjective.

12.8 Theta Divisor and Theta Functions The images of the symmetric powers C (d) of a curve C in the Jacobian J (C) under the Abel–Jacobi map uq for d = 1, . . . , g −1 are distinguished subvarieties in J (C) of dimension d. They are defined invariantly up to translation: changing the initial point q results in a translation of the image of the Abel–Jacobi map. In particular, there is a natural mapping from the curve C itself and its (g − 1)th symmetric power C (g−1) to the Jacobian. The image of the latter mapping is a multidimensional divisor, a subvariety of codimension 1 in a variety of dimension g. It is called the theta divisor.

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We have seen that, given a divisor on a curve (more exactly, a class of linearly equivalent divisors), one can construct a line bundle on this curve. In a similar way, given a divisor on a variety of large dimension, one can construct a line bundle over this variety. In particular, the theta divisor in J (C) determines a line bundle over J (C); it is called the theta bundle. Multivalued sections of the theta bundle are called theta functions; they can be regarded as functions on the space (Ω 1 (C))∨ , whose quotient is the torus J (C). Theta functions have numerous applications and are the subject of an extensive literature.

Chapter 13

Examples of Moduli Spaces

In Sect. 6.3, we have already discussed what does the moduli space of elliptic (i.e., genus 1) curves look like. This is a rather typical example, which allows one to observe many features common for all moduli spaces. In this chapter, we will study it in more detail and consider examples of other moduli spaces. From the very beginning, we deal with moduli spaces of curves with marked points: properties of such spaces can be described “in layman’s terms” already for curves of small genus, and these properties are highly nontrivial. As the genus grows, the geometry of the moduli space becomes complicated, and a complete description of this space is usually beyond reach. Nevertheless, one can compute many important geometric characteristics of moduli spaces of curves of high genus.

13.1 First Examples We begin constructing moduli spaces Mg;n of curves of genus g with n marked points. We assume that n ≥ 3 for g = 0 and n ≥ 1 for g = 1. With these restrictions, the automorphism group of a curve with marked points is finite. The reason for introducing them will be discussed below. The points are considered to be labelled, i.e., we take the equivalence relation determined by biholomorphisms of curves sending each marked point to the marked point with the same label. We usually denote the marked points by x1 , . . . , xn . According to Riemann’s calculation (see Sect. 9.5), the dimension of the space Mg = Mg;0 for g ≥ 2 is equal to 3g − 3. Adding each marked point increases the dimension of the moduli space by one, hence dim Mg;n = 3g − 3 + n. The last formula is valid for arbitrary values of g and n satisfying the above restrictions. The moduli space M0;3 has a very simple structure: as we know from Sect. 6.1, any two triples of points on the projective line CP1 can be mapped to each other by an automorphism of CP1 , i.e., a linear fractional transformation. Hence M0;3 © Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_13

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consists of a single point. This point can be imagined as the projective line with a fixed coordinate z, the marked points having the coordinates 0, 1, and ∞. The moduli space M0;4 can be naturally identified with the projective line CP1 with three punctures (by the above discussion, the choice of these punctures is irrelevant). Given fixed points x1 , x2 , x3 on CP1 , the fourth point x4 can occupy any position different from the positions of the first three points. It is the position of this fourth point that is the parameter on M0;4. Choosing an arbitrary coordinate on CP1 , if the points x1 , x2 , x3 , x4 have coordinates z1 , z2 , z3 , z4 , respectively, then the value of this parameter is given by the cross ratio (see Sect. 3.2) z3 − z1 z4 − z1 : . z3 − z2 z4 − z2 If the points x1 , x2 , x3 have the coordinates z1 = 0, z2 = 1, and z3 = ∞, then the coordinate z4 of the fourth marked point x4 cannot take the values 0, 1, and ∞, but does take all the other values. The case of elliptic (i.e., genus 1) curves with n = 1 is more complicated. We discussed it in Sect. 6.3. Such a curve is the quotient of the complex line by a two-dimensional lattice. The marked point coincides with the image of the lattice points under the quotient mapping. A lattice is determined by a pair of generators, complex vectors τ1 , τ2 such that the ratio τ1 /τ2 is not real. We assume that the orientation determined on CP1 by the ordering of these vectors coincides with the usual complex orientation. Different lattices may determine the same curve. For instance, the group C∗ of nonzero complex numbers under multiplication acts on the set of lattice generators as λ : (τ1 , τ2 ) → (λτ1 , λτ2 ),

λ ∈ C∗ .

This action determines a biholomorphism of the curves corresponding to these two lattices. Besides, the set of lattice generators is acted upon by the group SL2 (Z) (Klein modular group) consisting of linear fractional transformations with integer coefficients: z →

az + b , cz + d

a, b, c, d ∈ Z,

ad − bc = 1.

This action changes the basis of a lattice, preserving the lattice itself. Hence it preserves also the corresponding curve. Multiplying a basis (τ1 , τ2 ) by c = 1/τ1 , we obtain a basis of the form (1, τ ), where τ = τ2 /τ1 is a complex vector lying in the upper half-plane. Now, taking the quotient by the action of the Klein modular group allows one to identify the space of vectors τ with a modular curve, which is a fundamental domain of the action of

13.2 The Space M1;1

159

the group SL2 (Z) on the upper half-plane. As a fundamental domain we can take a strip bounded by the unit circle and two lines:   1 1 z ∈ C| − ≤ Re z ≤ , |z| ≥ 1, Im z > 0 . 2 2 But some points on the boundary of the fundamental domain should be identified, namely, the boundary points should be glued together pairwise according to the following rule: x + iy ≡ −x + iy. The resulting space is exactly the moduli space of elliptic curves with one marked point. It is not compact and has two singularities, which are points corresponding to elliptic curves with additional symmetries. These are the point τ = eπi/2 and the pair of points τ = eπi/3 and τ = e2πi/3 glued together.

13.2 The Space M1;1 The moduli space M1;1 is the quotient of the upper half-plane H by a discrete action of the group SL(2, Z). However, in order to study this space, it is more convenient to represent it as the quotient of another curve by an action of a finite group. Proposition 13.1 Consider the following action of the group Z2 × S3 on the projective line CP1 punctured at the three points {0, 1, ∞}: • the factor Z2 acts trivially; • the factor S3 acts by linear fractional transformations, performing all possible permutations of the triple of points {0, 1, ∞}. The space M1;1 is the quotient of the punctured projective line by this action. Proof Consider in SL(2, Z) the subgroup SL(2, Z)[2] of matrices congruent to the unit matrix modulo 2 (the 2-modular subgroup). It is normal in SL(2, Z), and the quotient SL(2, Z)/SL(2, Z)[2] ∼ = SL(2, Z2 ), which is the group of 2 × 2 matrices with determinant 1 over the field of two elements, is isomorphic to the permutation group S3 . The kernel of the homomorphism SL(2, Z) → SL(2, Z2 ) that sends an integer matrix to the matrix of residues modulo 2 is the group Z2 × PSL(2, Z)[2]. The action of the first factor on the upper half-plane is trivial, the action of the second factor is free, and the space H /PSL(2, Z)[2] is the projective line with three punctures, as required.

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Exercise 13.1 Show that the action of the group PSL(2, Z)[2] on the upper halfplane is indeed free. Find a fundamental domain of this action and verify the last claim of the proof. As we know, every elliptic curve can be represented as a ramified covering of CP1 of degree 2 with four ramification points. Moreover, a quadruple of points of CP1 , regarded up to a linear fractional transformation, uniquely determines the covering elliptic curve. The ramification points are not labelled, which allows us to represent the moduli space M1;1 as the quotient of the moduli space M0;4 by an action of a permutation group. The action of the group PSL(2, Z2 ) ∼ = S3 on CP1 \{0, 1, ∞} is generated by two linear fractional transformations z →

1 z

and z → 1 − z,

which interchange the points (0, ∞) and (0, 1), respectively. Almost all orbits of this action have length 6. The exceptions are two orbits, of length 3 and 2, respectively. The first one consists of the points {−1, 12 , 2}, and the second one, of the points {eπi/3 , e−πi/3 }. The orbit of length 3 corresponds to a square lattice, and the orbit of length 2, to a triangular lattice.

13.3 The Universal Curve Over M1;1 To every point τ of the upper half-plane H there corresponds an elliptic curve, which is the quotient of C by the integer lattice spanned by the vectors 1 and τ . On the direct product C × H we can define an action, fibered over H , of the group Z ⊕ Z by the formula (m, n) : (z, τ ) → (z + m + nτ, τ ). The quotient by this action will be denoted by CH and called the universal elliptic curve over H . There is a natural projection CH → H of the universal elliptic curve to the upper half-plane, the fiber over a point τ ∈ H being exactly the elliptic curve C/Z + τ Z. The action of the group Z2 = Z⊕Z on the fibers of the direct product C×H can be combined with the action of the group SL(2, Z) on the upper half-plane. Namely, consider the semidirect product G = SL2 (Z)  Z2 .

13.4 The Cohomology of the Space M1;1

161

The set of elements of this group coincides with SL(2, Z) × Z2 , and the multiplication is defined as follows: (γ1 , v1 )(γ2 , v2 ) = (γ1 γ2 , v1 γ2 + v2 ), where γi ∈ SL(2, Z), vi ∈ Z2 , i = 1, 2. Exercise 13.2 Show that the group G = SL2 (Z)  Z2 is isomorphic to the group of 3 × 3 matrices of the form ⎫ ⎧⎛ ⎞ ⎬ ⎨ a b 0   a b 2 ⎝ c d 0⎠ ∈ SL(2, Z), (m, n) ∈ Z  cd ⎭ ⎩ mn1 under matrix multiplication. Now we define an action of the group G on the direct product C × H by the formula ⎛ ⎞ a b0   ⎝ c d 0⎠ : (z, τ ) → mτ + n + z , aτ + b . cτ + d cτ + d mn1 In other words, ⎛ ⎞ ⎛ ⎞⎛ ⎞ τ a b0 τ 1 ⎝1 ⎠ → ⎝ c d 0⎠⎝ 1⎠. cτ + d z mn1 z With this definition, the action of SL(2, Z) on the upper half-plane is accompanied by a corresponding action on the space of lattices. The quotient C1;1 = C × H /Γ is endowed with a natural projection C1;1 → M1;1 = H /SL(2, Z) and is called the universal curve over the moduli space M1;1 . The fiber of this projection over a point of M1;1 is the quotient of the elliptic curve corresponding to this point by its automorphism group.

13.4 The Cohomology of the Space M1;1 We will not go into details of constructing the cohomology of orbifolds. The aim of this section is just to illustrate, by the example of the orbifold M1,1 , how the corresponding constructions can be carried out in principle. It can be omitted without loss of understanding. The orbifold M1;1 is the quotient of the projective line with three punctures by an action of a finite group Γ . Denote this punctured line by X and consider the

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direct product EΓ × X. Here EΓ is an arbitrary contractible topological space freely acted upon by Γ . Such a space always exists: one can take, for instance, an infinite-dimensional simplicial complex whose n-dimensional simplices are in a one-to-one correspondence with the points of the set Γ n+1 , n = 0, 1, 2, . . .. On such a simplicial complex, the group Γ acts diagonally. The quotient BΓ of the space EΓ by the action of the group Γ is called a classifying space of Γ , and the (co)homology of the space BΓ is called the (co)homology of Γ . Example 13.1 The group Z acts freely on the line R1 , which is a natural choice for EZ, by translations. In this case, the classifying space BZ is the circle S 1 . Example 13.2 As a classifying space for the group Z2 of residues modulo 2, one can take the infinite-dimensional projective space BZ2 = RP ∞ . It is natural to define the (co)homology of the quotient of a smooth variety X by an action of a group Γ as the (co)homology of the quotient of the direct product EΓ × X by the diagonal action of Γ . For the orbifold M1;1 , this approach yields the following results. Proposition 13.2 The orbifold M1;1 has the following homology and cohomology groups in small dimensions: H1 (M1;1 , Z) = Z12 ,

H 1 (M1;1 , Z) = 0,

H 2 (M1;1, Z) = Z12 .

Chapter 14

Approaches to Constructing Moduli Spaces

Constructing moduli spaces is a technically complicated task, involving the analysis of many subtleties. In this chapter, we will discuss, without going into details, one of the possible methods of carrying out such a construction. We will describe the general sequence of steps; numerous results justifying it are either stated without proofs or left as exercises.

14.1 Requirements on Moduli Spaces In the previous chapter, we gave examples of moduli spaces of curves. However, we still have no rigorous definition of what this term means. At the same time, it is clear that such a definition should cover the above examples. So, let us discuss what requirements on moduli spaces seem to be natural. 1. The points of a moduli space must be in a one-to-one correspondence with the biholomorphic equivalence classes of complex curves. Although natural, this requirement does not seem to be very strong, since we have not yet imposed any special conditions on the one-to-one correspondence in question. 2. A moduli space must be endowed with a topology and a complex structure. This requirement is not very strong either, since the above structures are not yet in any way related to points themselves, moduli of curves. At the same time, we are entitled to expect that complex curves given by close equations in the projective space determine close points of the moduli space. The formalization of this requirement is given in the next paragraph. 3. Assume for the moment that g ≥ 3, n = 0. We are entitled to expect that a moduli space of curves exists not by itself, but with a universal curve, i.e., along with a complex variety Cg;0 of dimension dim Cg;0 = dim Mg;0 + 1 and a mapping π : Cg;0 → Mg;0 such that the total preimage π −1 (c) of a generic point of Cg;0 is a complex curve C whose equivalence class coincides © Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_14

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with c ∈ Mg;0. Nongeneric points of the moduli space are curves having nontrivial automorphisms. Note that the absence of nontrivial automorphisms for a generic curve is guaranteed by the condition g ≥ 3 on the genus. The fiber of the universal curve over such a nongeneric point is the quotient of the curve itself by the action of its automorphism group. By the way, it is the existence of curves with nontrivial automorphisms that is an obstruction to the existence of a fine moduli space of curves, for which every fiber of the universal curve is the corresponding curve itself, rather than the quotient of this curve by its automorphism group. 4. A moduli space must be universal, i.e., contain all possible families of curves. Let E, B be two complex varieties and p : E → B be a holomorphic mapping such that the total preimage p−1 (b) of every point b ∈ B is the quotient of a complex curve of genus g by its automorphism group. Then we require the existence of holomorphic mappings E → Cg;0 and B → Mg;0 that together with p and π form a commutative square

5. As illustrated by the example of the moduli space of elliptic curves with one marked point, we cannot expect a moduli space to be a variety. Indeed, the modular curve has two special points which do not have neighborhoods isomorphic to the disk. However, this curve is an orbifold: every its point has a neighborhood isomorphic to the quotient of the disk by an action of a finite group. We postpone a discussion of orbifolds, observing only that the introduction of orbifolds is a necessary element of the construction of moduli spaces. Now we can give the definition. Definition 14.1 A coarse moduli space of curves of genus g is a triple (Cg;0 , Mg;0, π : Cg;0 → Mg;0 ) consisting of two complex orbifolds and a mapping from the first orbifold onto the second one such that 1) every fiber π −1 (b), b ∈ Mg;0, is the quotient of a smooth complex curve of genus g by its automorphism group; 2) every curve of genus g occurs exactly once as a fiber of π; 3) for every holomorphic family p : E → B whose fibers are quotients of smooth complex curves of genus g by their automorphism groups there exist holomorphic mappings E → Cg;0 and B → Mg;0, the first one being a fiberwise isomorphism, that together with π and p form a commutative square.

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In the case of marked curves, the definition must be slightly modified. Namely, the triple must be augmented by a fourth element, a collection of n sections σi : Mg;n → Cg;n corresponding to the marked points: the marked point xi in the fiber π −1 (b) is the value σi (b) of the section σi at the point b. Of course, different sections should be disjoint. Exercise 14.1 Give the definition of a coarse moduli space of curves with marked points. Exercise 14.2 Describe the universal rational curve a) with three marked points; b) with four marked points. c) Describe the universal elliptic curve with one marked point.

14.2 A Naive Attempt to Construct a Moduli Space One could try to construct a moduli space of curves of genus 3 as follows. As we know, every such curve C that is not hyperelliptic can be represented by a plane curve of degree 4 (the image of C under the canonical map ϕ : C → CP2 ). Conversely, every smooth plane curve of degree 4 is a curve of genus 3. Hence we can consider the space of all homogeneous quartic polynomials in three variables, projectivize it, and in the resulting space take the (open) subset of polynomials corresponding to nonsingular curves. The projectivized space of polynomials is acted upon by the group PGL(3, C) of projective transformations of CP2 . The quotient by this action is exactly the desired moduli space. The moduli space thus constructed can also be easily endowed with a universal curve. For this, consider the direct product of the space of quartic polynomials and the projective plane, and in the fiber over each polynomial take the curve determined by it. The result is a hypersurface in the direct product. The action of the group PGL(3, C) on the base can be extended to an action of this group on the whole direct product. Restricting it to the constructed hypersurface and taking the quotient of the hypersurface by the resulting action, we obtain a universal curve. One could try to proceed in a similar way in the case of curves of genus 2; this time, in the space of homogeneous polynomials of degree 4 one should take the subset of polynomials that define curves with one double point. The construction of a universal curve in this case is complicated by the need to resolve singularities of fibers. What difficulties are encountered in trying to follow this line? If we want to cover the case of hyperelliptic curves of genus 3 (as we know, the dimension of the space of such curves is one less than the dimension of the space of all curves and equals 5), then the suggested scheme no longer works. The reason is that the image of a hyperelliptic curve under the canonical map is a projective line, and hence is not isomorphic to the curve itself. Another universal difficulty is that we must take the quotient by an action of a large continuous group, the group of projective

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transformations of CP2 . The problem is that even if such a group acts on a good space, the quotient can have very bad singularities, much worse than those we expect from an orbifold (which is locally a quotient by a discrete group action). Proving that this construction cannot cause the appearance of nonalgebraic singularities is a difficult and crucially important problem. Besides, for g > 3, by no means every genus g curve can be realized as a smooth plane curve. Another difficulty is related to the problem of compactifying a moduli space. When constructing a moduli space, we mean to eventually construct its compactification, i.e., a compact space containing it as an open dense subset. Every noncompact space has many different compactifications. It is natural to look for a modular compactification, whose points are in a one-to-one correspondence with moduli of some curves. Since all smooth curves are already exhausted, these points must correspond to singular curves. When compactifying a space of plane curves, a natural compactification is again a space of plane curves of given degree. However, the simplest examples show that such a compactification cannot exist. Example 14.1 Consider the following family of plane elliptic curves depending on a parameter t: y 2 − x 3 − t = 0. For all nonzero values of t, the corresponding curves are pairwise biholomorphic (prove this!). Hence the limiting curve as t → 0 must be also biholomorphic to them. However, the curve y 2 = x 3 corresponding to t = 0 is rational and has a cusp-type singularity, which shows that one cannot construct a compactification in this way. In what follows, we compactify moduli spaces by moduli of curves that have only singularities of the simplest kind, ordinary self-intersection points. For this, we have to embed smooth curves not into the plane, and even not into the projective space of dimension g − 1, as in the case of the canonical map, but into a high-dimensional space, where a greater freedom makes it possible for curves to degenerate without acquiring complicated singularities. In this case, the spaces of curves we are interested in can no longer be described by spaces of homogeneous polynomials, since in the multidimensional case the set of zeros of a homogeneous polynomial is not a curve, but a hypersurface. Such spaces of curves are described by an object we have not yet encountered, a Hilbert scheme. To define a Hilbert scheme, we first need to introduce the notion of Hilbert polynomial.

