Tangents and secants of algebraic varieties [Reprint. ed.] 9780821838372, 0821838377, 9780821845851, 0821845853

Table of contents :
Theorem on tangencies and Gauss maps Projections of algebraic varieties Varieties of small codimension corresponding to orbits of algebraic groups Severi varieties Linear systems of hyperplane sections on varieties of small codimension Scorza varieties References Index of notations.

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Recent Titles in This Series 127 F . L. Zak, Tangent s and secants of algebrai c varieties, 1 99 3 126 M . L. Agranovskii, Invarian t functio n space s on homogeneous manifolds o f Li e groups and applications , 1 99 3 125 Masayosh i Nagata , Theor y of commutative fields, 1993 124 Masahis a Adachi, Embedding s and immersions, 1 99 3 123 M . A. Akivis and B. A. Rosenfeld, Eli e Cartan (1 869-1 951 ) , 1 99 3 122 Zhan g Guan-Hou, Theor y of entir e and meromorphic functions : Deficien t an d asymptotic values and singula r directions, 1 99 3 121 LB . Fesenko and S. V. Vostokov, Loca l fields and their extensions: A constructive approach, 1 99 3 120 Takeyuk i Hida and Masuyuki Hitsuda, Gaussia n processes , 1 99 3 119 M . V. Karasev and V. P. Maslov, Nonlinea r Poisso n brackets. Geometry an d quantization, 1 99 3 118 Kenkich i Iwasawa, Algebrai c functions, 1 99 3 117 Bori s Zilber, Uncountabl y categorica l theories, 1 99 3 116 G . M. Fel'dman, Arithmeti c of probability distributions , and characterizatio n problem s on abelian groups, 1 99 3 115 Nikola i V. Ivanov, Subgroup s o f Teichmuller modula r groups, 1 99 2 114 Seizolto , Diffusio n equations , 1 99 2 113 Michai l Zhitomirskii, Typica l singularitie s o f differential 1 -form s and Pfaffia n equations, 1 99 2 112 S . A. Lomov, Introductio n t o the general theory o f singula r perturbations, 1 99 2 111 Simo n Gindikin, Tub e domains and the Cauchy problem, 1 99 2 110 B . V. Shabat, Introductio n t o complex analysi s Part II . Functions of several variables, 1992 109 Isa o Miyadera, Nonlinea r semigroups , 1 99 2 108 Take o Yokonuma, Tenso r spaces and exterior algebra, 1 99 2 107 B . M. Makarov, M. G. Golnzina, A. A. Lodkin, and A. N. Podkorytov, Selecte d problem s in real analysis, 1 99 2 106 G.-C . Wen, Conforma l mapping s and boundar y value problems, 1 99 2 105 D . R. Yafaev, Mathematica l scatterin g theory: Genera l theory, 1 99 2 104 R . L. Dobnishin, R. Kotecky, and S. Shlosman, Wulf f construction : A global shape fro m local interaction, 1 99 2 103 A . K. Tsikh, Multidimensiona l residue s and thei r applications , 1 99 2 102 A . M. II'in, Matchin g of asymptoti c expansions of solution s of boundary valu e problems, 1 99 2 101 Zhan g Zhi-fen, Ding Tong-ren, Huang Wen-zao, and Don g Zhen-xi, Qualitativ e theory of differentia l equations , 1 99 2 100 V . L. Popov, Groups , generators, syzygies, and orbit s in invariant theory , 1 99 2 99 Nori o Shimakura, Partia l differentia l operator s of ellipti c type, 1 99 2 98 V . A. Vassiliev, Complement s o f discriminants o f smoot h maps : Topolog y an d applications, 1 99 2 97 Itir o Tamura, Topolog y o f foliations: A n introduction, 1 99 2 96 A . L Markushevich, Introductio n t o the classica l theor y of Abelia n functions , 1 99 2 95 Guangchan g Dong, Nonlinea r partia l differential equation s o f secon d order, 1 99 1 94 Yu . S . Il'yashenko, Finitenes s theorems fo r limi t cycles, 1 99 1 93 A . T. Fomenko and A. A. Tuzhilin, Element s of the geometry an d topolog y o f minima l surfaces i n three-dimensiona l space , 1 99 1 (Continued in the back of this publication)