14.3 Hilbert Polynomial

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14.3 Hilbert Polynomial Consider a finite collection X ⊂ CP1 consisting of |X| = d distinct points of the projective line. Let (x0 : x1 ) be a projective coordinate system in CP1 such that X does not contain the point (0 : 1). In the ring C[x0, x1 ] of polynomials in two variables, consider the ideal IX of polynomials vanishing on X. It decomposes into a direct sum of finite-dimensional subspaces of homogeneous polynomials: IX = IX(1) ⊕ IX(2) ⊕ IX(3) ⊕ . . . , where IX(n) stands for the subspace in IX of homogeneous polynomials of degree n, n = 1, 2, . . .. The dimensions of the spaces IX(n) are ! (n) dim IX

=

0

for n < d

n−d +1

for n ≥ d.

Indeed, a homogeneous polynomial p belongs to IX if and only if all points of X are roots of p. Hence the degree of such a polynomial cannot be less than d. Fix a homogeneous polynomial pX (x0 , x1 ) of degree d whose all roots are points of X (any two such polynomials differ by a nonzero multiplicative constant). Every polynomial from IX of degree n ≥ d can be obtained by multiplying pX (x0 , x1 ) by a homogeneous polynomial of degree n−d. The space of homogeneous polynomials of degree n − d has dimension n − d + 1, which implies the above formula. Now consider the sequence dim C(n) [x0, x1 ]/IX(n) of the dimensions of the quotients of the spaces of homogeneous polynomials of degree n in x0 , x1 by the subspaces of polynomials of the same degree lying in IX . For n = 1, 2, . . . , d − 1, the elements of this sequence are equal to n + 1; for n ≥ d, they are equal to (n + 1) − (n − d + 1) = d. Thus, (n)

dim C(n) [x0 , x1 ]/IX = d

for n ≥ d.

A similar statement holds also for a finite collection of points on the plane. Let X ⊂ CP2 be a collection of d pairwise distinct points, (x0 : x1 : x2 ) be projective coordinates in CP2 , and IX be the ideal in C[x0 , x1 , x2 ] of polynomials vanishing on X. In this case, the dimensions of the spaces IX(n) ⊂ C(n) [x0 , x1 , x2 ] of homogeneous polynomials of degree n already depend on the relative positions of the points of X. For instance, if all d points of X lie on the same line, then (1) (1) dim IX = 1; otherwise, dim IX = 0. Nevertheless, if n is sufficiently large, then dim C(n) [x0 , x1 , x2 ]/IX(n) = d.

(14.1)

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Indeed, for large values of n, the coset of IX in C(n) [x0 , x1 , x2 ] containing a polynomial p is uniquely determined by the collection of values of p at the points of X, which can be arbitrary. The value of n for which (14.1) starts to be valid depends on the particular relative positions of points. Exercise 14.3 Show that for n ≥ d equality (14.1) holds whatever the relative positions of the points of X in CP2 . Exercise 14.4 Let X be a set of d pairwise distinct points in the projective space of dimension N ≥ 2. Show that for n ≥ d the equality dim C(n) [x0 , x1 , . . . , xN ]/IX(n) = d holds whatever the relative positions of the points of X. Now let X be a line in CP2 . Then IX ⊂ C[x0, x1 , x2 ] is the ideal of polynomials in three variables vanishing on X. The dimensions of the homogeneous components of this ideal are   n(n + 1) n+1 (n) . dim IX = = 2 2 Indeed, choosing projective coordinates in which X is given by the equation x0 = 0, we see that IX is the ideal of polynomials divisible by x0 . The dimension of the space of homogeneous polynomials of degree n divisible by x0 is equal to the dimension of the space of homogeneous polynomials of degree n − 1 in three variables, exactly as stated. Thus, dim C(n) [x0 , x1 , x2 ]/IX(n)     (n + 1)(n + 2) n(n + 1) n+2 n+1 − = n + 1. = − = 2 2 2 2 If we take X to be a smooth conic in CP2 , then dim IX(n) =

  n(n − 1) n = 2 2

for n ≥ 2

(since IX(n) is the ideal of homogeneous polynomials of degree n divisible by the defining quadratic polynomial of the conic), and for the same values of n "

dim C(n) [x0 , x1 , x2 ]/IX(n)

# " # (n + 1)(n + 2) n(n − 1) n+2 n = − = − = 2n + 1. 2 2 2 2

14.3 Hilbert Polynomial

169

We have considered several simple examples of algebraic subvarieties in projective spaces. In all these examples, the value dim C(n) [x0, x1 , . . . , xN ]/IX(n) for large values of n turns out to be a polynomial in n of degree equal to the dimension of the subvariety: for a subvariety of dimension 0, the polynomial is constant, while for subvarieties of dimension 1, it is linear. Besides, the obtained polynomial depends not only on the variety X itself, but also on the embedding of X into the projective space: in CP2 , both a line and a smooth conic are rational curves, but the polynomials associated with them are different. It turns out that the properties listed above hold for every algebraic subvariety in a projective space. Theorem 14.1 (Hilbert) Let X be an algebraic subvariety in CPN , (x0 : . . . : xN ) be projective coordinates, and IX(n) ⊂ C(n) [x0 , x1 , . . . , xN ] be the subspace of polynomials vanishing on X in the space of homogeneous polynomials of degree n. Then, for sufficiently large values of n, (n)

dim C(n) [x0 , x1 , . . . , xN ]/IX

is a polynomial in n. The degree m = mX of this polynomial is equal to the d dimension of X, and the coefficient of nm is equal to m! , where d = dX is the degree of X. The polynomial hX (n) whose existence is stated in the theorem is called the Hilbert polynomial of X. We do not give a proof of this theorem, referring the reader, e.g., to [4]. Exercise 14.5 Find the Hilbert polynomials of the following subvarieties: a line in CP3 ; a smooth cubic in CP2 ; a plane in CP3 ; the union of a line and a point in CP2 ; the union of a line and a smooth conic in CP2 that intersect transversally; the union of a line and a smooth conic in CP2 that are mutually tangent; a smooth hypersurface of degree d in CPN (such a hypersurface is given by one irreducible homogeneous equation of degree d in N + 1 variables); • a twisted cubic in CP3 ; • a pair of intersecting lines in CP3 ; • a pair of skew (i.e., disjoint) lines in CP3 . • • • • • • •

Exercise 14.6 Let X ⊂ CP2 be a “double point” in the plane. By definition, IX ⊂ C[x0 , x1 , x2 ] is the ideal of polynomials with a zero of order at least 2 at this point. Find the Hilbert polynomial hX (n). Does it coincide with the Hilbert polynomial of a pair of distinct points in the plane?

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14.4 Hilbert Schemes A Hilbert scheme is a moduli space of subvarieties in a given projective space with a given Hilbert polynomial. The word “scheme” used in this section should be understood as “algebraic variety.” The reason for using the term “scheme” is that a moduli space of projective varieties with a given Hilbert polynomial may exist, but fail to be an algebraic variety, while the more general notion of scheme covers all possible situations. However, we do not need this more general construction for building moduli spaces of curves. Given a polynomial p(n), the Hilbert scheme of subvarieties in CPN with Hilbert polynomial p(n) is constructed as follows. Denote by QN (n) the dimension of the space of homogeneous polynomials of degree n in N + 1 variables. This is a polynomial of degree N in n:   (n + N)(n + N − 1) . . . (n + 1) N +n , QN (n) = = N N! the Hilbert polynomial of the whole projective space. For every subvariety X ⊂ CPN with Hilbert polynomial hX (n) = p(n) there exists n0 such that for n > n0 the dimension of the quotient C(n) [x0 , . . . , xN ]/IX(n) is equal to the value p(n) of this polynomial. For arbitrary n > n0 , the homogeneous component IX(n) determines a plane of dimension QN (n) − p(n) in the space C(n) [x0 , . . . , xN ] of dimension QN (n). One can show that any two different subvarieties in CPN with given Hilbert polynomial p(n) give rise to different planes. Thus we have associated with p(n) a subset in the Grassmannian Gr(QN (n), QN (n) − p(n)) of planes of dimension QN (n) − p(n) in the space of dimension QN (n). This subset is exactly the Hilbert p(n) scheme HilbN of subvarieties in CPN with Hilbert polynomial p(n). As in the case of other moduli spaces, a Hilbert scheme is endowed with a p(n) universal bundle. Namely, consider the direct product CPN × HilbN , and over p(n) each point of HilbN take the corresponding subvariety X in CPN . The union of all these subvarieties is exactly the desired universal bundle.

14.5 Pluricanonical Embeddings In order to place curves into high-dimensional spaces, we use pluricanonical embeddings. As we know (see Sect. 9.4), the canonical map of a curve of genus g ≥ 2 is an embedding if and only if the curve is not hyperelliptic. Theorem 14.2 Given a curve C of genus g ≥ 2, consider the third tensor power (T ∨ )⊗3 (C) of the canonical line bundle. Then the corresponding

14.6 The Quotient by the Action of the Group of Projective Transformations. . .

171

mapping ϕ3K from C to the projectivization of the space dual to the space of its holomorphic sections is an embedding. Recall that the mapping ϕL is constructed from a line bundle L as follows (see Sect. 8.5). Every point x of the curve C defines a linear functional on the space of sections of L up to a factor: the ratio of the values of this functional at two sections σ1 , σ2 : C → L is equal to σ1 (x) : σ2 (x). Hence every point x of C defines a point of the projectivization of the space dual to the space of holomorphic sections of L. Remark 14.1 In the statement of the theorem, the third tensor power of T ∨ (C) can be replaced by any larger power. Exercise 14.7 Find the dimension N of the projective space of holomorphic sections of the third tensor power of the canonical bundle on a curve of genus g. Exercise 14.8 Give an example of a curve of genus g = 2 for which the mapping ϕ2K corresponding to the tensor square of the cotangent bundle is not an embedding. Proof of the Theorem To prove the theorem, one can use the following result. Lemma 14.1 Let D be a divisor on a given curve C. The mapping ϕD : C → P (H 0 (L(D))∨ ) from this curve to the projectivization of the space dual to the space of holomorphic sections of the bundle L(D) is an embedding if and only if for any pair of distinct points x, y of C, l(D − x) = l(D) + 1,

l(D − x − y) = l(D) + 2.

We leave it to the reader to prove this lemma and to deduce the theorem from it. Exercise 14.9 Let C be a curve of genus g with pairwise distinct marked points x1 , . . . , xn (here n ≥ 3 if g = 0, and n ≥ 1 if g = 1). Denote by X the divisor on C consisting of these points taken with multiplicity one: X = x1 + . . . + xn . What tensor power k of the line bundle L(K + X) is sufficient for the mapping ϕk(K+X) to be an embedding for any curve of genus g with n marked points?

14.6 The Quotient by the Action of the Group of Projective Transformations and Stability We have prepared the ground for constructing moduli spaces Mg;0 for g ≥ 2. The p (n)

g . Here pg (n) = 3(2g − 2)n − construction is based on the Hilbert scheme Hilb5g−6 g + 1 is the Hilbert polynomial of a 3-canonical curve of genus g. At this stage, pg (n) however, we are interested not in the whole space Hilb5g−6 , but in the open subspace of Hilbert points of smooth curves. Note that every smooth curve in CP5g−6 with this Hilbert polynomial is a 3-canonical curve.

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Exercise 14.10 Check that the Hilbert polynomial of a 3-canonical curve is equal to pg (n) = 3(2g − 2)n − g + 1, and show that every smooth curve in CP5g−6 with this Hilbert polynomial is a 3-canonical curve of genus g. The space Mg;0 is the quotient of a subspace of the Hilbert scheme by the action of the group of projective transformations PGL(5g − 6). Usually, the quotient by an action of a large continuous group has bad nonalgebraic singularities. However, in the case we are interested in, that of the action of the group PGL(5g − 6) on an pg (n) open subset in Hilb5g−6 , this does not happen. Let us try to understand where such singularities come from and why they do not appear in our case. Assume that a Lie group G acts by linear transformations on a vector space V , and let M ⊂ V be a G-invariant subset in V . The quotient space M/G has nonalgebraic singularities in the case where two different orbits Gx and Gy of points x, y ∈ M have intersecting closures (in this case, the corresponding points of the quotient space cannot be separated). The closures of two different orbits have a nonempty intersection if and only if both closures contain the origin. A point x ∈ M ⊂ V is said to be stable if the closure Gx of its orbit Gx does not contain the origin. Example 14.2 Let V = C2 be the complex plane, and let the group C∗ of nonzero complex numbers under multiplication act on V by linear transformations. Every such action decomposes into a direct sum of one-dimensional actions. In turn, every one-dimensional action of C∗ has the form λ : t → λw t for some integer w called the weight of the action. Thus, with an action of the group C∗ on V we associate a pair of integers (w1 , w2 ) (which may coincide): in appropriate coordinates, the action has the form λ : (t1 , t2 ) → (λw1 t1 , λw2 t2 ). The stability of points of V depends on the signs of w1 and w2 . If both w1 and w2 are positive, then every point of V approaches the origin as λ → 0, and, consequently, the closures of the orbits of any two points have a nonempty intersection (see Fig. 14.1a). If both these numbers are negative, then every point of the plane approaches the origin as λ → ∞. On the contrary, if one of the numbers w1 , w2 is positive and the other one is negative, then the plane contains an open dense stability domain, which consists of the points that do not lie on the coordinate axes. The closure of the orbit of any point of this domain does not contain the origin, see Fig. 14.1c, and the quotient of this domain by the action of the group C∗ has no singularities. The closure of the orbit of a point that lies on an axis (where one of the coordinates becomes zero) contains the origin. The above example is general, and it allows one to state a stability criterion for an action of an arbitrary group on a vector space by linear transformations. Theorem 14.3 (Hilbert–Mumford Numerical Criterion) A point x ∈ V is stable with respect to an action of a group G on V by linear transformations if for every

14.6 The Quotient by the Action of the Group of Projective Transformations. . .

173

Fig. 14.1 Orbits of the action of the group C∗ on the plane given by the formula λ : (x, y) → (λw1 x, λw2 y) for (a) w1 , w2 > 0; (b) w1 , w2 < 0; (c) w1 > 0, w2 < 0

one-parameter subgroup of G, the expansion of x in terms of its eigendirections contains nonzero coordinates with both positive and negative weights. Now, the construction of a moduli space Mg = Mg;0 for g ≥ 2 looks as follows. p (n)

g 1. Construct the Hilbert scheme Hilb5g−6 of subvarieties in CP5g−6 with Hilbert polynomial pg (n) = (6g − 6)n − g + 1. It is embedded into the Grassmannian Gr(QN (n), QN (n) − pg (n)). 2. Embed the Grassmannian Gr(QN (n), QN (n) − pg (n)) into the projective space ∧pg (N) (C(N) [x0 , . . . , x5g−6 ]) via the Plücker embedding1 and consider the cone

1 The Plücker embedding sends a point of the Grassmannian to the exterior product of the elements of an arbitrary basis in the plane corresponding to this point regarded up to multiplication by a constant. The image of the Plücker embedding is given by quadratic equations (the Plücker relations).

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over the Grassmannian in the corresponding vector space of one dimension more. The action of the group of projective transformations PSL(5g − 6) on CP5g−6 can be lifted to an action of the group of linear transformations of the space embracing the cone. pg (n) 3. Using the Hilbert–Mumford numerical criterion, verify that points of Hilb5g−6 corresponding to the images of smooth curves of genus g under the 3-canonical map are stable with respect to the linear action of the above group. Therefore, the corresponding quotient is a smooth orbifold, and this is exactly the moduli space Mg;0. 4. Each of the previous steps is accompanied by the construction of the corresponding universal curve.

14.7 Moduli Spaces of Curves with Marked Points Moduli spaces Mg;n of curves with marked points can be constructed similarly to moduli spaces Mg = Mg;0, but instead of 3-canonical embeddings one should consider embeddings determined by an appropriate power of the line bundle L(K + X), where X is the divisor of marked points. Another method of constructing moduli spaces of curves with marked points is inductive. If we have already constructed a moduli space Mg;n , then we can try to construct a moduli space Mg;n+1 of curves of the same genus with one marked point more. In fact, as Mg;n+1 we can take the universal curve Cg;n with removed structural sections. Indeed, every point of this space determines a curve of genus g with n + 1 marked points: the first n of them are given by the structural sections, and the (n + 1)th point is the chosen point itself. This space has all necessary properties, but we need to construct a universal curve over it. Note that in this interpretation, the projection π from the definition of a universal curve becomes the forgetful map πg;n : Mg;n+1 → Mg;n : it “forgets” the (n + 1)th marked point, converting a curve with n + 1 marked points into a curve with n marked points. For rational curves, this construction looks especially simple. Consider the direct product M0;n × CP1 , where the second factor is the projective line with a fixed coordinate z. With every point of the moduli space M0;n associate the rational curve in which the marked points x1 , x2 , and x3 have the coordinates 0, 1, and ∞, respectively. Then the coordinates of all other marked points are uniquely determined, and we obtain n pairwise disjoint sections σ1 , . . . , σn : M0;n → M0;n × CP1 . The complement to these sections in the direct product is exactly the required universal curve C0;n ≡ M0;n+1 . Remark 14.2 Each moduli space of rational curves is a subset in the direct product of the previous moduli space and the projective line. However, this space itself is not a direct product of nontrivial factors. The forgetful map converts it into a bundle over the previous moduli space whose fibers have the same topology (that of a sphere with several punctures), but different complex structures.

14.7 Moduli Spaces of Curves with Marked Points

175

Of course, the resulting construction coincides with the following one, which at first sight seems to be simpler. Take the space Cn−3 with the coordinates z4 , . . . , zn acquired by the last n − 3 marked points after the coordinates of the first three points are chosen to be 0, 1, and ∞, and remove from this space the hyperplanes zi = 0, zi = 1, i = 4, . . . , n, and zi = zj for i = j , corresponding to the possible coincidences of marked points. However, it is much better adapted to the subsequent construction of a compactification we are interested in.

Chapter 15

Moduli Spaces of Rational Curves with Marked Points

In the previous chapter, we have described a procedure for constructing moduli spaces of curves with marked points. These spaces are usually not compact. The key point in the understanding of the geometry of noncompact spaces is constructing their compactifications. Every noncompact space has many different compactifications, but only few of them are convenient to work with. In contrast to higher genera, the moduli space M0;n is a smooth manifold rather than orbifold. This property simplifies a bit investigation of moduli spaces in genus 0 case. In this chapter, we describe a geometric construction of the moduli space M0;n due to M. Kapranov. In this construction, every rational curve with n marked points is realized as a Veronese curve in the projective space of dimension n − 2 passing through a fixed collection of n points in general position. Degenerations of Veronese curves are no longer smooth, and the points corresponding to these singular curves are being added to the moduli space of smooth curves to compactify it. Kapranov’s geometric construction does not provide any tool for computing homological characteristics of compactified moduli spaces. A detailed computation requires a more abstract approach, and we give a description of the cohomology ring due to Keel. It is complemented by a result of Kontsevich and Manin, which produces a collection of generators and linear relations on them in the cohomology spaces of all dimensions. However, both these descriptions are not very efficient, and there are ongoing attempts to find more explicit descriptions of the cohomology rings of the spaces M 0;n .

© Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_15

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15.1 Kapranov’s Construction In this section, we present a simple realization, due to Kapranov [12], of moduli spaces of rational curves as families of Veronese curves in projective spaces. Every curve of such a family passes through a fixed collection of points, which are exactly the marked points on the corresponding curve. First consider the projective plane CP2 . A collection of points in CP2 is said to be in general position if no three points are collinear. A conic is a second-order curve in the plane, i.e., a curve given by a homogeneous equation ax 2 + by 2 + cz2 + dxy + exz + fyz = 0. A conic meets every line in two points (which coincide if the line touches the conic). Recall that a smooth conic is rational, i.e., can be defined as the image of a singlevalued mapping from CP1 . Indeed, it suffices to choose a point on the conic and associate with every line passing through it the second intersection point of the line with the conic. Given five points in general position, there is a unique conic passing through them: such five points give five linear equations on the coefficients a, b, c, d, e, f of the conic equation, from which they can be uniquely recovered up to multiplication by a common nonzero factor. Given four points x1 , x2 , x3 , x4 in general position, there is a one-parameter family of smooth conics passing through them. This family can be naturally identified with the moduli space M0;4. To prove this, we must show that every rational curve with four marked points occurs in this family exactly once. Recall that any ordered quadruple of points in general position in CP2 can be mapped to any other such quadruple by a unique projective transformation of CP2 . Thus, from the viewpoint of conics passing through them, any two quadruples of points in CP2 in general position are not different from each other. Consider the projective plane dual to CP2 , i.e., the space of lines in CP2 . Taking the intersection of a line in CP2 with a conic C, we associate with this line a pair of points of C, thus identifying the dual projective space with the symmetric square C (2) of C. An isomorphism of curves C and C  induces an isomorphism of their symmetric squares, i.e., a projective transformation of the dual projective plane. The transformation of CP2 dual to this projective transformation preserves four points, and hence is the identity. Thus the curves C and C  coincide, as required. Unfortunately, this argument cannot be directly extended to projective spaces of higher dimension. The problem is that for m ≥ 3, a curve in CPm is, in general, no longer a complete intersection, i.e., cannot be defined by a system of m − 1 independent polynomial equations. So, it is convenient to regard curves in spaces of higher dimension as images of embeddings. The Veronese embedding of the projective line CP1 into the projective space CPn−2 is the mapping (t0 : t1 ) → (t0n−2 : t0n−2 t1 : t0n−3 t12 : . . . : t1n−2 ),

15.1 Kapranov’s Construction

179

given by the monomials of degree n − 2. The degree of such a curve is equal to n − 2, i.e., coincides with the dimension of the target space. In an affine chart, the Veronese embedding has the form t → (t, t 2 , . . . , t n−2 ). Clearly, it is an isomorphism onto the image. It is also clear that the images of any n distinct points of CPn−2 under the Veronese embedding are in general position, i.e., no n − 1 of them lie in the same hyperplane. (This immediately follows from the fact that the Vandermonde determinant does not vanish.) Curves obtained by applying projective transformations to the image of the Veronese embedding will be called Veronese curves (they are also called rational normal curves). For instance, smooth conics in the plane are Veronese curves for n = 4. For n = 5, Veronese curves are called twisted cubics in CP3 . The above reasoning about conics in CP2 can be extended without change to Veronese curves in spaces of higher dimension. Fix n points x1 , . . . , xn in general position in CPn−2 . There is an (n − 3)-dimensional family of Veronese curves passing through the points x1 , . . . , xn , which form a collection of marked points on each curve of the family. Theorem 15.1 Given points x1 , . . . , xn in general position in CPn−2 , the family of Veronese curves passing through them is isomorphic to the moduli space M0;n . Proof Since every collection of n points in general position in CPn−2 can be mapped to any other one by a projective transformation, it is clear that every rational curve with n marked points can be realized by a Veronese curve containing x1 , . . . , xn . Repeating our arguments for the case of CP2 , we prove that no curve occurs in this family twice. Consider the dual projective space to CPn−2 . It consists of hyperplanes in CPn−2 . Taking the intersection of a hyperplane in CPn−2 with a Veronese curve C, we associate with this hyperplane a collection of n − 2 points of C, thus identifying the dual projective space with the (n − 2)th symmetric power C (n−2) of C. An isomorphism of curves C and C  induces an isomorphism of the (n − 2)th symmetric powers C (n−2) and C (n−2) , i.e., a projective transformation of the dual projective space. The transformation of CPn−2 dual to this projective transformation preserves n points x1 , . . . , xn , and hence is the identity. Thus the curves C and C  coincide, as required. Exercise 15.1 Let x1 , . . . , xn , where n ≥ 3, be pairwise distinct points of CP1 . Consider the vector space Ω 1 (x1 + . . . + xn ) of rational 1-forms on CP1 with poles of order at most one at the points x1 , . . . , xn . a) Show that the dimension of this space is equal to n − 1. b) Show that the image of the mapping from CP1 to the projectivization of the space dual to Ω 1 (x1 + . . . + xn ) that sends a point x to the functional ω → ω(x) (see Sect. 8.5) is a Veronese curve.

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15.2 Compactification of Moduli Spaces Kapranov observed that Veronese curves allow one to construct a compactification of the moduli space of rational curves with marked points. Example 15.1 In the example with n = 4, consider the family of all (rather than only smooth) plane conics passing through the given four points. This means that the family considered above must be augmented by singular conics, each of them being a pair of lines connecting the given points pairwise. There are three such conics, depending on the point connected with x1 . Thus we have constructed the closure of the space M0;4 , by identifying this closure with a projective line in the space of homogeneous forms of degree 2 on the plane. This projective line consists of the forms vanishing at each of the points x1 , x2 , x3 , x4 . We will denote this compactified space by M 0;4. The example shows that in order to construct a compactification of a moduli space of rational curves with marked points, this space must be augmented by points corresponding to singular curves. Besides, in the example, the singular curves to be added are pairs of projective lines meeting in a single point, these lines contain two marked points each, and the marked points do not coincide with the intersection point of the lines. Assume that we have a finite collection of rational curves, some of them intersecting. These curves constitute a singular curve and are its irreducible components. The disjoint union of the irreducible components is called the normalization of a singular curve. With a singular curve we associate a graph (called the graph of irreducible components, or the modular graph), whose vertices are in a one-to-one correspondence with the irreducible components and two vertices are connected by an edge if the corresponding lines intersect (the number of edges connecting two vertices is equal to the number of intersection points of the corresponding lines). The original collection of rational curves is called a rational nodal curve if the resulting graph is a tree. Definition 15.1 A rational nodal curve C with n marked points is said to be modularly stable if • none of these points coincides with an intersection point of components of C; • the automorphism group of C is finite, i.e., every irreducible component contains at least three special points (marked points or points of intersection with other components). In what follows, we omit the word “modularly” and simply speak of stable curves. In the case of conics in CP2 , the curves added to the moduli space to compactify it are stable: each of them is a pair of intersecting lines with three special points. In the case of arbitrary dimension, a compactification is constructed in a similar way: the space of Veronese curves is augmented by the limiting singular curves. They are stable, but, in contrast to the two-dimensional case, admit no simple

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definition. The reason is that Veronese curves in higher dimension are no longer complete intersections. Example 15.2 Let x1 , x2 , x3 , x4 , x5 be points in general position in the three-dimensional projective space. Let us see what singular curves must be added to the moduli space M0;5 to compactify it. Draw a line through the points x1 , x2 . It intersects the plane passing through the points x3 , x4 , x5 in some point different from the original ones. To construct the closure of the space of smooth curves, we must add the singular curves one of which is the line passing through x1 , x2 and the other one is a conic passing through x3 , x4 , x5 and the intersection point of the line and the plane. As we know, the family of such conics (and hence the family of singular curves of this form) is one-dimensional. When a conic degenerates into a pair of lines, the singular curve degenerates into a triple of lines one of which intersects the other two. This middle line contains one marked point, and each of the remaining two lines contains two marked points. The distribution of marked points among the line and the conic, as well as among the lines of the triple, can be arbitrary, only the number of marked points must be preserved. Theorem 15.2 Every stable rational curve with n marked points can be realized in a unique way as a stable curve passing through a given collection of points x1 , . . . , xn in general position in CPn−2 . To prove this, we need to extend the notion of Veronese embedding to singular stable curves. We already know (see Exercise 15.1) that for smooth rational curves with marked points, the Veronese embedding coincides with the mapping determined by meromorphic 1-forms with poles of order at most one at the marked points. Let us extend this construction to singular curves. Let (C; x1 , . . . , xn ) be a stable rational curve. Consider the space of meromorphic 1-forms on C with poles of order at most one at the points x1 , . . . , xn and no other poles except possibly simple poles at double points. Each such 1-form determines a collection of meromorphic 1-forms on the irreducible components of the singular curve. We require that the residues of these 1-forms at the two preimages of each double point sum to 0. The space of such meromorphic 1-forms is (n − 1)-dimensional, and the standard construction from Sect. 8.5 defines a mapping from the curve to the projectivization CPn−2 of the space dual to this space of 1-forms. It is this mapping that will be called the Veronese mapping for singular curves. One can easily see that the images of the points x1 , . . . , xn under this mapping are in general position. The remaining part of the proof follows the same lines as the proof of Theorem 15.1. For details, we refer the reader to [12]. Exercise 15.2 Show that the canonical image of a hyperelliptic curve of genus g is a Veronese curve in CPg−1 . Exercise 15.3 Consider a stable rational curve in CP3 that is the union of a line with two marked points and a conic with three marked points intersecting transversally

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in one point. Construct a holomorphic one-parameter family (parametrized by the punctured disk D ∗ ⊂ C) of twisted cubics passing through the five marked points whose limiting element (corresponding to the puncture) coincides with this singular curve. Remark 15.1 A formal proof that the above definition of the compactified moduli space M 0;n is correct requires the formalism of Hilbert schemes, see the previous chapter. The Hilbert polynomials of all smooth Veronese curves in CPn−2 are equal, and Kapranov defines the compactified moduli space M 0;n as the closure in the corresponding Hilbert scheme of the space of Veronese curves passing through the chosen n points in general position. The subsequent argument requires checking that singular Veronese curves have the same Hilbert polynomial as smooth ones, and that all subvarieties with this Hilbert polynomial are of this form. Exercise 15.4 Find the Hilbert polynomial of a Veronese curve in CPn−2 . Show that singular Veronese curves have the same Hilbert polynomial. Check that every curve (in fact, every subvariety) with this Hilbert polynomial is a stable Veronese curve.

15.3 Poincaré Polynomials of Moduli Spaces Let X be a compact topological space. Its Poincaré polynomial is the polynomial PX (t) = bd t d + . . . + b0 , where bi = bi (X) = dim(Hi (X, C)) is the ith Betti number of X and d = dim X. Thus, the Poincaré polynomial gives a rough description of the topology of X. For example, the Poincaré polynomial of the complex projective line (which, from the topological point of view, is the two-dimensional sphere) is PCP1 (t) = t 2 + 1. Since M 0;4 = CP1 , this is also the Poincaré polynomial PM 0;4 (t) of the moduli space of stable rational curves with four marked points. Now let X be a topological space representable in the form X = X1 \ X2 where X2 ⊂ X1 are two compact algebraic varieties. We define the motivic Poincaré polynomial of X by the formula PX (t) = PX1 (t) − PX2 (t). For instance, the Poincaré polynomial of the moduli space M0;4 is PM0;4 (t) = PM 0;4 (t) − 3P{·} (t) = (t 2 + 1) − 3 = t 2 − 2.

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It turns out that the notion of motivic Poincaré polynomial thus introduced is well defined: any two representations of X as a difference of compact algebraic varieties lead to the same motivic Poincaré polynomial. The Poincaré polynomial behaves naturally with respect to taking unions and direct products of varieties. For example, PCP1 × CP1 (t) = (1 + t 2 )2 = 1 + 2t 2 + t 4 . Let us use this property to find the Poincaré polynomial of the moduli space M0;n for every n. To this end, recall the inductive construction of the space M0;n+1 . Fix a coordinate on the projective line CP1 and consider the direct product M0;n × CP1 . Puncture each one-dimensional fiber of its projection to the first factor at the points x1 , . . . , xn , assuming that x1 , x2 , x3 have the coordinates ∞, 0, 1, respectively. Then the coordinates of the other points are uniquely determined by the projective equivalence class of the whole collection x1 , . . . , xn . Thus, the n marked points determine n sections σi : M0;n → M0;n × CP1 . The complement to these sections in the direct product is naturally isomorphic to the moduli space M0;n+1 . Indeed, every point of this complement, being added to the n punctures of the fiber, determines a collection of n + 1 points on the rational curve up to projective equivalence. Conversely, every collection of n + 1 pairwise distinct points can be uniquely obtained in this way: one should consider the fiber corresponding to the collection of the first n points. Thus, for every n there is a forgetful map π : M0;n+1 → M0;n fibering the moduli space over the previous one. Every fiber of this map is a rational curve with n punctures, which allows us to find the Poincaré polynomials of moduli spaces of smooth curves. Proposition 15.1 The Poincaré polynomial of the space M0;n is equal to PM0;n (t) = (t 2 − 2)(t 2 − 3) . . . (t 2 − n + 3). Indeed, the Poincaré polynomial of a fibered space is the product of the Poincaré polynomials of the fiber and the base. Compactified moduli spaces are glued out of pieces, each of them being a product of noncompactified moduli spaces. This construction allows one to find the Poincaré polynomials of compactified spaces. Example 15.3 Let us find the Poincaré polynomial of the moduli space M 0;5 . This space is the disjoint union of the following spaces: • the moduli space of smooth curves M0;5 ; • moduli spaces of stable curves with one singular point; each of them is the direct product M0;3 ×M0;4 (one irreducible component of a curve contains two marked

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points, the other one contains three marked points, and each component contains also the point of intersection with the other component); • moduli spaces of stable curves with two singular points; each of them is the direct product M0;3 × M0;3 × M0;3. To find the Poincaré polynomial of the space M 0;5 , we must also know the number of spaces of each type it contains. This number can be computed combinatorially, and in this case the computation is easy: • there is one space of the first type; • there are 10 spaces of the second type: this is the number of ways to choose two marks out of five; • there are 15 spaces of the third type: there are 5 ways to choose a mark on the middle irreducible component, and then there are 3 ways to divide the remaining 4 marks into two pairs for the extreme components. In total, we have 3 PM 0;5 = PM0;5 + 10PM0;3 PM0;4 + 15PM 0;3

= (t 2 − 2)(t 2 − 3) + 10(t 2 − 2) + 15 = t 4 + 5t 2 + 1. As the number of marked points grows, the combinatorics involved quickly becomes cumbersome, so we present only one more example. Example 15.4 The space M 0;6 is the disjoint union of the following spaces: • the moduli space of smooth curves M0;6 ; • moduli spaces of stable curves with one singular point; each of them is either the direct product M0;3 × M0;5 (one irreducible component of a curve contains two marked points and the other one contains four marked points), or the direct product M0;4 × M0;4 (three marked points on each component); • moduli spaces of stable curves with two singular points; each of them is the direct product M0;3 × M0;3 × M0;4, the middle component containing either two marked points or one marked point; • moduli spaces of stable curves with three singular points; each of them is the direct product M0;3 × M0;3 × M0;3 × M0;3 , the tree of irreducible components being either a chain of four vertices, or a star with a center of valence 3. Let us find the number of spaces of each type: • there is one space of the first type; • the number of spaces of curves with one singular point and the 2–4 distribution of marked points is equal to 15, and the number of such spaces with the 3–3 distribution is equal to 10; • the number of spaces of curves with two singular points and two marked points on the middle component is equal to 45, and the number of such spaces with one marked point on the middle component is equal to 60;

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• the number of spaces of curves with three singular points for which the tree of irreducible components is a chain is equal to 90, and the number of such spaces for which the tree of irreducible components is a star is equal to 15. Thus, 2 PM 0;6 = PM0;5 + 15PM0;3 PM0;5 + 10PM 0;4 2 4 + (45 + 60)PM P + (90 + 15)PM 0;3 M0;4 0;3

= (t 2 − 2)(t 2 − 3)(t 2 − 4) + 15(t 2 − 2)(t 2 − 3) + 10(t 2 − 2)2 + 105(t 2 − 2) + 105 = t 6 + 16t 4 + 16t 2 + 1. Note that, as in the previous case, we have obtained a polynomial with nonnegative coefficients. This polynomial is self-reciprocal: the sequence of its coefficients is symmetric about the midpoint, as must be the case (by Poincaré duality) for every compact nonsingular variety. It turns out that the laborious combinatorial argument illustrated by the above examples can be encoded in a compact and efficient form. Namely, we introduce two generating functions for the Poincaré polynomials of moduli spaces of smooth and stable rational curves with marked points: P(x, t) = x −

∞ 

PM0;n+1 (t)

xn , n!

PM 0;n+1 (t)

yn . n!

n=2

P(y, t) = y +

∞  n=2

Theorem 15.3 The generating functions P(x, t) and P(y, t) are mutually inverse with respect to composition, i.e., P(P(y, t), t) ≡ y;

P(P(x, t), t) ≡ x.