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Tangents an d Secant s of Algebraic Varietie s

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10.1090/mmono/127

Translations o f

MATHEMATICAL MONOGRAPHS Volume 1 2 7

Tangents an d Secant s of Algebraic Varietie s F. L . Za k

American Mathematical Societ y § Providence , Rhode Island

$EHOP JIA3APEBH H 3A K KACATEJItHLIE H CEKYHIH E AJirEBPAH^ECKHX MHOrOOBPA3H H T r a n s l a t e d b y t h e a u t h o r fro m a n origina l R u s s i a n m a n u s c r i p t T r a n s l a t i o n edite d b y Simeo n Ivano v 2000 Mathematics Subject

Classification.

Primar

y 1 4Jxx .

ABSTRACT. Thi s boo k i s devote d t o geometr y o f algebrai c varietie s i n projectiv e spaces . Amon g the object s considere d i n som e detai l ar e tangent an d secan t varieties , Gaus s maps , dua l varieties , hyperplane sections , projections , an d varietie s o f smal l codimension . Emphasi s i s mad e o n th e study o f interplay betwee n irregula r behavio r o f (higher ) secan t varietie s an d irregula r tangencie s t o the origina l variety . Classificatio n o f varieties wit h unusua l tangentia l propertie s yield s interestin g examples man y o f whic h aris e a s orbit s o f representation s o f algebrai c groups .

Library o f Congres s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a Zak, F . L . [Kasatel'nye i sekushchi e algebraicheskik h mnogoobrazii . English ] Tangents an d secant s o f algebrai c varieties/F . L . Za k p. cm . — (Translation s o f mathematica l monographs ; v . 1 27 ) Includes bibliographica l references . ISBN 0-821 8-4585- 3 (har d cover ) ISBN 0-821 8-3837- 7 (sof t cover ) 1. Algebrai c varieties . I . Title . II . Series . QA564.Z35131 99 3 93- 750 516.3'5—dc20 CI

2 P

C o p y i n g an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given . Republication, systemati c copying , o r multipl e reproductio n o f an y materia l i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addresse d t o th e Acquisition s Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail t o [email protected] . Copyright © 1 99 3 b y th e America n Mathematica l Society . Al l right s reserved . Reprinted b y th e America n Mathematica l Society , 2005 . Translation authorize d b y th e All-Union Agenc y fo r Authors ' Rights , Mosco w T h e America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 21 1

0 09 0 8 07 06 0 5

Contents Introduction Chapter I . Theore m o n Tangencies 1 and Gaus s Maps §1. Theorem o n tangencie 1 s and it s applications §2. Gauss maps o f projective varietie s 2 §3. Subvarieties o f comple x tori 2

5 5 0 8

Chapter II . Projection s o f Algebraic Varieties 3 §1. An existence criterio n fo r goo d projection s 3 §2. Hartshome's conjectur e o n linear normalit y an d it s relativ e analogs 4

7 7

Chapter III . Varietie s o f Smal l Codimension Correspondin g t o Orbit s of Algebrai c Group s 4 §1. Orbits o f algebrai c groups, null-forms, an d secan t varieties 4 §2. //F-varietie s o f smal l codimension 5 §3. i/F-varietie s a s birational image s of projectiv e space s 6

2 9 9 5 6

Chapter IV . Sever i Varieties 7 1 §1. Reduction t o the nonsingular cas e 7 1 §2. Quadrics on Sever i varieties 7 4 §3. Dimension o f Sever i varieties 8 0 §4. Classification theorem s 8 5 §5. Varieties of codegre e three 9 1 Chapter V . Linea r System s of Hyperplan e Section s on Varieties of 1 Small Codimensio n 1 § 1. Higher secan t varieties §2. Maximal embedding s o f varieties o f smal l codimension 1 1

05 05 2

1 Chapter VI . Scorz a Varieties 2 1 1 § 1. Properties o f Scorz a varieties 2 1 1 §2. Scorza varieties wit h S = 1 2 5 1 §3. Scorza varieties wit h 5 = 2 2 9 1 §4. Scorza varieties with 3 = 4 3 5 §5. The end o f the classification o f 1 Scorz a varieties 4 9 References 5