The proof of Theorem 15.3 can be found in [7]. Since we know the Poincaré polynomials of moduli spaces of smooth curves, we obtain an efficient mechanism for computing the Poincaré polynomials of compactified spaces. This mechanism does not require considering graphs of irreducible components and can be easily implemented in every computer algebra system. The Euler characteristic of every compact manifold is the result of substituting the value t = −1 into its Poincaré polynomial. Similarly to how we have defined motivic Poincaré polynomial, we can define motivic Euler characteristic. For the difference of two compact smooth algebraic varieties, it is defined as the difference

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of the corresponding Euler characteristics. The motivic Euler characteristic of a noncompact variety is also the result of substituting t = −1 into its Poincaré polynomial. In particular, Theorem 15.3 can be carried over to Euler characteristics. Corollary 15.1 The exponential generating functions P(x, −1) = x +

∞ 

(−1)n−1

n=2

xn n(n − 1)

and P(y, −1) for the Euler characteristics of noncompactified and compactified moduli spaces are mutually inverse with respect to composition. Since the Euler characteristics of the noncompactified moduli spaces are known, χ(M0;n ) = (−1)n−1 (n − 1)!, this result allows one to extend the list of computed values of Euler characteristics of compactified spaces: n

3

4

5

6

7

8

9

10

χ(M 0;n )

2

7

34

213

1630

14,747

153,946

1,821,473

15.4 Compactified Spaces: Keel’s Description of Cohomology Except in the simplest cases, knowing the Betti numbers, i.e., the coefficients of the Poincaré polynomial, is not sufficient for understanding the cohomology of a space. In order to describe the finer structure of the cohomology ring, one must not only know the pieces the space consists of, but also how exactly they are glued together. Example 15.5 We know that the moduli space M 0;5 is glued out of M0;5, ten copies of the direct product M0;3 × M0;4, and fifteen copies of the direct product M0;3 × M0;3 × M0;3. To understand how all these spaces are glued together, we present an explicit construction of the space M 0;5 . Consider the direct product M 0;4 × CP1 . Fix a coordinate on CP1 in which the marked points x1 , x2 , x3 have the coordinates ∞, 0, 1, respectively. These points determine three pairwise disjoint structural sections σi : M 0;4 → M 0;4 × CP1 , i = 1, 2, 3. The fourth section σ4 , corresponding to the marked point x4 , intersects each of the first three sections transversally in one point, see Fig. 15.1. These intersection points lie over points of the boundary ∂M 0;4 . To construct the moduli space M 0;5 , it suffices to blow up each of the three intersection points of σ4 with σ1 , σ2 , σ3 on the surface M 0;4 × CP1 , see Fig. 15.2. Thus, to the direct product M 0;4 × CP1 = CP1 × CP1 , with Poincaré polynomial (1 + t 2 )(1 + t 2 ) = 1 + 2t 2 + t 4 , we add a triple of two-dimensional cycles, obtaining the Poincaré polynomial PM 0;5 (t) = 1+5t 2 +t 4 , which agrees with the computation

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Fig. 15.1 The direct product M 0;4 × CP1 and structural sections in it

Fig. 15.2 The result of blowing up the intersection points of structural sections in the direct product M 0;4 × CP1 and structural sections in it

above. So, we get a realization of the space M 0;5 as the universal curve C 0;4 over the moduli space M 0;4. The fiber of C 0;4 over a point of the space of smooth curves M0;4 is still a rational curve with four marked points. The fiber over each point of the boundary is a pair of transversally intersecting rational curves, with two marked point each. One of the curves in such a pair is a fiber in the direct product, and the other one appeared in the process of blowing up. In particular, the space M 0;5 is thus identified with the universal curve C 0;4 over the moduli space M 0;4. This means that every fiber of the projection M 0;5 → M 0;4 is exactly the curve with four marked points whose module is the corresponding point of the base. In the geometric representation of M 0;5 thus obtained, this space turns out to be divided into the parts prescribed earlier. Indeed, it consists of • the space M0;5, the complement to the four sections σi in M0;4 × CP1 ; • 10 copies of the space M0;3 × M0;4, represented by the four sections σi and three punctured special fibers of the projection of the blown-up surface to the factor M 0;4 , each being the union of two punctured lines; • 15 pairwise intersection points of the closures of these ten punctured lines. Remark 15.2 Blowup, used in the construction of the moduli space M 0;5, is a standard tool for resolving singularities in algebraic geometry. In this case, the singularity we wanted to get rid of is the intersection point of two structural sections. A blowup can be performed in a smooth manifold of arbitrary dimension

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along a smooth submanifold with codimension at least two. We are interested primarily in the case of blowing up a two-dimensional surface along a point. If X is a two-dimensional surface and p ∈ X, then a blowup is a pair consisting of a two-dimensional surface Y and a holomorphic mapping Σp : Y → X such that Σp is an isomorphism over X \ {p} and the preimage of p is a projective line. This projective line should be understood as the projectivization of the tangent plane Tp X to X at the point p. Blowup is a local operation which does not affect the structure of the surface outside the point being blown up. In local coordinates x, y in a neighborhood of the point p and local coordinates u, v in a neighborhood of its preimage, the blowup has the form x = uv, y = v. Clearly, under such a mapping, every line x = ay is the image of the line u = a, i.e., lines intersecting at p are images of parallel lines. The result of blowing up the plane X = C2 at the origin can be naturally identified with the total space of the line bundle O(−1) over the projective line. Indeed, every point of C2 except the origin uniquely determines a line in C2 passing through this point and the origin, i.e., a fiber of the bundle O(−1) over CP1 , and also a point in this fiber. The fiber of O(−1) over the origin is identified with the projective line introduced in the process of blowing up: the points of this fiber are in a one-to-one correspondence with the lines in C2 passing through the origin. Under a blowup of a point on a surface, the full preimage of a neighborhood of this point is biholomorphic to a neighborhood of the fiber of O(−1) over the origin. In a similar way, for every positive integer n ≥ 2, the result of blowing up the origin in the space Cn is identified with the total space of the line bundle O(−1) over the projective space CPn−1 . Exercise 15.5 Describe a typical stable rational curve corresponding to each of the components of the space M 0;5 in the above description. Exercise 15.6 The surface M 0;5 is obtained by blowing up three points in CP1 × CP1 . Is it true that every triple of points in CP1 × CP1 can be mapped to any other such triple by a biholomorphic transformation? Does the cohomology ring of the surface CP1 × CP1 blown up in three points depend on what points are being blown up? Using a generalization of this construction, Keel managed to find a simple description of the cohomology rings of the spaces M 0;n . Denote by D = A  B an unordered partition of the set of indices of marked points {1, 2, . . . , n} into two disjoint subsets A and B, each containing at least two elements. (The partition A  B coincides with B  A.) Consider all stable rational curves in M 0;n consisting of two irreducible components, one containing all points with indices from A, and the other one containing all points with indices from B. Such stable curves form a subvariety of (complex) codimension 1 in M 0;n . Its closure is a smooth compact complex subvariety of codimension 1. It is isomorphic to the direct product of M 0;|A|+1 and M 0;|B|+1 (the intersection point of two irreducible components of a generic curve is an additional distinguished point on each of the components). Denote by [D] the second cohomology class Poincaré dual to this subvariety: [D] ∈ H 2 (M 0;n ).

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Theorem 15.4 The classes [D] generate the cohomology ring H ∗ (M 0;n ). In particular, these classes linearly span the second cohomology space H 2 (M 0;n ), the whole cohomology ring is generated by H 2 (M 0;n ), and the odd cohomology vanishes. The generators [D] are not linearly independent. For example, for n = 4 all three generators, corresponding to the partitions {1, 2}  {3, 4},

{1, 3}  {2, 4},

{1, 4}  {2, 3},

represent the same cohomology class, namely, the class dual to a point. More generally, fix an arbitrary quadruple i, j, k, l of pairwise distinct integers between 1 and n. Denote by [ij Dkl] ∈ H 2 (M 0;n ) the sum of all generators [D] such that the indices i, j belong to one part of the partition D and the indices k, l belong to the other part. Then the generators [D] satisfy the system of linear relations Rij kl : [ij Dkl] = [ikDj l] = [ilDj k]. These relations can be easily deduced by considering the forgetful map M 0;n → M 0;4 that forgets all points except xi , xj , xk , xl and shrinks all components of the curve that have become unstable to a point. Theorem 15.5 The relations Rij kl linearly span the space of linear relations in H 2 (M 0;n ). Finally, let us say that two partitions D, D  are compatible if the set of indices {1, 2, . . . , n} admits a partition into three pairwise disjoint sets A, B, C such that D = (A  B)  C,

D  = A  (B  C).

One can easily see that if partitions D and D  are not compatible, then the corresponding cycles are disjoint. Hence the product of the generators [D] and [D  ] in the cohomology ring vanishes. Theorem 15.6 The cohomology algebra H ∗ (M 0;n ) is generated by the classes [D] modulo the additive relations Rij kl and multiplicative relations [D][D  ] = 0 for any incompatible partitions D and D  . Keel also establishes an isomorphism of the Chow ring of M 0;n (i.e., the intersection ring of rational equivalence classes of subvarieties in M 0;n ) with the cohomology ring M 0;n , thus obtaining a description of the Chow ring. Proof of Theorem 15.6 In Keel’s construction, moduli spaces of stable rational curves are built inductively. Assume that the space M 0;n is already constructed. Then the space M 0;n+1 is obtained by a sequence of blowups of the direct product M 0;n × M 0;4 along smooth subvarieties of codimension 2.

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Denote by p1 the projection of M 0;n × M 0;4 to the first factor; as usual, by π : M 0;n+1 → M 0;n we denote the mapping that forgets the last marked point. Then there exists a mapping π1 : M 0;n+1 → M 0;n × M 0;4 such that the corresponding triangle is commutative:

We turn to constructing this mapping as a composition of successive blowups. Let σ1 , . . . , σn : M 0;n → M 0;n+1 be the structural sections of the universal curve. Composing them with π1 , we obtain mappings π1 (σi ) : M 0;n → M 0;n × M 0;4 . Each of them sends every divisor [D] ⊂ M 0;n to a smooth subvariety of codimension 2 in M 0;n × M 0;4 . Denote this subvariety by D. Let Dk  be the union of all subvarieties D corresponding to ordered partitions D = A  B in which B consists of k elements, k = 2, 3, . . . , n − 2. Setting X1 = M 0;n × M 0;4 and successively performing blowups of the variety X1 along D2 , D3 , . . . , Dn−2 , we obtain Xn−2 = M 0;n+1 . In more detail, the variety X2 is the result of blowing up X1 along D2 . Observe that D2  is a disjoint union of pairwise isomorphic subvarieties indexed by the pairs of distinct indices from {1, . . . , n}. Let f1 : X2 → X1 be the blowup mapping. Every preimage f1−1 (Dk ), k ≥ 3, is a union of pairwise isomorphic irreducible smooth components. Moreover, irreducible components in f1−1 (D3 ) are pairwise disjoint. Indeed, the intersection of two irreducible components of D3  is nonempty if the parts B of the partitions corresponding to these components differ by one element. After the blowup along D2 , the preimages of these components no longer intersect. In the same way, all subsequent blowups are performed along subvarieties that are disjoint unions of smooth pairwise isomorphic components. Exercise 15.7 Using Keel’s theorem, give a complete description of the cohomology rings H ∗ (M 0;5 ), H ∗ (M 0;6 ). Exercise 15.8 Is it true that the cohomology classes dual to any five boundary strata of codimension 1 in M 0;5 are linearly independent, i.e., form a basis in H 2 (M 0;5 )? Exercise 15.9 Find the Sn -symmetric part in the cohomology rings of the spaces M 0;5 and M 0;6 .

15.5 Compactified Spaces: Kontsevich and Manin’s Description of Cohomology

191

15.5 Compactified Spaces: Kontsevich and Manin’s Description of Cohomology For large values of n, Keel’s description of the cohomology spaces H k (M 0;n ) gets cumbersome due to a large number of generators and relations. Kontsevich and Manin suggested a description of these spaces in terms of modular trees. With a stable rational curve with n marked points we associate its modular tree whose vertices have labels from the set {1, . . . , n}; each vertex is labelled by the numbers of the marked points lying in the corresponding component (a vertex can have several labels). With every modular tree T we can associate a subvariety in M 0;n , the closure of the set of all curves with modular tree T (this subvariety will be called a modular stratum), and the cohomology class [T ] ∈ H ∗ (M 0;n ) dual to this subvariety. For example, the tree consisting of a single vertex gives rise to the whole variety M 0;n and the identity class in H 0 (M 0;n ). The dimension of the cohomology class [T ] corresponding to a modular tree T is equal to twice the number of edges in T , i.e., to the number of singular points of a curve with this modular tree. Exercise 15.10 Draw all labelled modular trees with two edges for n = 6. Theorem 15.7 The classes [T ] corresponding to trees with k edges linearly span the space H 2k (M 0;n ). These classes are not independent and satisfy linear relations which can be constructed as follows. Fix a tree T , a quadruple of labels i, j, k, l of vertices of T , and a vertex v separating these labels. The latter requirement means that all four shortest paths connecting v with each of the vertices labelled by i, j , k, l start with different edges. On modular trees, we have the operation of contracting an edge T  → T , which decreases the number of edges and vertices by 1. We say that a contraction T  → T separates the labels i, j from the labels k, l if the vertex v of T is obtained by contracting an edge in T  such that the shortest path from one endpoint of this edge to the vertices with labels i, j and the shortest path from the other endpoint to the vertices with labels k, l do not contain the edge itself. Lemma 15.1 The cohomology classes corresponding to modular trees satisfy the linear relations   [T  ] = [T  ], ij T  kl

ikT  j l

where the sum in the left-hand side is over all contractions T  → T separating the labels i, j from the labels k, l, and the sum in the right-hand side is over all contractions T  → T separating i, k from j, l. Proof Consider the result of contracting in T all edges that do not contain the labels i, j, k, l on their endpoints. We obtain a mapping from the corresponding

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stratum to M 0;4 , the left- and right-hand side of the above equality corresponding to two different points of the boundary in M 0;4 . The preimages of these two points in M 0;n represent the same second cohomology class and give the same cohomology class in H 2(k+1)(M 0;n ) when intersected with the class [T ] ∈ H 2k (M 0;n ). Theorem 15.8 The linear relations described in the lemma span the whole space of linear relations between the classes [T ]. Exercise 15.11 Write down all linear relations between classes of modular strata in a) H 2 (M 0;6 ), b) H 4 (M 0;6), c) H 4 (M 0;7 ) and find the dimensions of the corresponding spaces by a direct calculation.

Chapter 16

Stable Curves

In the previous chapter, we introduced the notion of a stable rational curve with marked points. In this chapter, we extend this notion to curves of higher genus. The (modular) stability of a curve means that it has a finite group of automorphisms. Curves of small genus (g = 0 and g = 1) become stable only after marking several (at least three in the rational case, and at least one in the elliptic case) points. For g ≥ 2, smooth curves are stable even without marked points. However, in order to compactify moduli spaces of smooth stable curves, we have to consider moduli of singular curves. The space M g;n of stable nodal curves of a given genus g with a fixed number n of marked points is a natural compactification of the space of smooth curves. Like a moduli space of smooth curves of positive genus, it is usually not a variety, since stable curves may have nontrivial, though finite, automorphism groups. However, this does not prevent the space M g;n from being a smooth orbifold.

16.1 Definition and Examples of Stable Curves The need to compactify moduli spaces of curves and the modularity condition (which requires that the points being added to a moduli space must also correspond to some curves) forces one to consider not only smooth, but also singular curves. However, we want the new curves to have the simplest possible singularities and still finite automorphism groups. These conditions are satisfied by so-called stable curves. Adding them yields a desired compactification. Definition 16.1 A stable curve is a nodal curve with a finite automorphism group. This definition needs to be clarified, but first let us give a couple of examples.

© Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_16

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Example 16.1 Let C1 , C2 be two elliptic curves and p1 , p2 be points on C1 and C2 , respectively. Consider the curve obtained by identifying p1 and p2 . It is stable. Indeed, its automorphism group is generated by the automorphisms of C1 preserving p1 , the automorphisms of C2 preserving p2 , and, possibly, the automorphisms permuting C1 and C2 and sending p1 to p2 (the latter exist only if C1 and C2 are isomorphic). $ Consider the curve C Example 16.2 Let p, q be points of an elliptic curve C. obtained by identifying p and q. It is stable. Indeed, its automorphism group is $ obtained from C generated by the automorphisms of the original elliptic curve C, by “splitting” the points p and q, preserving the pair of points p and q. The group of such automorphisms is finite. Recall that a nodal curve is an algebraic curve allowed to have not only smooth points, but also points with a neighborhood biholomorphic to a pair of disks with a common point (which can be assumed to be the center of both disks). Such singular points are called double points, or nodes. The number of double points on a nodal curve is finite. We speak about abstract curves, but it is convenient to imagine a double point as a point of transversal self-intersection of a plane or spatial curve. Given a nodal curve C, one can define a normalization of C. This is a smooth $ along with a mapping C $ → C that is a local biholomorphism complex curve C everywhere except the preimages of double points. Each double point has exactly two preimages, and each of them has a neighborhood that is mapped biholomorphically onto one of the disks of the neighborhood of the double point (each preimage $ of the nodal is assigned to its own disk). In Example 16.1 the normalization C curve C is the disjoint union of the smooth curves C1 and C2 , and in Example 16.2 it is a connected smooth elliptic curve. The image of every connected component under a normalization mapping is called an irreducible component of C, so the curve in Example 16.1 consists of two irreducible components, and the curve in Example 16.2 has one irreducible component. Example 16.3 For t = 0, the curves of the family y 2 = x 3 + x 2 + t are smooth plane elliptic curves. For t = 0, the corresponding curve is singular, having a node at the origin. Its normalization is a rational curve, and the degenerate cubic itself is obtained by gluing together two points of this rational curve. As we will see below, it is natural to view it also as an elliptic curve. A formal definition of a nodal curve requires defining the ring of meromorphic functions on such a curve (this is necessary, for example, to distinguish between transversal intersection and tangency). We define the ring of meromorphic functions on a nodal curve as the ring of meromorphic functions on its normalization taking the same finite value at both preimages of every double point. The stability condition imposes certain restrictions on nodal curves. They are as follows: • every rational component of the normalization of a stable nodal curve must contain at least three preimages of double points;

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• every elliptic component of the normalization must contain at least one preimage of a double point; Indeed, the automorphism group of a curve is infinite only if its genus is less than two, and such components must be stabilized by preimages of double points. Now we can give a definition of a stable curve with marked points. Definition 16.2 A stable curve with marked points is a nodal curve with marked points whose nodes do not coincide with the marked points and whose automorphism group is finite. The last condition means that • every rational component of the normalization of a stable nodal curve must contain at least three special points (i.e., points that are either marked or preimages of double points); • every elliptic component of the normalization must contain at least one special point.