5

Index o f Notations 6 1

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Introduction During th e las t twent y year s algebrai c geometr y ha s bee n experiencin g a remarkable shif t o f interest fro m developmen t o f abstract theories to investigation o f concrete properties of projective varieties . Man y problems o f classical algebrai c geometr y concentrate d aroun d th e notion s o f linea r systems , projections, (embedded ) tangen t spaces , etc . B y using modern technique s i t has lately become possible to make considerable progress toward the solutio n of som e of thes e problems . Among recent achievement s in the field of multidimensional projective geometry w e mention result s o f Hironaka , Matsumura , Ogus , and Hartshorn e on formal neighborhood s and local cohomology, theorems of Barth, Goresky, and MacPherson on the topology of projective varieties, classification o f Fario varieties given by Iskovskih, Mori, and others , and various versions of Schu bert's enumerativ e geometry . On e o f th e mos t importan t result s o f th e las t decade i s th e connectedness theorem o f Fulto n an d Hanse n (cf . [26] , [27]), which i s essentially use d i n Chapte r I o f thi s book. A powerful incentiv e t o further researc h was given by Hartshorne's repor t [33] , where several conjec tures concerning properties of projective varieties of small codimension wer e put forwar d (som e of them ar e proven i n this book), and the relationship o f this circl e o f problem s wit h loca l algebr a an d th e theor y o f vecto r bundle s was considered. Th e main thrust of Hartshorne's conjectures i s that the lower the codimensio n o f a nonsingular projectiv e variet y X n c P ^ th e close r ar e its propertie s t o thos e o f complet e intersection s (som e result s o f thi s typ e were known before , e.g. , Bart h an d Larse n showe d tha t fo r i n). The n r < m - n . The theorem on tangencies allows us to study the structure of higher Gauss maps (cf . § 2 of Chapte r I) . For example , fo r a nonsingular variet y X n c P ^ the Gaus s ma p y:X -* G(N , n) associatin g t o a point x e X th e poin t i n the Grassman n variet y correspondin g t o th e embedde d tangen t spac e T x x is finite (i n th e cas e when th e groun d field has characteristi c zer o y i s bira tional, s o that y: X - • y(X) i s the normalizatio n map) . Anothe r importan t application concern s dual varieties, viz., we prove the classical conjecture ac cording t o whic h fo r a nonsingula r variet y X c P th e dimensio n o f th e dual variet y X* c P^ * i s no t les s than th e dimensio n o f X. Applyin g th e theorem o n tangencies t o hyperplanes , w e obtain som e results strengthenin g classical Bertin i theorems ; fo r example , fo r a nonsingular variet y X n c P ^ each hyperplane section is reduced for N 4 , wit h hn'°(X) < 1 , Fujit a [24 ] used result s o n relativ e secan t varietie s fro m § 1 of Chapter I i n th e classificatio n o f threefold s wit h smal l secan t varieties , an d Lange [52 ] deduced th e well-known Nagat a theore m o n minima l section s o f ruled surfaces fro m Exampl e 1 . 6 in Chapter V. Applications have been foun d even outside algebrai c geometry, e.g. , i n codin g theory (Skorobogatov) . The book consists of this introduction an d si x chapters divided into nine teen sections . W e turn t o a detailed descriptio n o f th e content s o f the book . Chapter I is devoted to a study of tangent space s to subvarieties of projec tive spaces and comple x tori . In § 1 we show that i f Y i s an irreducibl e r-dimensiona l subvariet y o f a n irreducible variety X n c¥ N, S(Y , X) i s the closure of a union o f points of P^ lyin g on chords joining points of Y wit h point s of X , an d T\Y , X) i s the variety swep t ou t b y the limit s o f chord s (x , x) whe n x , x - > y € Y, x # x , the n eithe r

Tf(Y,X) =

S(Y,X)

or dim f{Y, X)

= r + n9 dimS(Y

, X) = r + n + 1

(Theorem 1 .4) . Thi s result ha s two important corollaries .