16.2 The Genus of a Nodal Curve In order to compactify the moduli space of curves of genus g by stable curves, we must know what is the genus of a nodal curve. The genus must be defined in such a way as to be preserved under degenerations of smooth curves. One alternative is to introduce the space of holomorphic 1-forms on a nodal curve. Then the genus of the curve can be defined as the dimension of this space. Clearly, a holomorphic 1-form on a nodal curve must be holomorphic at its smooth points. It remains to prescribe the behavior of such a form at singular points. We allow it to have simple poles at double points on each branch, and require that the residues at these points sum to zero. So, the geometric genus of a nodal curve is the dimension of the space of meromorphic 1-forms on its normalization whose all poles are located at preimages of double points, the order of every pole is at most one, and the residues at the two preimages of every double point sum to zero. A typical example illustrating the behavior of holomorphic (and, more generally, meromorphic) 1-forms on a degenerate curve, which actually leads to the above definition, is as follows. Consider the one-parameter family xy = t of rational curves in the plane. The restriction of the 1-form dx/x to a curve of this family is a meromorphic 1-form on this curve with two simple poles, both at infinity (the residues at these poles do not depend on t). This restriction coincides with the restriction to the same curve of the 1-form −dy/y. (Indeed, the difference of these 1-forms is equal to dx dy + = d(xy)/xy, x y

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and its restriction to the curve xy = const is identically zero.) When the parameter t vanishes, the conic xy = t degenerates into two coordinate lines, and the limiting 1-form looks as follows: it coincides with the form dx/x on the line y = 0, and with the 1-form −dy/y on the line x = 0. At the intersection point, the residues of these 1-forms are opposite in sign. The fact that the residues at the preimages of a double point sum to zero immediately follows from the fact that the residues of every meromorphic 1-form on a smooth curve sum to zero. Here is another definition of the same notion of genus. A neighborhood of a double point is isomorphic to a pair of disks intersecting in one point. The boundary of this neighborhood is a pair of circles. Let us smoothen this pair of disks, converting it into a cylinder whose boundary is the same pair of circles, and do this for each double point of the curve. The result is a smooth compact two-dimensional surface. Its genus is called the genus of the corresponding nodal curve. Exercise 16.1 Show that the above two definitions of the genus of a nodal curve give the same result. Exercise 16.2 Show that if a nodal curve is obtained by gluing together a pair of points of a smooth curve of genus g, then its genus is equal to g + 1. The same is true for gluing together a pair of points of a nodal curve of geometric genus g. Exercise 16.3 Find the genus of the nodal curve obtained by gluing together two nodal curves of genera g1 and g2 , respectively, in a pair of points. The notion of the modular graph of a rational nodal curve can be extended to curves of arbitrary genus. With a nodal curve we associate the graph whose vertices are its irreducible components and two vertices are joined by an edge if and only if the corresponding components intersect. The number of edges connecting two vertices is equal to the number of intersection points of the corresponding irreducible components. To every double point of an irreducible component there corresponds a loop (an edge whose endpoints coincide) in the graph. Let us label each vertex by the genus of the normalization of the component corresponding to this vertex. The obtained graph is called the modular graph of the curve. Exercise 16.4 Draw the modular graphs of all stable curves of genus 2 and 3 without marked points. Vertices of modular graphs of marked nodal curves are usually labelled by the labels of the marked points lying in the corresponding irreducible components. Exercise 16.5 Draw the modular graphs of all stable curves of genus 1 with one, two, and three marked points, and of genus 2 with one marked point. Exercise 16.6 Is it true that the order of the automorphism group of a stable curve of genus g (possibly with marked points) cannot exceed 84(g − 1)?

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16.3 Degenerations of Smooth Curves Points of the Deligne–Mumford compactification of the moduli space of curves of genus g with n marked points are biholomorphic equivalence classes of stable curves. If a compact moduli space of stable curves M g;n does indeed exist, then it has the following property: every holomorphic morphism from a punctured disk to M g;n can be extended to the puncture. This means that every holomorphic family of smooth curves over the punctured disk can be augmented by a stable curve. Let us see how this works in practice. Example 16.4 Let C be a nodal curve with marked point x1 obtained by gluing together two points, different from x1 , of a rational curve. The geometric genus of C is equal to 1. Indeed, the space of holomorphic 1-forms on C is the space of meromorphic 1-forms on CP1 with simple poles at the points being glued together (the residues of such a 1-form at these points automatically sum to zero). The dimension of the latter space is equal to 1. The automorphism group of the curve C is finite, so C is stable. As one can easily see, there are no other singular stable elliptic curves with one marked point. Adding a point corresponding to C to the moduli space M1;1, we obtain the compactified moduli space M 1;1 . Locally, this looks like the result of the degeneration of the family of elliptic curves described in Example 16.3 (one may assume that the point x1 marked on all curves of the family is at infinity). In terms of the representation of M1;1 as a fundamental domain for the action of the group PSL(2, Z) on C, the point being added describes the limiting structure of the elliptic curve corresponding to the lattice spanned by the vectors 1 and τ = a + bi as b → ∞. Exercise 16.7 What is the automorphism group of the nodal elliptic curve with one marked point from the previous example? Example 16.5 Let C be a curve of genus g ≥ 1, and let x1 be a fixed point of C and x2 be a variable point of C. Denote by Cx2 the curve C with the points x1 and x2 marked. We want to know what point of the moduli space this curve approaches as x2 → x1 . Consider the two-dimensional surface C × D (which we interpret as an identity one-dimensional family of curves), the direct product of the curve C and the complex disk D, fibered over D. The disk D can be identified with a neighborhood of the point x1 in C; the points of D are the possible positions of the point x2 . This fiber space has two distinguished sections: the constant section s1 : D → C × D, corresponding to the point x1 , and the variable section s2 : D → C × D, corresponding to the point x2 . These two sections intersect transversally in a single point over the center of the disk. To construct the stable limit of this family, we blow up the surface C × D at the intersection point of s1 and s2 . Then a projective line gets glued into the surface, the sections intersecting it transversally. This line intersects transversally the fiber over the puncture. Together with the fiber it constitutes the limiting curve. The preimages

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of the sections under the blowup leave traces on the glued-in projective line. These traces are the limiting positions of the points x1 and x2 . Their exact location is irrelevant: all triples of points on a projective line are equivalent. So, the limiting stable element of our family is the curve C with a projective line, containing both marked points, glued in at the fixed point (before the degeneration, this point is the marked point x1 ). The question arises whether one can obtain another limit using another construction. It is exactly the existence of a moduli space of stable curves that guarantees the uniqueness of the limit. Exercise 16.8 Describe the stable limit of the one-parameter family of curves of genus g + 1 (where g ≥ 1) obtained from a given smooth curve C of genus g by gluing together a fixed point p and a variable point q as q tends to p.

16.4 Compactification of Moduli Spaces by Stable Curves and Pluricanonical Embeddings Now we can give a rigorous definition of a moduli space of stable curves. The right way to think of this space is as a universal family of stable curves of given genus with given number of marked points. Definition 16.3 A family of stable curves of genus g with n marked points is a collection (B, X, π, {s1 , . . . , sn }) consisting of • • • •

a smooth orbifold B; a smooth orbifold X of complex dimension dim B + 1; a holomorphic mapping π : X → B; a collection of holomorphic mappings si : B → X

and satisfying the following properties: • the images si (B) of si are disjoint and do not pass through singular points of fibers of π; moreover, the composition π ◦si for every i = 1, . . . , n is the identity mapping, i.e., si are sections of π; • the fibers of π are the quotients of stable complex curves of genus g with n marked points by their automorphism groups, the marked points being intersections with the images of si . The moduli space Mg;n of smooth curves of genus g with n marked points is a smooth complex orbifold of complex dimension 3g − 3 + n. There is also a family Cg;n of smooth stable curves over Mg;n , whose dimension is equal to 3g − 2 + n. The compactification M g;n of Mg;n is a compact smooth complex orbifold of the same dimension. The universal curve Cg;n over Mg;n also admits a compactification. More exactly, the following theorem holds. Theorem 16.1 (Deligne–Mumford) For any nonnegative integers g and n satisfying the condition 2g − 3 + n ≥ 0 there exists a family (M g;n , C g;n , πg;n ,

16.4 Compactification of Moduli Spaces by Stable Curves and Pluricanonical. . .

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{σ1 , . . . , σn }) of stable curves of genus g with n marked points such that for every family (X, B, π, {s1 , . . . , sn }) of stable curves of genus g with n marked points there exists a unique pair of holomorphic mappings F : X → C g;n , G : B → M g;n such that the diagram

is commutative and F ◦ si = σi ◦ G for all i = 1, . . . , n. The moduli space M g;n is defined uniquely up to biholomorphism and irreducible. By abuse of language, in what follows we speak about the moduli space M g;n of stable curves with n marked points, meaning the whole collection of spaces and mappings from the corresponding definition. Remark 16.1 The conclusion of the theorem means that the moduli space is coarse. It would be fine if the spaces C g;n and M g;n were smooth varieties rather than orbifolds and the fibers of πg;n were stable curves rather than quotients of stable curves by the actions of their automorphism groups. The moduli space M 0;n is fine for every n ≥ 3, but for g ≥ 1 fine moduli spaces of stable curves do not exist for any n. On the other hand, for every g the noncompactified moduli space Mg;n is fine for sufficiently large n: it suffices to choose n greater than the maximum number of fixed points of a nontrivial automorphism of a smooth curve of genus g. The moduli space of smooth curves Mg;n is a Zariski dense subset in M g;n . Its complement ∂M g;n = M g;n \ Mg;n is called the boundary of the moduli space of stable curves. For example, the boundary ∂M 1;1 consists of a single point, namely, the rational curve CP1 with one marked point and two points glued together. Points of the boundary ∂M g;n correspond to singular stable curves. As in the case of rational stable curves, this boundary is stratified by degenerations of curves: every stratum is determined by the number of singular points on a typical curve, the genera of its irreducible components, the mutual configuration of irreducible components, and the distribution of marked points among the irreducible components. The strata are in a one-to-one correspondence with the modular graphs. For g ≥ 1, such a graph is no longer necessarily a tree, and each its vertex (associated with an irreducible component of a generic curve of the stratum) is assigned, apart from a collection of marked points, a nonnegative integer, the genus of this irreducible component. The proof of Theorem 16.1 follows the same lines as the proof of the existence of a moduli space of smooth curves. Stable curves with marked points are sent into a high-dimensional projective space by a pluricanonical map of appropriate degree (it suffices to take the fifth degree of the canonical bundle). Their images in this

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space have the same Hilbert polynomial. The set of pluricanonical images of stable curves coincides with the closure in the corresponding Hilbert scheme of the set of images of smooth stable curves. Then M g;n can be defined as the quotient of this closure by the action of the group of projective transformations of the ambient space. Following this approach, one encounters many difficulties. In particular, we do not know any direct proof of the fact that Hilbert points of singular stable curves are stable. Below we give a number of exercises that outline the general strategy of the proof. Exercise 16.9 State and prove the Riemann–Roch formula for stable curves. Exercise 16.10 Check that the degree of the canonical class on a stable curve of genus g is equal to 2g − 2. Exercise 16.11 Find the dimension of the space of holomorphic sections of a) the canonical bundle; b) the r-canonical bundle on a stable curve. Exercise 16.12 Show that the 5-canonical map of a stable curve of genus g ≥ 2 without marked points is an embedding. Exercise 16.13 Find the Hilbert polynomial of the image of a stable curve of genus g ≥ 2 under a 5-canonical embedding. Show that every algebraic subvariety in the projective space of the corresponding dimension with this Hilbert polynomial is the image of a stable curve under the 5-canonical embedding. Exercise 16.14 Generalize the constructions from the previous exercises to the case of stable curves with marked points.

Chapter 17

A Backward Look from the Viewpoint of Characteristic Classes

The introduction of characteristic classes allows one to see many calculations carried out above in a new light and simplify them. Characteristic classes are a universal tool for computing topological characteristics of algebraic varieties, both smooth and singular. We begin with discussing definitions and general properties of Chern classes of vector bundles, and then show how one can use them to obtain some results we already know and their generalizations. In the next chapter, we will speak about characteristic classes that arise naturally in the study of the topology of moduli spaces of curves.

17.1 The First Chern Class of a Line Bundle It is easiest to define characteristic classes for line bundles (i.e., vector bundles with one-dimensional fibers). We have already encountered line bundles over curves, see Chap. 7. An example of a line bundle over a manifold of higher dimension is the tautological bundle O(−1) over the projective space CPn . The fiber of this bundle over a point of CPn corresponding to a line in Cn+1 passing through the origin is the line itself. The simplest line bundle over a given manifold M is the trivial bundle, which is the direct product M × C equipped with the projection to the first factor. The dual of a given line bundle is also a line bundle. For instance, this is true for the bundle O(1) over CPn dual to the tautological bundle. The tensor product L1 ⊗ L2 of two line bundles over a given manifold M is also a line bundle over M. The dth tensor powers of the bundles O(1) and O(−1) are denoted by O(d) = (O(1))⊗d and O(−d) = (O(−1))⊗d , respectively. The first Chern class c1 (L) ∈ H 2 (M) of a line bundle L over a manifold M is an obstruction to the existence of an everywhere nonzero holomorphic section of L. If a holomorphic line bundle has an everywhere nonzero holomorphic section, then

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this line bundle is trivial. The first Chern class of a trivial line bundle is zero, so the first Chern class measures the deviation of a given line bundle from triviality. To define the first Chern class of a line bundle, it suffices to take an arbitrary nonzero meromorphic section of this bundle: the cohomology class represented by the divisor of zeros and poles of such a section does not depend on the choice of the section. Moreover, the divisor of zeros and poles is defined uniquely up to rational equivalence of subvarieties: the difference of two such divisors corresponding to different sections is a principal divisor, i.e., the divisor of zeros and poles of some meromorphic function (this function is the ratio of the sections, which is well defined in every line bundle). In other words, the first Chern class of a line bundle is well defined not only as an element of the second cohomology group, but also as an element of the Chow ring, the ring of rational equivalence classes of subvarieties. Usually, the Chow ring is a finer invariant of an algebraic variety than the cohomology ring, but it is also more difficult to compute. For example, the first Chern class of the line bundle O(1) over CPn is the class dual to a hyperplane in CPn . Exercise 17.1 Show that the first Chern class is well-behaved with respect to taking tensor products and duals of line bundles: c1 (L1 ⊗ L2 ) = c1 (L1 ) + c1 (L2 ) and c1 (L∨ ) = −c1 (L).

17.2 Chern Classes of Vector Bundles Characteristic classes of vector bundles measure to what extent these bundles differ from trivial ones. In general, a characteristic class χ is a rule that associates with every vector bundle L over a manifold M a cohomology class χ(M) ∈ H ∗ (M) and behaves naturally under base change. The naturalness condition admits a rigorous mathematical formulation. Namely, the equality χ(f ∗ L) = f ∗ χ(L) must hold for every mapping f : N → M between two manifolds. In the left-hand side, f ∗ L is the vector bundle over N obtained by lifting L via f ; in the right-hand side, f ∗ χ(L) is obtained by lifting the cohomology class χ(L) ∈ H ∗ (M) to H ∗ (N) via the pullback mapping f ∗ : H ∗ (M) → H ∗ (N). Depending on what manifolds and bundles (topological, smooth, or complex) and what mappings between manifolds (continuous, smooth, or holomorphic) we consider, one distinguishes between topological, smooth, and complex characteristic

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classes. In what follows, we are primarily interested in holomorphic bundles over complex manifolds and the naturalness with respect to holomorphic mappings between these manifolds. However, it should be noted that in practice the naturalness condition is usually satisfied whenever a characteristic class is defined by a construction that is natural in the “general mathematical” sense. Exercise 17.2 Show that the naturalness condition holds for the first Chern class of holomorphic line bundles over complex manifolds introduced in the previous section. Example 17.1 Euler class. An analog of the first Chern class for vector bundles of rank greater than 1 is the Euler class. Let E → M be a vector bundle of rank r where M is a complex manifold. Choose an arbitrary nonzero holomorphic section in general position in this vector bundle. The zeros of such a section form a submanifold Z of real codimension 2r in M. The Euler class e(E) of the vector bundle E is the cohomology class e(E) = [Z] ∈ H 2r (M) Poincaré dual to Z. The Euler class can also be defined in the case where a holomorphic vector bundle E has no holomorphic sections. In this case, instead of a holomorphic section, one can take a C ∞ -smooth section in general position. The Euler class is represented by the submanifold Z of its zeros. This is a real smooth submanifold of real codimension 2r. To define e(E) as an integral class, one must also introduce a coorientation of Z. The existence of a natural coorientation of Z follows from the existence of a natural orientation of every complex vector bundle. The chosen coorientation is determined by the complex structure. One can easily check that the Euler class does not depend on the choice of a section and satisfies the naturalness condition. A more algebraic way to define the Euler class is to pass from the base space M of the vector bundle to its total space E. The projection p : E → M has contractible fibers, hence the pullback homomorphism p∗ : H ∗ (M) → H ∗ (E) is an isomorphism. The pullback bundle p∗ E on E has the natural “tautological” holomorphic section, and its zeros form the submanifold M ⊂ E itself embedded as the zero section. Thus, the Euler class of E is defined as the preimage under the isomorphism p∗ of the class dual to the zero section of E. In the case where E is the tangent bundle T M and the base M is compact, the value of the Euler class on the fundamental class of the base is equal to the Euler characteristic χ(M) of M. Denoting the Euler class by e(M), we can write  χ(M) =

e(M). M

More generally, by



c we denote the value of a characteristic class c on the

M

fundamental class of M. In general, the definition of the Euler class can be applied to an arbitrary fiber bundle. Moreover, it can be naturally extended to fiber bundles whose base is not

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necessarily smooth (and even not necessarily an algebraic variety, but an arbitrary cellular space). The Chern classes ci (E) ∈ H 2i (M), i = 1, 2, . . . , dim M, are distinguished characteristic classes of holomorphic vector bundles. Every characteristic class is a polynomial combination of Chern classes. Below we give a definition of the Chern classes through a number of properties (axioms) sufficient for computing them in concrete examples. We prove neither that these classes are well-defined, nor that the constructed theory of characteristic classes of complex vector bundles is complete. However, the mere fact that this theory is consistent (the independence of the Chern classes on the chosen strategy for computing them) provides an efficient tool for solving many geometric problems. Like the first Chern class of a line bundle, the Chern classes of a vector bundle of arbitrary rank can be viewed as elements of the Chow ring, rather than the cohomology ring. A vector bundle E → M of rank r, where r ≥ 0, is said to be split if it has a complete flag of subbundles 0 = V0 ⊂ V1 ⊂ . . . ⊂ Vr = E,

rk Vk = k.

The successive quotients Ik = Vk /Vk−1 are line bundles over M, and the first Chern classes for them are already defined. Definition 17.1 If E is a split bundle, then the kth Chern class ck (E) of E is the kth elementary symmetric function of the classes ti = c1 (Ii ), i = 1, . . . , r, i.e., ck (E) = ek (c1 (I1 ), . . . , c1 (Ir )). This definition can be rewritten as follows: 1 + c1 (E) + . . . + cr (E) =

r % (1 + ti ). i=1

The inhomogeneous class in the left-hand side is called the total Chern class of E and denoted by c(E); the classes ti ∈ H 2 (M) are called the Chern roots of E. It turns out that the naturalness condition allows one to extend the definition of Chern classes in a unique way from split vector bundles to arbitrary ones. In other words, every vector bundle can be split by an appropriate base change. The standard construction of such a “virtual” splitting is as follows. Let E → M be a vector bundle. Denote by p : F (E) → M the (locally trivial) fiber bundle of complete flags. This is no longer a vector bundle: the fiber p−1 (x) over a point x is the variety of complete flags in the vector space Ex Cr . Exercise 17.3 Give a rigorous definition of the fiber bundle of complete flags which would describe the topology of the space F (E). In particular, show that F (E) is a complex manifold whenever E → M is a holomorphic vector bundle over a complex base.

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The space F (E) has the following two important properties. First, the pullback p∗ E of E is, obviously, a split vector bundle over F (E), and hence the Chern classes ck (p∗ E) are well defined. Second, the pullback homomorphism p∗ : H ∗ (M) → H ∗ (F (E)) is injective, and hence every cohomology class on M is uniquely defined by its pullback in the cohomology of F (E). We define ck (E) as the class whose pullback is ck (p∗ E). One should check that the above definition of Chern classes is consistent (namely, that the Chern classes of a split bundle do not depend on the splitting, and also that the class ck (p∗ E) lies in the image of the homomorphism p∗ ). We omit this verification. The fact that every vector bundle can be reduced to a split one is formalized in the splitting principle. It says that performing computations with Chern classes, we can formally assume that all involved vector bundles are split. The result of these computations is a symmetric polynomial in the Chern roots of the original vector bundle, hence it can be expressed in terms of elementary symmetric functions of these roots, i.e., in terms of Chern classes. The answer obtained in this way under the splittability assumption holds also in the general case. As an application of the splitting principle, we prove the Whitney formula for the Chern classes of a direct sum of bundles:  c(E  ⊕ E  ) = c(E  ) c(E  ), ck (E  ⊕ E  ) = ci (E  ) cj (E  ), i+j =k

where we set c0 (E) = 1 and ck (E) = 0 for k < 0 or k > rk E. Indeed, let t1 , . . . , tn  and t1 , . . . , tn be the Chern roots of the vector bundles E  and E  , respectively. Then all these classes together are the Chern roots of the direct sum E  ⊕ E  , and we obtain 



n n % % (1 + ti ) (1 + tj ) = c(E  ) c(E  ). c(E ⊕ E ) = 



i=1

j =1

The Whitney formula remains valid if a direct sum is replaced with a skew sum: c(E) = c(E  ) c(E  ) for every short exact sequence of vector bundles 0 → E  → E → E  → 0. From the topological point of view, every short exact sequence of vector bundles can be split into a direct sum, but the embedding E  ⊂ E that implements such a splitting is not necessarily algebraic even if the original vector bundles are. With the Whitney formula in mind, one often denotes the skew sum by + instead of ⊕,

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writing, for example,1 c(E) = c(E  +E  ) = c(E  ) c(E  ) and c(E  ) = c(E−E  ) = c(E)/c(E  ). Another important corollary of the splitting principle is as follows. Proposition 17.1 The top Chern class cr (E) where r = rk(E) coincides with the Euler class. Indeed, this is obviously true if E is a direct sum of line bundles. Then, by the splitting principle, this is true in the general case too. Let us summarize the main properties of Chern classes ck (E) of vector bundles, which can be viewed as axioms. 1. The classes ck are characteristic classes of real grade 2k: they behave naturally under base change. 2. We have ck (E) = 0 for k < 0 and k > rk(E); c0 (E) = 1. 3. If r = rk(E), then cr (E) = e(E) is the Euler class. 4. For a short exact sequence of vector bundles, the Whitney formula holds. The listed properties (axioms) uniquely define the Chern classes of vector bundles. As we will see, the mere consistency of these axioms implies a large number of corollaries.

17.3 Other Approaches to Defining Chern Classes Let us briefly describe other possible approaches to constructing Chern classes. Let p : P (E) → M be the projectivization of a vector bundle E → M; its fibers are projective spaces, the projectivizations of the fibers of E. In the same way as one constructs the tautological line bundle O(−1) over the projectivization CPn of the vector space Cn+1 , one can construct the tautological line bundle O(−1) over the projectivization P (E) of the vector bundle E, which is the union of fiberwise bundles over projective spaces. The tautological line bundle O(−1) over P (E) is a subbundle in the induced bundle p∗ E. Denote by Q = p∗ E/O(−1) the corresponding quotient bundle of rank r − 1, and set t = c1 (O(1)) = −c1 (O(−1)) ∈ H 2 (P (E)). The collection of classes 1, t, t 2 , . . . , t r−1 restricted to a fiber of p is a basis in the cohomology space of the fiber (isomorphic to a projective space). The well-known Dold principle implies that these classes are free generators of the cohomology space of P (E) regarded as a module over the cohomology ring of M: H ∗ (P (E))

1 This

H ∗ (M) 1 ⊕ H ∗ (M) t ⊕ . . . ⊕ H ∗ (M) t r−1 .

notation originates from K-theory.

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In other words, every cohomology class on P (E) can be uniquely represented as a linear combination of the classes 1, t, t 2 , . . . , t r−1 whose coefficients are cohomology classes of the base. In particular, this property applied to the class −t r means the existence of classes c1 , . . . , cr in H ∗ (M) such that t r + c1 t r−1 + . . . + cr−1 t + cr = 0 ∈ H ∗ (P (E)). One can easily see that ci are nothing else but the Chern classes of E, and this equation can be taken as an alternative definition of these classes. Indeed, by the Whitney formula, we get c(Q) =

1 + c1 + . . . + cn c(p∗ E) = = (1 + c1 + . . . + cr )(1 + t + t 2 + . . .). c(O(−1)) 1−t

In particular, the left-hand side of the previous relation is nothing else but the rth Chern class of the vector bundle Q, which vanishes since Q has rank r − 1. Another possible approach to defining Chern classes uses the pushforward homomorphism (“fiberwise integration”) p∗ : H k (P (E)) → H k−2(r−1)(M) sending the class Poincaré dual to an algebraic subvariety Z ⊂ P (E) to the class Poincaré dual to the image p(Z) of this subvariety. The projection preserves the dimension of a subvariety, while the codimension decreases by the dimension of the fiber. This causes the shift in the grading. The pushforward homomorphism is not multiplicative; instead, the projection formula p∗ (p∗ (a) b) = a p∗ (b) holds for any cohomology classes a ∈ H ∗ (M), b ∈ H ∗ (P (E)). In other words, p∗ is a homomorphism of H ∗ (M)-modules. Consider the action of the homomorphism p∗ on a monomial t k . This monomial is the kth term of the geometric progression (1 − t)−1 = c(−O(−1)), whence t k = ck (−O(−1)) = ck (p∗ E − O(−1) − p∗ E) = ck (Q − p∗ E)  = ci (Q)cj (−p∗ E). i+j =k

Applying p∗ and using the projection formula, we obtain p∗ (t k ) =

r−1  i=0

ck−i (−E) p∗ (ci (Q)).

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17 A Backward Look from the Viewpoint of Characteristic Classes

The class p∗ (ci (Q)) vanishes for i < r − 1 by counting dimensions, while for i = r − 1 we obtain a class p∗ (cr−1 (Q)) ∈ H 0 (M) which is a universal constant. Considering an arbitrary fiber of the vector bundle, we see that this constant is equal to 1. (The equation p∗ (cr−1 (Q)) = 1 expresses the fact that every nonzero vector in Cr is contained in a unique line). Finally, we obtain p∗ (t k ) = ck−r+1 (−E). The class c(−E) = c(E)−1 is called the (total) Segre class of the vector bundle E. The last equation can be taken as an independent definition of Segre classes and, consequently, of Chern classes. Exercise 17.4 Let E → M be a rank r vector bundle, c(E) = 1 + c1 + . . . + cr , and let L be a line bundle with the same base and first Chern class c1 (L) = h. Find the Chern classes of the vector bundle E ⊗ L. Solution. Let t1 , . . . , tr be the Chern roots of E; then t1 + h, . . . , tr + h are the Chern roots of E ⊗ L. Hence c(E ⊗ L) =

r % (1 + h + ti ) = (1 + h)r + c1 (1 + h)r−1 + . . . + cr . i=1

In particular, for the top Chern class we obtain cr (E ⊗ L) = hr + c1 hr−1 + . . . + cr = cr (E + L). Exercise 17.5 Find the Chern classes of the tangent bundle of CPn . Solution. An infinitesimal translation of a line in Cn+1 is determined by a linear mapping from this line to Cn , with homotheties giving rise to the trivial infinitesimal translation. Hence there is a canonical isomorphism T CPn Hom(O(−1), Cn+1 /O(−1)) Hom(O(−1), Cn+1 )/C. It follows that c(T CPn ) = c(O(1) ⊗ Cn+1 ) = (1 + t)n+1 =

 n   n+1 i=0

i

t n,

where t = c1 (O(1)) is the hyperplane class. In particular, the Euler characteristic of CPn is equal to  χ(CPn ) =

cn (T CPn ) = CPn

  n+1 = n + 1, n

17.4 The Genus of a Smooth Plane Curve

209

which agrees with the fact that CPn can  ben divided into n + 1 even-dimensional cells. Here we use the obvious equality t = 1 (which means that n hyperplanes CPn

in general position in CPn meet in a single point).

17.4 The Genus of a Smooth Plane Curve Let C be a smooth plane curve of degree d. In Sect. 3.2 we found the genus of such a curve using the Riemann–Hurwitz formula. Now let us solve the same problem using the Chern classes methods developed above. The Euler characteristic of C is nothing else but the value of the first  Chern class c1 (T C) of the tangent bundle of C on its fundamental class: χ(C) = c1 (T C). C

To find this value, we use the short exact sequence of vector bundles 0 → T C → T CP2 |C → ν → 0. Here ν is the normal bundle to the plane curve C, and we regard the tangent bundle of C as a subbundle in the restriction to C of the tangent bundle of CP2 . Denote by t the generator of the cohomology ring H ∗ (CP2 ) represented by the hyperplane section, t ∈ H 2 (CP2 ). All Chern classes we are interested in can be expressed in terms of this cohomology class. We have c(T CP2 ) = (1 + t)3 . On the other hand, a curve of degree d is the variety of zeros of a homogeneous polynomial f of degree d, i.e., of a section of the bundle O(d) over CP2 . Hence • this curve represents the cohomology class d · t; • the total Chern class of its normal bundle is equal to c(ν) = 1 + d · t. Therefore, c(T C) =

1 + 3t + 3t 2 (1 + t)3 = = 1 + (3 − d)t 1 + dt 1 + dt

(higher degrees of the class t vanish when restricted to the curve C). The value of the Euler class c1 (T C) = (3 − d)t on the class [C] = dt is the Euler characteristic χ(C) = 2 − 2g of C, and we obtain the equality (3 − d)d = 2 − 2g,

210

17 A Backward Look from the Viewpoint of Characteristic Classes

whence g=

(d − 1)(d − 2) , 2

in agreement with the results of Sect. 3.2. The above calculation can be easily extended to the case of the Euler characteristic of an arbitrary hypersurface M of degree d in the projective space CPn of an arbitrary dimension n. An exact sequence of vector bundles over a curve turns into an exact sequence of vector bundles over a surface: 0 → T M → T CPn |M → ν → 0. Using the fact that c(T CPn ) = (1 + t)n+1 , c(ν) = 1 + dt, we obtain c(T M) =

(1 + t)n+1 . 1 + dt

Since [M] = dt, the Euler characteristic of the hypersurface takes the form χ(M) = [t n ]dt

(1 + t)n+1 , 1 + dt

i.e., it is equal to the coefficient of t n in the power series expansion at the origin of the rational function in the right-hand side: χ(M) =

(1 − d)n+1 − 1 + (n + 1) d . d

17.5 The Genus of a Complete Intersection A hypersurface in CPn is a special case of a complete intersection, a smooth variety of codimension k represented as the transversal intersection of k smooth hypersurfaces in CPn . Let us find the genus of the intersection of two generic hypersurfaces of degrees m and n in CP3 . Applying the adjunction formula, we get c(T M) =

(1 + t)4 = 1 + (4 − m − n)t + . . . . (1 + m t)(1 + n t)

17.6 Plücker Formulas

211

It follows that  2 − 2g = χ(M) =

 m n t 2 c1 (T M) = m n (4 − m − n),

c1 (T M) = M

CP 3

i.e., g = m2n (m + n − 4) + 1. In particular, the intersection of two quadrics has genus 1, the intersection of a quadric and a cubic has genus 4, etc.

17.6 Plücker Formulas Using Chern classes, we can give another, perhaps more correct conceptually, interpretation of the Plücker formulas, see Chap. 5. Let C ⊂ CP2 be a complex curve having only double points and semicubical cusps as singularities, and assume that its dual C ∨ ⊂ (CP2 )∨ also has singularities only of these two types. Then the Plücker formulas say that the geometric invariants of the curves C and C ∨ (the numbers of double points δ and δ ∨ , as well as the numbers of cusps k and k ∨ , respectively) are in fact topological invariants and can be uniquely expressed in terms of basic topological characteristics of these curves, say, their degrees n and n∨ and the genus g of their normalization (or the Euler characteristic χ = 2 − 2g of the normalization). Denote by M ⊂ CP2 ×(CP2 )∨ the incidence hypersurface formed by the pairs (x, ) such that x is a point of a line . The natural projections p : M → CP2 and p∨ : M → (CP2 )∨ to the first and second factor identify M with the projectivization spaces of the cotangent bundles P T ∗ CP2 and P T ∗ (CP2 )∨ , respectively. Recall that the conormal unfolding Cˆ ⊂ M, formed by the pairs (x, ) such that x ∈ C and  is a tangent line to C at x, is a common normalization of the curves C and C ∨ . The idea behind computing the numbers k and k ∨ is to identify the cusps of the ˆ curves C and C ∨ with the zeros of sections of appropriate vector bundles over C; one must only properly determine these bundles to find their degrees. The tangent bundle of M has an important subbundle of rank 2, called the contact bundle, which is defined as the direct sum L1 ⊕ L2 of two line bundles formed by the tangent vectors to the fibers of the projections p and p∨ , respectively. By definition, c1 (L1 ) = c1 (T M − T CP2 ) = c1 (T (CP2 ×(CP2 )∨ ) − νCP2 ×(CP2 )∨ M − T CP2 ), ˆ so we can find the degree of the restriction of L1 to C: deg L1 = 3 (n + n∨ ) − (n + n∨ ) − 3 n = 2 n∨ − n; analogously, deg L2 = 2 n − n∨ .

212

17 A Backward Look from the Viewpoint of Characteristic Classes

A tangent line to the curve Cˆ ⊂ M always lies in the contact plane, and the cusps ˆ are the points at which the tangent direction coincides with of its image C = p(C) the direction of the projection p, i.e., with the line of L1 . In other words, the cusps of C are the zeros of the projection T Cˆ ⊂ L1 ⊕ L2 → L2 , whence k = deg Hom(T C, L2 ) = deg L2 − deg T C = 2 n − n∨ − χ and, analogously, k ∨ = 2 n∨ − n − χ. To find the number of double points of each of the curves, first assume that the curve C ⊂ CP2 is immersed, i.e., k = 0. Let us find the number of intersection points of the curve C with a small perturbation of this curve (obtained, for example, by applying a projective transformation of CP2 close to the identity transformation). On the one hand, by Bézout’s theorem, this number is equal to n2 . On the other hand, the appearance of some of these points is caused by the nontriviality of the normal bundle. The number of such points is equal to the degree of the normal bundle and can be found by the adjunction formula: there are 3 n − χ of them. The other part is caused by the existence of double points, and there are 2δ of them. Finally, we obtain the equation n2 = (3n − χ) + 2δ. Now consider the general case, i.e., a curve with cusps. Let us represent it as the image of a smooth curve under a projection from a high-dimensional projective space. A small perturbation of the projection direction yields an immersed curve (of the same degree and with normalization of the same genus as the original one) with n + δ double points. Hence n2 = (3n − χ) + 2(δ + k) and, analogously, (n∨ )2 = (3n∨ − χ) + 2(δ ∨ + k ∨ ). The obtained two equations, along with the above expressions for the numbers k and k ∨ , are equivalent to the Plücker formulas.

Chapter 18

Moduli Spaces of Stable Maps

In this chapter, we show how moduli spaces of maps can be applied to compute topological characteristics of various varieties. The notion of a stable map was introduced by Kontsevich. He applied it to solving the classical problem of enumerating rational curves of a given degree in the plane passing through a given collection of points. The methods suggested by Kontsevich turned out to be applicable to a wide circle of problems of enumerative geometry, being now the main tool for computing Gromov–Witten invariants. Moduli spaces of stable maps are very closely related to moduli spaces of stable curves (although a map from a curve can be stable even if the curve itself is not stable). Usually, spaces of stable maps are fibered over spaces of stable curves, sometimes even in several different ways. Thus, when computing intersection numbers in spaces of stable maps, we can rely on knowing the geometry of spaces of stable curves, and vice versa.