INTRODUCTION

5

1) A nonsingular variet y X n c P ca n b e isomorphicall y projecte d t o a projective space F M, M < In , if and only if it can be projected to ¥ M with out ramificatio n ( a specia l cas e o f thi s corollar y wa s proved b y K . Johnso n [45]). 2) (Theorem on tangencies). I f a linear subspac e L m c F N i s tangen t t o a variet y X n c P ^ alon g a close d subvariet y Y r c X n, the n r < m - n (Corollary 1 .8) . The theore m o n tangencie s ha s man y interestin g applications . A s a n ex ample, here we give one of the simples t o f the m (Corollar y 1 .1 5) . If X n c P ^ i s a nonsingula r variety , the n fo r N < 2n al l hyperplan e sections o f X ar e reduce d an d fo r N < 2n - 1 al l hyperplan e section s ar e normal. Actually all results of § 1 o f Chapter I are proved for varieties with arbitrar y singularities, and w e give examples showin g that al l our bounds ar e sharp . In § 2 we us e result s o f § 1 for th e stud y o f Gaus s map s o f projectiv e va rieties. Th e classica l Gaus s ma p associate s t o eac h poin t o f a nonsingula r real affin e hypersurfac e th e uni t vecto r o f th e externa l norma l a t thi s point . It i s convenient t o formulat e man y result s o f classica l differentia l geometr y in term s o f Gaus s maps . I f X n c A ^ (resp. , X n c P^ ) i s a n arbitrar y irreducible affin e (resp. , projective ) variety , the n i t i s natura l t o defin e th e rational Gaus s ma p y\X -~ • Gras(iV , n) (resp. , y:X --- > G(N, n)) , wher e Gras(N, n) (resp. , G(N 9 n)) i s th e Grassman n variet y o f /^-dimensiona l vector (resp. , linear ) subspace s o f a n N-dimensiona l vecto r (resp. , projec tive) space , b y associatin g t o a nonsingula r poin t x e X th e poin t i n th e Grassmannian correspondin g t o th e embedde d tangen t spac e t o X a t x. More generally , fo r a nondegenerat e (i.e. , no t containe d i n a hyperplane ) variety X w e set &>m = {(x, a) e SmX x G(N, m)\L aD T

Xx},

n Fn which i s a n isomorphis m of f X n T sx z , wher e th e tangen t spac e T sx z to SX a t z i s tangen t t o X alon g a n f-dimensiona l quadri c i n P z (cf . Theorem 2.4) . In § 3 we sho w tha t o n a n n-dimensiona l Sever i variet y X tw o distinc t |-dimensional quadric s Y x, Y 2 fro m th e famil y constructe d i n § 2 intersect along a linear subspace , an d i f dim(Y j n Y 2) > 0, the n Y { n Y 2 = P 4 . Fur thermore, w e describe th e se t of al l quadrics intersectin g a quadric Y fro m the abov e famil y alon g a fixed linea r subspac e o f maxima l dimension . Fi nally, usin g propertie s o f spino r varietie s parametrizin g linea r subspace s o f maximal dimensio n o n even-dimensiona l quadrics , w e sho w tha t th e onl y possible values fo r th e dimensio n o f Sever i varieties ar e 2 , 4 , 8 , and 1 6 (cf . Theorem 3.1 0) . In §4 we complete the classification o f Severi varieties. A detailed analysi s of the birational projection X — • P" constructe d i n §2 and the characteriza tion of HV Sever i varieties given in § 3 of Chapter II I show that eac h Sever i variety i s a n H F-variety, s o that ther e exis t exactl y fou r Sever i varieties — one in each o f th e dimensions 2 , 4, 8, and 1 6 (cf. Theore m 4.7) . It i s curiou s tha t th e classificatio n o f Sever i varietie s ove r K turne d ou t to be intimately connecte d wit h the classification o f composition an d Jorda n algebras ove r K . Mor e precisely , X n c P N i s a Sever i variety i f an d onl y if p ^ = p(3) , wher e 3 i s the Jorda n algebr a o f Hermitia n ( 3 x 3)-matrice s over a composition algebr a 21 , an d X correspond s to the cone of Hermitia n matrices of ran k < 1 (i n that case SX correspond s to the cone of Hermitia n matrices wit h vanishin g determinant ; cf . Theore m 4.8) . I n othe r words , X is a Severi variety if and only if X i s the "Veronese surface" over one of th e composition algebras over the field K (Theore m 4.9). Thus , the classificatio n of Sever i varietie s give n i n Chapte r I V allow s u s t o giv e a ne w unexpecte d proof o f th e well-know n Jacobso n theore m o n th e structur e o f compositio n algebras (cf. , e.g. , [43] , [44]). The abov e descriptio n o f Sever i varieties a s rank on e Hermitia n ( 3 x 3 ) matrices ove r compositio n algebra s immediatel y show s tha t thei r dua l va rieties ar e define d b y cubi c equation s (b y vanishing o f determinant s o f th e