18.1 Rational Curves in the Plane The number of parameters determining a plane rational curve of degree d is equal to 3d − 1. Indeed, in order to define a parametrized rational curve CP1 → CP2 , (u : v) → (x : y : z), one needs 3d + 2 parameters: d + 1 parameters for each of the coordinates x, y, z, determined up to a common factor. The group of parameter changes in the preimage CP1 is three-dimensional, hence the space of nonparametrized rational lines in the plane has dimension (3d + 2) − 3 = 3d − 1. Another way to compute this dimension is to consider the space of homogeneous polynomials of degree d in three variables. Such a polynomial is determined by (d+1)(d+2) coefficients, hence the plane curves of degree d form a projective 2 space Qd of one dimension less. A smooth such curve has genus (d−1)(d−2) , and 2 since each double point reduces the genus of the normalization by one, a rational © Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_18

213

214

18 Moduli Spaces of Stable Maps

curve must have exactly this number of double points. The condition of the existence of a singular point has codimension 1, and we see that the dimension of the space of rational curves is equal to (d + 1)(d + 2) (d − 1)(d − 2) −1− = 3d − 1. 2 2 Since the dimension of the space of rational curves of degree d is equal to 3d −1, there are finitely many rational curves of degree d passing through 3d − 1 points in general position in the plane. For example, through 2 points there is a unique line and through 5 points there is a unique conic. A rational cubic must have one singular point. As shown by the following example, there are 12 rational cubics passing through 8 points in general position, each of them having a double point. Example 18.1 Let us find the number of rational and cubic curves passing through 8 given points in general position in the plane. Every plane cubic curve is the set of zeros of a homogeneous polynomial of degree 3 in three variables, unique up to a nonzero factor. The vector space of such polynomials has dimension 10, and its projectivization has dimension 9; denote this projectivized space by Q3 . The closure of the set of polynomials determining rational cubic curves is a hypersurface (a variety of dimension 8) in this projective space; denote it by R3 , R3 ⊂ Q3 . The condition for a curve to pass through a given point is a linear condition on the coefficients of the corresponding polynomial, and for a generic configuration of points these linear conditions are linearly independent. Hence the number of curves passing through given 8 points in general position coincides with the degree of the subvariety of rational cubics, i.e., with the number of intersection points of the subvariety R3 and a generic line in the space of cubics Q3 . Take this generic line to be the set of linear combinations uF + vG, u, v ∈ C, of two nondegenerate cubic polynomials F, G such that the corresponding curves F = 0 and G = 0 intersect transversally in 9 points. Denote by S the subvariety of zeros of the polynomial uF + vG in the direct product CP1 × CP2 , where (u : v) is a homogeneous coordinate on the factor CP1 . Let us find the Euler characteristic of the surface S in two different ways. On the one hand, the surface S is fibered over the projective line CP1 . A generic fiber of this bundle is a smooth elliptic curve. However, CP1 contains several points (u : v) for which the fiber is a rational cubic. Since the Euler characteristic of a smooth elliptic curve is 0 and the Euler characteristic of a rational cubic is 1 (topologically, a rational cubic is a two-dimensional sphere with two points glued together), the Euler characteristic of the surface S is equal to the number of special fibers. On the other hand, the projection to the second factor determines a mapping from S to the projective plane CP2 . One can easily see that this mapping is one-toone everywhere except the nine intersection points of the curves F = 0 and G = 0. A local analysis shows that in a neighborhood of such a point the projection is nothing else but a blowup of CP2 at this point: choosing local coordinates (x, y)

18.1 Rational Curves in the Plane

215

in CP2 in a neighborhood of a transversal intersection of the curves F = 0 and G = 0 so that F = x and G = y, we obtain a projection of the hypersurface ux + vy = 0 to the plane (x, y). Thus S is the result of blowing up the projective plane at 9 points. Each blowup replaces a point with a projective line, thus increasing the Euler characteristic of the surface by one. Hence the Euler characteristic of S equals the Euler characteristic of the projective plane increased by 9, i.e., 12. We conclude that the degree deg R3 of the hypersurface of rational cubics R3 , and hence the number N3 , is equal to 12. Another method to find the number N3 is as follows. In the direct product Q3 × CP2 consider the variety X of pairs of the form (a curve, a singular point of this curve). A singular point is determined by the condition that all three partial derivatives of the corresponding polynomial (a point of Q3 ) vanish. The expression for a partial derivative is linear in the coefficients of the polynomial and quadratic in the coordinates in CP2 . Hence every partial derivative is a section of the line bundle OQ3 (1)×OCP2 (2) with the first Chern class t +2h. Here t and h are the classes dual to hyperplanes in the cohomology rings H ∗ (Q3 ) and H ∗ (CP2 ), respectively. Thus X is the intersection of three hypersurfaces of bidegree (1, 2), and the cohomology class in H ∗ (Q3 × CP2 ) dual to X is equal to (t + 2h)3 . The variety of singular cubics R3 is the image of X under the projection Q3 × CP2 → Q3 to the first factor. The associated pushforward homomorphism maps the class h2 to 1 and the other powers of h to 0 (which corresponds to the fact that two lines in CP2 meet in a point, and a larger number of lines usually have an empty intersection). We conclude that the degree of the variety of singular cubics R3 equals the coefficient of th2 in (t + 2h)3 , i.e., 12. For d ≥ 3, every plane rational curve of degree d is singular: a smooth curve of degree d must have genus (d−1)(d−2) > 0. The problem of finding an efficient 2 method for computing the number of rational curves of degree d in the plane has been solved by Kontsevich fairly recently. The answer is given by the following recurrence relation. Theorem 18.1 The number Nd of plane rational curves of degree d passing through given 3d − 1 points in general position in CP2 is given by the formula Nd =

d−1  k=1

     3d − 4 3d − 4  Nk Nd−k k 2 (d − k)2 − k 3 (d − k) . 3k − 2 3k − 1

The first values of Nd are presented in the following table: d Nd

1 1

2 1

3 12

4 640

5 87 304

6 26 312 976

216

18 Moduli Spaces of Stable Maps

18.2 Moduli Spaces of Stable Maps Kontsevich’s calculation is an easy corollary of a quite general result on the associativity of the so-called quantum cohomology. To define it, we need the notions of a stable map and a moduli space of stable maps. Let M be a compact complex manifold of arbitrary dimension. Consider the space H = H ev (M, C) of even degree cohomology of M. This is a finitedimensional graded vector space equipped with a nondegenerate inner product (the Poincaré pairing (·, ·)) and a multiplication. These two structures agree with each other, i.e., (a · b, c) = (a, b · c) for every triple of elements a, b, c ∈ H , and also with the grading. Besides, the algebra H contains an identity element. Example 18.2 In the case M = CP2 , the space H ∼ = C3 is generated by three classes e0 , e1 , e2 , of degrees 0, 2, 4, respectively. Here e0 is the class dual to the whole plane, e1 is the class dual to a line, and e2 is the class dual to a point. The inner product is given by the formula (e0 , e2 ) = (e1 , e1 ) = 1, all the other pairings vanishing. The class e0 is an identity element, e2 = e12 , and e13 = 0. Note that the projective plane has no nontrivial odd degree cohomology. The quantum cohomology of the manifold M is a deformation of the multiplication in the algebra H . The deformation coefficients enumerate rational curves in M, and the deformation parameters correspond to cohomology classes from H . Denote by Mg;n (M, d) the moduli space of holomorphic maps f from curves of genus g with n marked points to M such that the image f (C) represents a given second homology class d ∈ H2 (M, Z). In the case where the space Mg;n (M, d) is smooth, it has dimension 3g − 3 + n(c1 (M), d) + (1 − g) dim M. Here (c1 (M), d) is the value of the first Chern class c1 (M) ∈ H 2 (M) of the tangent bundle of M on the homology class d ∈ H2 (M). In much the same way as the moduli space Mg;n of smooth curves with marked points admits a compactification M g;n by stable curves, the space Mg;n (M, d) admits a compactification M g;n (M, d) by stable maps. A map is said to be Kontsevich stable if its automorphism group is finite. In particular, a Kontsevich stable map from a rational curve to a complex manifold M is a holomorphic map from the tree of rational curves to M such that all components contracted to a point have at least three special points. One can show that the space M g;n (M, d) exists. However, it is singular and contains components of different dimensions; the most interesting of them is the component containing the space of maps from smooth curves. Up to now, marked points have played only an auxiliary role. Now they come to prominence. Let A1 , . . . , An be submanifolds in M, i.e., Ai ⊂ M, i = 1, . . . , n. Consider the set of holomorphic maps f : (C; x1 , . . . , xn ) → (M; A1 , . . . , An ) such that f (xi ) ∈ Ai . Under some conditions on Ai this set is finite, and the number of its elements can be found as follows.

18.2 Moduli Spaces of Stable Maps

217

Let us introduce the “evaluation” maps evi : M g;n (M, d) → M,

i = 1, . . . , n,

where evi associates with every map f its value at the point xi : evi : f → f (xi ). Then the number of maps f such that f (xi ) ∈ Ai for i = 1, . . . , n depends only on the cohomology classes represented by Ai and is given by the formula 

ev∗1 (a1 ) . . . ev∗n (an ).

ϕg;d (a1 , . . . , an ) = M g;n (M,d)

Here ai = [Ai ] ∈ H is the cohomology class Poincaré dual to Ai . Since the space M g;n (M, d) is singular, defining an integral over it presents significant technical difficulties. To ensure that the integral is well defined, one must construct the so-called “virtual fundamental class” of this space, whose dimension coincides with the dimension of Mg;n (M, d) and is called the expected dimension of the space of maps. The above integral can be nonzero only in the case where the degree of the class to be integrated coincides with the expected dimension. Example 18.3 Let g = 0, M = CP2 , and let A1 , . . . , An be points in M. Then M0;n (CP2 , de1 ) is the space of maps from rational curves to CP2 such that the images of the marked points x1 , . . . , xn are the points A1 , . . . , An , respectively, and the image of the curve is a curve of degree d (i.e., the cohomology class of the image is the dth multiple of the line class e1 ). We have c1 (CP2 )(de1 ) = 3d, whence dim M0;n (CP2 , de1 ) = −3+n+d +2 = n+d −1. The condition that a given point of the curve is mapped to a given point of the plane has complex codimension 2, thus deg ev∗ (ai ) = 2, which implies that the integral can be nonzero only if 2n = n + 3d − 1, i.e., if n = 3d − 1, and ϕ0;d (e2 , . . . , e2 ) = Nd . & '( ) 3d−1

Along with maps such that the image of a given marked point is a chosen point, we can consider maps such that the image of a given marked point belongs to a chosen line. Since every line in general position meets a curve of degree d in d points, we conclude that ϕ0;d (e2 , . . . , e2 , e1 , . . . , e1 ) = Nd · d r . & '( ) & '( ) 3d−1

r

218

18 Moduli Spaces of Stable Maps

Note that adding the line class e1 as an argument increases by 1 both the dimension of the space M0;n (CP2 , de1 ) (due to the appearance of a new marked point, n is replaced by n + 1) and the degree of the class to be integrated (since π ∗ (e1 ) has degree 1). Hence this class can be added in any amount, this does not affect the condition on dimensions.

18.3 Quantum Cohomology and the Associativity Equation The values ϕ0;d (a1 , . . . , an ), called the Gromov–Witten invariants of genus 0, can be organized into the generating series 

Φ(t1 , . . . , tN ) =

n; k1 ,...,kn ; d

1 ϕ0;d (ek1 , . . . , ekn ) q d tk1 . . . tkn , n!

where e1 , . . . , eN is a basis in the space H of dimension N = dim H . If the space H2 (M, C) is r-dimensional, then d = (d1 , . . . , dr ) is an r-dimensional vector, and by q d we denote the monomial q d = q1d1 . . . qrdr , where q1 , . . . , qr are additional formal variables. In the case M = CP2 , we have N = 3, r = 1, and the series Φ, by the discussion above, has the following explicit form: ∞

Φ(t0 , t1 , t2 ) =

 t 3d−1 1 2 (t0 t2 + t0 t12 ) + q d edt1 . Nd 2 2 (3d − 1)! d=1

The variables t1 , . . . , tN can be regarded as coordinates in the space H ; then Φ is a (formal) function on this space. The quantum multiplication is an algebra structure in the tangent space Tt H of H at every point t ∈ H . We denote it by . Coordinatewise, the quantum multiplication is given by the formula (ei  ej , ek ) =

∂ 3Φ , ∂ti ∂tj ∂tk

where ei = ∂t∂ i is regarded as a tangent vector. The constructed multiplication is automatically commutative (by the equality of mixed partial derivatives). The main nontrivial property of the quantum multiplication is the following. Theorem 18.2 The quantum multiplication is associative. In the example M = CP2 we are interested in, the only nontrivial relation implied by the associativity condition is (e1  e1 , e2  e2 ) = (e1  e2 , e1  e2 ), which can be written in the form   Φ1,1,i Φj,2,2 = Φ1,2,i Φj,1,2 , i+j =2

i+j =2

18.3 Quantum Cohomology and the Associativity Equation

219

that is, 2 Φ1,1,1 Φ1,2,2 + Φ2,2,2 − Φ1,1,2 = 0,

where subscripts stand for third partial derivatives. The obtained equation is equivalent to the recurrence relation from Theorem 18.1. Proof of Theorem 18.2 The proof is based on a simple observation which we have already used in Keel’s description of the cohomology ring of the space M 0;n : in the cohomology of M 0;4, various components of the boundary divisor represent the same cohomology class. Indeed, M 0;4 = CP1 , and each component of the boundary (there are three of them in total) is represented by a point. Another useful observation, which follows from the Poincaré duality, is that ∗ the cohomology class of the diagonal in the Cartesian  i,jsquare H (M × M, C) ∗ ∗ = H (M, C)⊗H (M, C) is represented by the class η ei ⊗ej , where e1 , . . . eN i,j

is a basis in the cohomology space and the matrix ηi,j , whose components are denoted by superscripts, is inverse to the matrix with the components ηi,j = (ei , ej ). In more detail, consider the following integral, which generalizes the definition of the Gromov–Witten invariants:  p∗ ([pt]) ev∗1 (a1 ) ev∗2 (a2 ) ev∗3 (a3 ) ev∗4 (a4 ), M 0;4 (M,d)

where p : M 0;4 (M, d) → M 0;4 is the map that sends a map to its definition domain. The integral does not depend on the choice of a cycle [pt] representing a point. Take such a cycle to be the connected component of the boundary in which the marked points with labels 1 and 2 lie in a separate rational curve. Every nodal curve from this cycle can be represented by a pair of curves of degrees d1 and d2 such that d1 + d2 = d, and the condition that these curves are glued together at the additional pair of marked points allows one to represent this integral in the form  

ϕ0,d1 (a1 , a2 , ei )ηi,j ϕ0,d2 (a3 , a4 , ej ).

d1 +d2 =d i,j

In a similar way, if we choose [pt] to be another connected component of the boundary of M 0,4 , namely, the component in which the marked points with labels 1 and 3 lie in a separate rational curve, then we obtain another expression 



ϕ0,d1 (a1 , a3 , ei )ηi,j ϕ0,d2 (a2 , a4 , ej )

d1 +d2 =d i,j

for the same integral. The equality of these two expressions is equivalent to the associativity of the quantum multiplication in the tangent space T0 H at the origin.

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18 Moduli Spaces of Stable Maps

For a general point of H , the associativity follows from similar considerations, with different representatives of the point class in the cohomology of M 0;4 replaced by the corresponding representatives of the pullback of the point class in the cohomology of M 0;n under the map forgetting all marked points except four. Caporaso and Harris generalized Kontsevich’s calculation of the number of rational curves passing through a fixed collection of points to the case of plane curves of arbitrary genus. Exercise 18.1 Show that the number Np,q of rational curves of bidegree (p, q) in CP1 × CP1 passing through 2(p + q) − 2 points in general position satisfies the recurrence Np,q =

q−1 p−1   p1 =1 q1 =1

Np1 ,q1 Np−p1 ,q−q1 (p1 (q − q1 ) + q1 (p − p1 ))(q − q1 )

     2p + 2q − 4 2p + 2q − 4 = p1 − (p − p1 ) . 2p1 + 2q1 − 1 2p1 + 2q1 − 2

Chapter 19

Exam Problems

Below we give a list of problems used in final exams in the 1-year course repeatedly taught by M. E. Kazaryan and S. K. Lando at the Department of Mathematics of the Higher School of Economics in 2010–2014. Most of these problems were given as exercises in the main text, and we have collected them here for the reader’s convenience. Along with problems, we also give a list of exam questions.

19.1 First Term 1. Show that an orientable surface of genus g can be obtained by gluing the sides of a given 2n-gon pairwise if and only if n ≥ 2g. 2. Let h − 1 divide g − 1, where h ≥ 2. Show that there exists a covering of a surface of genus h by a surface of genus g. 3. Construct a smooth rational curve of degree 4 in CP3 . 4. Show that every curve of degree 3 in CP3 that is not contained in any hyperplane is rational. 5. Show that a curve of degree 4 in CP3 that is not contained in any hyperplane is either rational or a curve of genus 1. 6. Find the genus of a smooth intersection of two quadrics in CP3 . 7. Find the genus of a smooth intersection of a quadric and a cubic in CP3 . 8. Give an example of a meromorphic differential on an elliptic curve that has a zero of order 2 and two poles of order 1. 9. Draw the curve dual to the curve y = x 4 − x 2 , find its equation and singular points. 10. A circle of radius 1/3 rolls along the inside of a circle of radius 1. Find the equation of the curve (hypocycloid, deltoid) described by a point on the smaller circle, the genus of this curve, its singularities, and the dual curve.

© Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2_19

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222

19 Exam Problems

11. A circle of radius 1/4 rolls along the inside of a circle of radius 1. Find the equation of the curve (hypocycloid, astroid) described by a point on the smaller circle, the genus of this curve, its singularities, and the dual curve. 12. Show that the plane curve x 3 + y 3 − 3xy = 0 is rational and find a rational parametrization of this curve. 13. Show that the plane curve (x − 1)(x 2 + y 2 ) = x 2 is rational and find a rational parametrization of this curve. 14. Show that the plane curve x 4 + x 2 y 2 + y 4 = x(x 2 + y 2 ) is rational and find a rational parametrization of this curve. 15. Show that the plane curve x 4 = x 3 − y 3 is rational and find a rational parametrization of this curve. 16. Find and classify the singular points of the plane curve (y 2 − x 2 )(x − 1) (2x − 3) = 4(x 2 + y 2 − 2x)2 and find the genus of this curve. 17. In the plane with coordinates x, y, consider the curve consisting of the pairs (x, y) such that the polynomial t 4 − t 2 + xt + y in t has multiple roots. Find the equation of this curve, its singular points, and the dual curve. 18. Show that the irreducible curve given by the equation Pn (x, y) + Pn−1 (x, y) = 0, where Pn is a nonzero homogeneous polynomial of degree n and Pn−1 is a nonzero homogeneous polynomial of degree n − 1, admits a rational parametrization. 19. Find the automorphism group of the plane curve y 2 = x 3 + x 2 . 20. For what values of the parameter m is the curve x 3 + y 3 + z3 = mxyz smooth? Find its inflection points. 21. Let C be a curve in CP3 . Consider the projection of C from a point to a space of one dimension less. Show that when projecting from a generic point outside C, the projection has the same degree as C, and when projecting from a generic point on C, the degree of the projection is one less than the degree of C. 22. Show that every curve of genus 2 can be immersed into the projective plane with one ordinary double point. 23. Let an elliptic curve be realized as a ramified covering of degree 2 over the projective line with ramification points −1, 0, 1, ∞. Find the automorphism group of this elliptic curve. 24. Let f be a polynomial of degree n with n pairwise distinct roots a1 , . . . , an . 1 1 1 Show that f  (a + f  (a + . . . + f  (a = 0. n) 1) 2) 25. Let X be a compact algebraic curve of genus 5 and Y be a compact algebraic curve of genus 4. Show that every holomorphic mapping f : X → Y is constant. 26. Let X be a compact algebraic curve of genus 2 and Y be a compact algebraic curve of genus 1, and let f : X → Y be a nonconstant holomorphic mapping. Show that on X there are at most two ramification points of f . Can such a point be unique? 27. Let C be the plane algebraic curve given in an affine chart by the equation w4 = z3 − 3z. Find its genus. Find the poles and residues of the 1-form dz/w on C. Explicitly construct a basis in the space of holomorphic 1-forms on C.

19.2 Exam Questions

223

28. Consider the algebraic curve X = {(z, w) ∈ C2 | z2 + zw + w2 = 3} 29. 30. 31.

32.

33.

34. 35. 36. 37. 38.

and the point p = (1, 1) ∈ X. Find the residue of the 1-form wdz 2 −1 at p ∈ X. Show that there is no Hurwitz curve of genus g = 2. Show that the group of orientation-preserving symmetries of the icosahedron is isomorphic to PSL(2, 5). Find the poles, their orders, and their residues for the 1-form obtained by lifting to the curve y 2 = x 3 − x of the 1-form dx via the projection of the curve to the x axis. 3 dx Find the poles and residues of the meromorphic 1-form xx 2 −1 . What is the domain of definition of this form (describe the compact smooth Riemann surface on which it is defined)? 4 dx Find the poles and residues of the meromorphic 1-form xx 2 −1 . What is the domain of definition of this form (describe the compact smooth Riemann surface on which it is defined)? Find the poles and residues of the restriction of the meromorphic 1-form dx to the curve x 3 + y 3 + xy = 1. Find the poles and residues of the restriction of the meromorphic 1-form dx to the curve x 3 + y 3 + xy = 0. Find the poles and residues of the restriction of the meromorphic 1-form dx y to the curve x 3 + y 3 + xy = 1. Find the poles and residues of the restriction of the meromorphic 1-form dx y to 3 3 the curve x + y + xy = 0. Find the poles and residues of the restriction of the meromorphic 1-form dx to y2

the curve x 3 + y 3 + xy = 1. 39. Find the poles and residues of the restriction of the meromorphic 1-form

dx y2

to

the curve + xy = 0. 40. Determine the singular points of the following plane curves and find the genus of the normalization. In the genus 0 case, construct a rational parametrization. In the genus 1 case, construct a meromorphic function of degree 2, find its critical values, their cross ratio, and the j -invariant: x 3 + y 3 + xy = 1, y 2 − x 4 = xy 3 , y 2 = x 4 + x 2 , y 2 = x 5 + x 2 . 41. Describe all curves of genus 1 on which there exists a meromorphic function of degree 3 with exactly three critical values. x3

+ y3

19.2 Exam Questions 1. Gluing two-dimensional surfaces out of polygons. 2. Coverings and ramified coverings of two-dimensional surfaces. Constructing a ramified covering from a monodromy.

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19 Exam Problems

3. 4. 5. 6.

Riemann–Hurwitz formula. Bézout’s theorem and its applications. Rational parametrization of plane curves. The genus of a smooth plane curve. The genus of a normalization of a plane curve with ordinary self-intersections. Hessian and inflection points. Hyperelliptic curves. The genus of a hyperelliptic curve. Whitney’s theorem on embeddings and immersions of curves. Meromorphic functions on curves in projective spaces. Projective duality for curves. Plücker formulas for nonsingular curves. Plücker formulas for curves with singularities. Automorphisms of the Riemann sphere. Finite automorphism groups. Elliptic curves. Mappings from elliptic curves. Moduli of elliptic curves. Cubics as elliptic curves. The j -invariant of cubic curves. Automorphisms of curves. Hurwitz’s theorem. The tangent and cotangent bundle. Differential 1-forms on curves. The dimension of the space of holomorphic 1-forms on a plane smooth curve. Integration of 1-forms. Exact 1-forms. The dimension of the space of holomorphic 1-forms on a curve with ordinary self-intersections. Residues and integrals of meromorphic 1-forms.

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

19.3 Second Term 1. Show that for any two points on an elliptic curve there is a meromorphic function of degree 2 with simple poles at these points. Express this function in terms of the Weierstrass function. 2. Show that if for two points x1 , x2 on a curve C there exists a function with simple poles at these points and no other poles, then the canonical map of C sends them to the same point. 3. Show that the image of a hyperelliptic curve of arbitrary genus under the canonical map is a rational curve. 4. Every meromorphic function of degree 2 on a complex curve determines an involution on this curve—an automorphism of order 2 permuting the preimages of every point from CP1 . The fixed points of this involution are the critical points of the function. Find the dimension of the space L(D) of meromorphic functions on a hyperelliptic curve of genus g ≥ 2 for the following cases: a) D = 2x where x is a fixed point of the involution; b) D = 2x where x is not a fixed point of the involution;

19.3 Second Term

225

c) D = x1 + x2 where points x1 , x2 constitute an orbit of the involution; d) D = x1 + x2 where points x1 , x2 do not constitute an orbit of the involution. 5. Show that every nonhyperelliptic curve of genus 5 is biholomorphic to an intersection of three quadrics in CP4 . 6. Show that every curve of genus 2 can be immersed into the projective plane as a curve of degree 4 with one ordinary double point. 7. Prove Clifford’s theorem: if a curve C is not hyperelliptic, then for every point x ∈ C the inequality l(kx) < k2 + 1 holds for k = 1, . . . , 2g − 1. 8. Find and classify all Weierstrass points on the plane Fermat curve x 4 + y 4 = 1. Use this result to find the automorphism group of this curve. 9. Find the Hilbert polynomial of the union of a line and a point in CP2 . 10. Find the Hilbert polynomial of the union of a line and a smooth quadric in CP2 that intersect transversally. 11. Find the Hilbert polynomial of the union of a line and a smooth quadric in CP2 that are mutually tangent. 12. Find the Hilbert polynomial of a twisted cubic in CP3 . 13. Find the Hilbert polynomial of a pair of intersecting lines in CP3 . 14. Find the Hilbert polynomial of a pair of skew lines in CP3 . 15. Find the dimension N of the projective space of holomorphic sections of the third power of the canonical bundle on a curve of genus g. 16. Give an example of a curve of genus g = 2 on which the 2-canonical map ϕ2K corresponding to the tensor square of the cotangent bundle is not an embedding. 17. Check that the Hilbert polynomial of a 3-canonical curve is equal to pg (n) = 3(2g − 2)n − g + 1,

18.

19. 20. 21. 22. 23.

24.

and show that every smooth curve with this Hilbert polynomial in CP5g−6 is a 3-canonical curve of genus g. Find the Hilbert polynomial of a Veronese curve in CPn−2 . Show that singular Veronese curves have the same Hilbert polynomial. Check that every curve (in fact, every subvariety) with this Hilbert polynomial that is not contained in any hyperplane is a stable Veronese curve. Draw the modular graphs of all stable curves of genus 2 and 3 without marked points. Draw the modular graphs of all stable curves of genus 2 with one marked point. Draw the modular graphs of all stable curves of genus 2 with three marked points. Is it true that the order of the automorphism group of a stable curve of genus g cannot exceed 84(g − 1)? Describe the stable limit of the one-parameter family of curves of genus g + 1 (where g ≥ 1) obtained from a given smooth curve C of genus g by gluing together a fixed point p and a variable point q as q tends to p. State and prove the Riemann–Roch formula for stable curves.

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19 Exam Problems

25. Show that the 5-canonical map of a stable curve of genus g ≥ 2 without marked points is an embedding. 26. Find the Euler characteristic of the space M 1;2. 27. Find the Euler characteristic of the space M 2;0. 28. Find the Euler characteristic of the space M 2;1. 29. Find the intersection index ψ1 ψ2 = τ12 τ03  by constructing sections of the M 0;5

corresponding bundles.  ψ12 ψ2 = τ1 τ2 τ04  by constructing sections of 30. Find the intersection index M 0;6

the corresponding bundles.  ψ13 = τ3 τ05  by constructing sections of the 31. Find the intersection index M 0;6

corresponding bundles.

19.4 Exam Questions 1. The Riemann–Roch theorem with a proof. 2. Applications of the Riemann–Roch theorem: hyperelliptic curves and curves of genus 2. 3. Applications of the Riemann–Roch theorem: Riemann’s calculation. 4. Applications of the Riemann–Roch theorem: curves of genus 3. 5. Weierstrass points on curves of genus 3 and inflection points of plane quartics. 6. Weierstrass points: weights. 7. Weierstrass points: the finiteness of the automorphism group of a curve. 8. Jacobian and theta divisor. 9. Abel’s theorem. 10. The moduli space M1;1 and the universal curve over it. 11. Moduli spaces of smooth rational curves with marked points. 12. Canonical and pluricanonical embeddings. 13. The Hilbert polynomial of a smooth curve of genus g under the canonical and pluricanonical embedding. 14. Stable curves. The genus of a stable curve. 15. The Hilbert polynomial of a stable curve under a pluricanonical embedding. 16. The Euler characteristic of moduli spaces of rational curves with marked points. 17. The cohomology of moduli spaces of rational curves with a small number of marked points. 18. The combinatorics of the stratification of moduli spaces of rational curves with a small number of marked points.

References1

1. Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of Algebraic Curves, vol. I. Grundlehren der Mathematischen Wissenschaften, vol. 267. Springer, New York (1985) 2. Arbarello, E., Cornalba, M., Griffiths, P.A.: Geometry of Algebraic Curves, vol. II. With a contribution by J.D. Harris. Grundlehren der Mathematischen Wissenschaften, vol. 268. Springer, Heidelberg (2011) 3. Clemens, C.H.: A Scrapbook of Complex Curve Theory. Plenum Press, New York/London (1980) 4. Eisenbud, D.: Commutative Algebra. Springer, New York (1995) 5. Forster, O.: Lectures on Riemann Surfaces. Springer, New York/Heidelberg/Berlin (1981) 6. Fulton W.: Algebraic Curves: An Introduction to Algebraic Geometry. Addison-Wesley, Redwood City (1969) 7. Getzler, E.: Operads and moduli spaces on genus 0 Riemann surfaces. In: The Moduli Spaces of Curves, pp. 199–230. Birkhäuser, Boston (1995) 8. Griffiths, P.A.: Variations on a theorem of Abel. Invent. Math. 35, 321–390 (1976) 9. Griffiths, P.A.: Introduction to Algebraic Curves. American Mathematical Society, Providence, RI (1989) 10. Griffiths, P.A., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1978) 11. Harris, J., Morrison, I.: Moduli of Curves. Springer, New York (1998) 12. Kapranov, M.M.: Veronese curves and Grothendieck–Knudsen moduli space M 0;n . J. Algebr. Geom. 2, 239–262 (1993) 13. Keel, S.: Intersection theory on moduli space of stable N-pointed curves of genus zero. Trans. Am. Math. Soc. 330(2), 545–574 (1992) 14. Kontsevich, M., Manin, Y.: Gromov–Witten classes, quantum cohomology, and enumerative geometry. Commun. Math. Phys. 164(3), 525–562 (1994) 15. Krushkal, S.L., Apanasov, B.N., Gusevskii, N.A.: Uniformization and Kleinian Groups [in Russian]. Novosibirsk University Press, Novosibirsk (1979) 16. Miranda, R.: Algebraic Curves and Riemann Surfaces. American Mathematical Society, Providence, RI (1995) 17. Mumford, D.: Curves and Their Jacobians. The University of Michigan Press, Ann Arbor (1974) 18. Prasolov, V.V.: Polynomials. Springer, Berlin (2004)

1 References [1, 3, 8–10, 13–15, 17, 20, 21] not mentioned in the body of the present book can be used by the reader to enhance the knowledge of algebraic curves and their moduli.

© Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2

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References

19. Prasolov, V.V.: Elements of Combinatorial and Differential Topology. American Mathematical Society, Providence, RI (2006) 20. van der Waerden, B.L.: Einfürung in die algebraische Geometrie. Springer, Berlin (1939) 21. Walker, R.J.: Algebraic Curves. Princeton University Press, Princeton, NJ; Oxford University Press, London (1950)

Index

Abelian differential, 144 of the first kind, 144 of the second kind, 144 of the third kind, 144 Abel–Jacobi map, 141 Abel–Jacobi theorem, 141 Abel’s theorem, 141 Affine coordinates, 2 Astroid, 222 Automorphism of a curve, 71

Complex structure, 33 Conic, 178 Connected sum, 4 Conormal unfolding, 60 Contact bundle, 211 Convex curve, 60 Cotangent vector, 92 Covering, 6 Cross ratio, 40 Cubic, 77

Bézout’s theorem, 22 Biholomorphic mapping, 34 Bolza surface, 89 Brill–Noether duality, 125 Brill–Noether matrix, 154

Degree of a bundle, 105 of a covering, 6 of a curve, 13 of a divisor, 105 of a linear system, 111 of an algebraic curve, 52 Deligne–Mumford theorem, 198 Deltoid, 67 Derivation, 92 Differential form, 92 Divisor, 104 of a section of a line bundle, 103 of poles, 103 of zeros, 103 Double point, 16, 194 Double tangent, 38 Dual curve, 60 Dual vector space, 59

Canonical class, 106 Canonical curve, 109 Canonical divisor, 106 Canonical map, 109 Cayley’s theorem, 26 Characteristic class, 202 Chart, 2 Chern classes, 204 Chern roots, 204 Chow ring, 202 Classifying space, 162 Class of a plane curve, 61 Clifford’s theorem, 132 Closed surface, 3 Complete linear system, 110 Complex projective space, 2

Effective divisor, 109 Elementary differential, 144

© Springer Nature Switzerland AG 2018 M. E. Kazaryan et al., Algebraic Curves, Moscow Lectures 2, https://doi.org/10.1007/978-3-030-02943-2

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230 Elliptic curve, 43, 45 Elliptic function, 117 Euler characteristic, 4 Euler class, 203 Euler identity, 15 Exact form, 95

Family of stable curves, 198 Fermat curve, 35 First Chern class of a line bundle, 202 Forgetful map, 174

Gap, 132 Gelfand–Leray form, 98 Generic divisor, 153 Generic object, 154 Geometric genus, 195 Gromov–Witten invariants, 218

Handle, 4 Hessian, 41 Hilbert–Mumford numerical criterion, 172 Hilbert polynomial, 169 Hilbert scheme, 170 Holomorphic mapping, 34 Homogeneous coordinates, 2 Hurwitz’s theorem, 87 Hyperelliptic curve, 43, 48 Hyperelliptic involution, 44 Hyperplane divisor, 105

Immersion, 55 Induced orientation, 6 Inflection point, 38, 41 Involution, 44 Irreducible algebraic curve, 14 Irreducible component of a nodal curve, 194 of a singular curve, 180 Isomorphism of curves, 34

Jacobian, 140 Jacobi’s inversion theorem, 141 j -invariant, 40, 81

Klein modular group, 158 Klein quartic, 88 Kontsevich stable map, 216

Index Leibniz rule, 92 Lemniscate, 30 Lifting of a 1-form, 95 Linearly equivalent divisors, 104 Linear series, 110 Linear system, 110 Line bundle, 92, 107 Local branch of a curve, 68 Local coordinate, 33 Locally irreducible curve, 68

Macbeath surface, 89 Meromorphic function, 46 Modular curve, 158 Modular function, 82 Modular graph, 180, 196 Modular group, 75 Modularly stable curve, 180 Moduli space, 164 Monodromy, 7 Motivic Euler characteristic, 185 Motivic Poincaré polynomial, 182 Multiplicity of a singular point, 17

Nodal curve, 194 Node, 194 Non-gap, 132 Normalization of a curve, 56 of a nodal curve, 194 of a singular curve, 180 Normalized period matrix, 147 Normal Weierstrass point, 133

Ordinary inflection point, 41 Orientable surface, 4 Orientation, 3

Pencil of curves, 25 Period lattice, 140 Period matrix, 147 Picard group, 104 Picard variety, 141 Plane algebraic curve, 13 Plücker formulas, 66 Pluricanonical map, 109 Poincaré polynomial, 182 Point at infinity, 2 Point of tangency, 19 Points in general position, 23, 178

Index Poncelet’s closure theorem, 86 Principal congruence subgroup, 88 Principal divisor, 105 Principal part of a function, 114 Projection formula, 207 Projective space, 2 Projective transformation, 3 Quantum multiplication, 218 Ramification index, 10, 45 Ramified covering, 8 Rational curve, 2 Rational nodal curve, 180 Rational normal curve, 54, 179 Rational parametrization, 28 Real part of a real curve, 13 Reducible algebraic curve, 14 Residue, 102 Resultant, 22 Riemann–Hurwitz formula, 9 Riemann’s bilinear relations, 145 Riemann’s inequality, 125 Riemann sphere, 2 Riemann’s theorem, 115 Riemann surface, 34 Schoeneberg’s lemma, 138 Segre class, 208 Smooth curve, 14, 52 Sphere with handles, 4 Splitting principle, 205 Split vector bundle, 204 Stabilizer, 87 Stable curve, 193

231 Stable curve with marked points, 195 Stable point, 172 Symmetric power, 150 Symplectic matrix, 147

Tangent line, 16 Tangent vector, 91 Tautological line bundle, 106 Theta bundle, 156 Theta divisor, 155 Theta function, 156 Torus, 4 Total Chern class, 204 Transition function, 107 Transversal intersection, 19 Triangulation, 4 Twisted cubic, 54, 179 Two-dimensional surface, 3

Universal curve, 161, 163 Universal elliptic curve, 160

Vector field, 92 Veronese curve, 179 Veronese embedding, 18, 178

Weakly modular function, 82 Weierstrass function, 78, 117 Weierstrass point, 133 Weight of a point, 133 Whitney formula, 205 Wronskian, 134 Wronski matrix, 134