12

INTRODUCTION

corresponding ( 3 x 3)-matrices) . I n § 5 o f Chapte r I V w e sho w tha t Sev en varietie s (an d thei r isomorphi c projections ) ar e almos t characterize d b y this property. Mor e precisely, we classify al l nonsingular projectiv e varietie s whose dual varieties have degree three. In Chapter V we turn to the study of higher secant varieties S X , k > 1 , defined a s th e closur e o f th e unio n o f genera l secan t /c-spaces , i.e. , linea r subspaces spanned b y general collection s of k + 1 point s o f X . In § 1 we study the ascending chai n

XcSXc--cSk°X =

F N,

where k 0 = min{k \ S X = FN} (al l inclusion s ar e strict) . Fo r a generi c point u e S kX, 0 < k < k Q, th e tangen t spac e T skx u i s tangen t t o X along a subvariet y Y u, di m Y u = S k> define d a s th e closur e o f th e se t o f points of X lyin g on general secan t /c-space s (cf . Propositio n 1 .4) . These secant defects 8 k canno t b e arbitrary . Roughl y speaking , thei r rat e of growth is at least linear. I n particular, if a variety X ca n be isomorphicall y projected t o a projective spac e P r , the n dimSX< r , 8{=S >2n + l -s, 8k>k8, 0 2 , 8 < f , and N > 2n+2-d > j(3n + 4), i.e. , n < §(A^-2), and we obtain anothe r proof o f Hartshorne' s conjectur e o n linear normalit y (Corollar y 1 .1 3) . In § 2 of Chapter V we consider th e function M(n , d) , wher e M(n , S) i s the maxima l numbe rT V for whic h ther e exist s a nonsingular nondegenerat e projective variet y X n c F N suc h tha t d i m * = n , 8{X)

= In + 1 - dimSJ T = d.

The function M i s well defined (assume s finite values) fo r al l pairs (n , 3) e Z2 suc h tha t 1 < S < n , an d i n Propositio n 2. 2 w e establis h som e basi c properties o f thi s function . In Theore m 2.3 , whose proof i s based o n the result s o f §1 , w e show tha t

M{n,S) § w e have S Z = P ^ an d eac h variet y i s extrema l (thi s clai m i s equivalen t t o Hartshorne's conjectur e o n linear normality; i t is an immediate consequenc e of inequality (j|)) . I f 8 = § , the n X i s extremal if and only if X i s a Severi variety; thi s follow s fro m Theore m 4. 7 i n Chapte r I V or fro m formul a (fl) . To obtai n a generalization o f Sever i varieties t o th e cas e when 1 < 8 < \ , we consider nonsingula r nondegenerat e varietie s X n c P ^ fo r whic h n(n + 8 + 2) + e(8-e-2) 28 We call such varieties Scorza varieties in honor o f the Italian mathematicia n Gaetano Scorza , who obtained pioneering results on linear normalizations of low-dimensional varieties (cf . Definitio n 1 . 1 i n Chapter VI). From inequalit y (Jt) i t follow s tha t al l Scorza varieties ar e extremal . Chapter V I i s devoted t o a classificatio n o f Scorz a varieties ove r a n alge braically closed field of characteristic zero. I n §1 we show that on an arbitrary Scorza variety X n c P ^ ther e is a family o f Scorza subvarietie s Y* s , where 2 < k < k 0, 8 = 8(X) = 2n + 1 - dimSX > 0 . I n particular , fo r k = 2 , Yu i s a Sever i variety , an d fro m result s o f Chapte r I V i t follow s tha t th e secant defec t 8 fo r a Scorz a variet y ca n assum e onl y fou r values , viz. , 1 , 2, 4 , an d 8 . Th e correspondin g fou r case s ar e considere d i n §§ 2 through 5 of Chapte r VI . It turn s ou t (Theore m 5.6 ) tha t th e onl y Scorz a varieties ar e Veronese varieties ^(P* ) (8=1 ), Segr e varieties F m x F m an d F m x Pm + 1 (8 = 2), Grassman n varietie s G(m ,1) (8 = 4), an d th e variety £ 1 6 c P 26 constructed i n § 2 of Chapte r H I (8 = 8). Scorz a varietie s ar e constructe d as birational image s of P * unde r rationa l map s define d b y linear system s of quadrics passin g throug h a subvariet y ^ c P n - 1 c P " 5 an d i n th e proo f o f Theorem 5. 6 w e us e th e characterizatio n o f standar d HV Scorz a varietie s given i n § 3 of Chapte r III . Thus , al l Scorz a varietie s ar e 7/F-varietie s an d are obtained b y (birationally ) projectin g Verones e varieties ^ ( P " ) . From th e result s o f Chapter s V an d V I i t follow s tha t i f I " c P r i s a nonsingular variety , the n

h°(x,&x(i))min{(m-n)(N m) + n-b1 , (r a - n + l)(iV - ra) + 1}; c) if chari £ = 0 and y m = v m°ym is the Stein factorization of the morphism ym> then v m is a birational isomorphism and the generic fiberofthe morphism ym {and y m ) is a linear subspace of P of dimension d i m ^ m - dimX ^ . PROOF,

a ) immediatel y follow s fro m Theore m 1 .7 , an d sinc e d i m ^ m = dimX + dimG(N - n - 1 , m-n1 1 = n + (m-n)(N-m), (2.3.

) )

7

a ) follow s fro m a) . b) Suppose first that ra = N -1 . I t is clear that di m y~l_{(a) < n - 1 , an d it suffice s t o verify tha t ifn-l>b + 2, i.e. , / ? >Z ? -h 3, the n fo r a genera l point a 6 X* w e have dimy^ij(a ) ^ n - 1 . Suppos e that this is not so , and let A : be a general poin t o f X . Sinc e n - 1 > b + 1 , fro m Theore m 1 . 7 i t follows tha t th e syste m o f divisor s

Y^p^^y-^ia)) ,

aeTl

X,x

is no t fixed, an d therefor e X = \J aYa, wher e a run s throug h th e se t o f general point s o f T x x . Hence , fo r a genera l poin t y e l ther e exist s a hyperplan e A c T* x x suc h tha t fo r a genera l poin t / ? G A w e hav e L

/? D

T

x,y

Butthe

n

(Tx,x,TxJc(Ayy =

Fn+\

i.e., fo r a general pai r o f point s x, y e X w e have dim(7>i;cn7Viy) = n - l . From thi s i t follow s tha t eithe r al l ^-dimensiona l linea r subspace s fro m yn(X) ar e contained in an (n+ l)-dimensiona l linea r subspace P" +1 c P ^ o r

22

I. THEORE M ON TANGENCIES AND GAUSS MAPS

they all pass through an (n - l)-dimensiona l subspace P w _ 1 c P ^ . Bu t in the first case X i s a hypersurface an d by Theorem 1 .7 , di m Y a = n - 1 < b + 1 , contrary to our assumption, and in the second case the intersection of X wit h a genera l linea r subspac e p N~n+l c F N i s a nonsingula r strang e curv e (w e recall that a projective curv e of degree > 2 i s called strange if al l its tangent lines pass through a fixed point). I t i s well known (cf . [59] , [34, ChapterlV], [39], o r [75] ) tha t th e onl y nonsingula r strang e curve s ar e conie s i n charac teristic 2. Therefore , i n th e secon d cas e X i s a quadric, an d w e again com e to a contradiction. Thus , assertion b ) hold s fo r m — N - \ (i f cha r K = 0 , then on e ca n simplif y th e proo f usin g th e reflexivit y theore m accordin g t o which (Xy = X (cf . [96])) . Next we prove assertion b) fo r m - k unde r th e assumptio n tha t i t hold s for m = k + 1 . I t i s clear that fo r genera l points a k e X* k , a k+l e X k+l w e have dim7 < d i m 7 . (2.3.2 ) If b+ 1 >k + n-N, the

n fro m th e induction hypothesi s i t follow s tha t d i m F < d i m 7 0 , the n assertio n c ) o f Theore m 2. 3 is no longer true. A s an example, i t suffice s t o conside r the hypersurface i n Pw+1 define d by the equation X^= o *f +1 = 0 (i n this case y i s the Frobenius map). Th e case of positive characteristic is treated in [50]. 2.4. COROLLARY . / / char K = 0 , I ^ c P ^ is a nonsingular variety, and N-n+l b + 3 , then n preceding one if N >2n- b - 1

P* , X n # F n, and let n = dim** , b = - 1 . In particular, for a nonsingular variety, > N - n + 1 (this bound is better than the ).

The following example shows that both bounds in Corollary 2.5 are sharp. 2.6. EXAMPLE . Le t X 0 = P 1 x P""^ 2 c p 2 w " 2 ^ 3 , n > b + 2 , an d let X be a projective con e wit h vertex P * an d base X Q. The n X n c V N , N = 2«-Z>-2, dim(SingX ) = 6 , X* = X* - X Q, an d n =n-b~l = 7V-*+l . 2.7. REMARK . I n the case when cha r if = 0 an d b = - 1 , the inequalit y n* > N - n + 1 wa s independently prove d by Landman (cf . [50]) . Anothe r proof was earlier given by the author (cf . [96 , Proposition 1 ] for n = 2 ; the general case is quite similar). 2.8. COROLLARY . Let X n c P * , X n # F n, b = di m (Sing X). Then dimy(X) > n - b - 1 . In particular, for a nonsingular variety, diray(X) = dimJf and y is a finite morphism. If in addition charA ^ = 0 , then y is a birational isomorphism (i.e., y is the normalization morphism). 2.9. REMARK . I n the case when K = C an d b = - 1 , Griffiths an d Harris [29] prove d tha t 6imy n(X) = dimA" . Differen t proof s o f finiteness of y n in this case were later given by Ein [1 8 ] and Ran [68] . I n our first proo f o f Corollary 2.8 (and Theorem 1 .7 ) we used methods of formal geometry. Sinc e related techniques are used in §3, we give this proof here.

24

I. THEORE M ON TANGENCIES AND GAUSS MAPS

As in the proof o f Theore m 1 .7 , considerin g the intersection o f X wit h a general (N-b-1 )-dimensiona l linear subspace of ¥ N w e reduce everythin g to the case when b = - 1. Suppos e that the w-dimensiona l linear subspace L corresponding to a point a L e G(N , n) i s tangent to X alon g an irreducibl e subvariety 7 , dimY > 0, i.e. , Y c y~\a L). Le t X = X, Y b e th e comple tion of X alon g Y , an d let< & = y(X), b e the formal neighborhoo d o f th e point a L i n th e variet y y{X) c G(N, n) . Sinc e X n ± P* , 6xmy{X) > 0. Hence, H°(e,ffi e) an d /f°(3£ , ^ ) D # ° ( ©, ^ ) ar e infinite-dimensiona l vector space s over the field K . On the other hand, let M c P N b e a linear subspace, di m M = N - n - 1 , M n L = 0 , an d le t 7t : Z - • P " b e the projectio n wit h cente r a t M . The n n,Y: X -• P/wy ) i s an isomorphism o f forma l spaces , and therefor e # ° ( X , d y c ,1 / / 0 ( £ , ^ £ ) , (2.9.

)

where £ = L, Y ~ P ^ ( r ) i s the completio n o f L alon g Y . Bu t b y the wellknown theorem on formal function s (cf . [31 , Chapter V], [36]), H° (£, ^f £) = K, whic h i s impossible, sinc e H° (X, r. (2. 6. r^

)

Hence, y(Y)

= y{Y n S m I ) c{ a e G(N , n) \ d i m La H L > r}=S(L, r)cG(N , n),

where S{L, r) i s the correspondin g Schuber t cel l and y\ X —> G(N , n) i s the Gauss map. Sinc e by our assumption m-r < N-n, i.e. , n + m-r < N, from (2.1 6.1 ) i t follow s tha t fo r eac h poin t y e Y n S m I ther e exist s a hyperplane M containin g L an d tangen t t o X a t y . Pu t S{M9 L, r) = {ae G(N , n) \ La e M, dimL Then S(M, L,r)cS(L, r)

a

n l > r}.

an d

dimS{M, L , r) = (r + l)( m - r) + (n - r)(N -n-l), dimS(L, r) = {r+ \){m -r) + (n- r)(N - n) , codim 5(L r ) S(M, L , r) = n - r = codim^ Y.

§2. GAUS S MAPS OF PROJECTIVE VARIETIES 2

7

Replacing r , i f necessary , b y min^ y jdi m (T X DLJ \ w e ma y assum e that y{Y)nS(M, L, r ) n S m ( 5 ( L , r) ) ^ 0 . Then dim(y(Y)f)S(M, L,r))>

dimy(7

) - codim 5(L>r) 5(Af, L , r)

= (r-f)-(n-r) =

2r-n-f, 1 (2. 6.2

)

where / i s the dimension o f a general fiber of y\ Y . O n th e othe r hand , y(Y) n S ( M , L , r) = y{{y € 7 n S mI |

7^ c M}) 1 , (2. 6.3

)

and Theore m 1 . 7 implie s tha t dim(y(r)nS(M,L,r))b + 5, n + b = 1 (mod 2) , be a general linear projectio n o f th e Grassman n variet y G( n~2+l ,1 ) i n P n ~2 ~ 5 , an d let X n c P * , N = 2n - b - 4 , b e a cone with vertex F b an d base X 0. The n XQ ~ G(*=f± i , 1 ) (cf . [33] , [38]) and dim(SingX ) = b . Furthermore, X n D Yr, wher e Y\ b < r < n , r = n (mod 2) , i s the con e with vertex P ove r Y^~b~x, an d 7 0 i s the projection o f a Grassmann subvariet y G( r~b^x , 1 ) c G(*=f±i, 1 ) . The n m = 6 + l + 2 ( r - 6 - l ) - 3 = 2 r - 6 - 4 , a n d m - r = r-b-4 n+l), then X does not contain projective hypersurfaces of dimension greater than The following example s sho w that th e bound i n Corollar y 2.2 0 is sharp. 2.21. EXAMPLES . a t ) Le t XQ l , « > fc -b 2, b e a nonsingular quadric , n N and le t X c F , N = n + 1 , b e a con e wit h verte x P 6 an d bas e X , .

I. THEORE M ON TANGENCIES AND GAUSS MAPS

28

Then di m (Sing X) = b , an d X contain s a linear subspac e Y r = P r , wher e r = b + 1 + [ n^] = [ ^ ] (cf . [28 , Volume 2 , Chapter 6] , [37]). a 2 ) Le t X 0 = P 1 x P " ^ "2 , b > 6 + 3, be a Segre variety, and let X n c P ^ , N = 2n-b-2,btd, con e with vertex P an d base X 0 . Then di m (Sing X) = b, an d X contain s a linear subspac e Y n~l = Pw _ 1. I n this case r = n - 1 = 2

"

b) I n th e assumption s o f Exampl e 2.1 9 , le t n = b + 7. The n A" " c P" +3 contains the quadratic cone Y n~ wit h vertex P whos e base is a nonsingular four-dimensional quadri c G(3 , 1 ) . Her e n - 2 = a±|±2 = ^ . Apparently, i t is hard t o construct example s of multidimensional varietie s containing a hypersurface o f dimensio n [^j 1 ] . §3. Subvarietie s o f complex tori Besides subvarieties of projective spac e there is another importan t clas s of varieties for whic h it is natural t o introduce Gaus s maps, viz., subvarieties of complex tori. Let A N b e a n ^-dimensiona l comple x torus , an d le t X n c A N b e a n analytic subset. Le t C b e the universal coverin g of A , and le t C — > A be th e correspondin g homomorphis m o f abelia n groups . Usin g shifts , on e can identif y th e tangen t spac e t o A N a t a n arbitrar y poin t z e A wit h C ^ , an d th e tangen t spac e t o I a t a poin t x e X ca n b e identifie d wit h a vector subspac e e ^ ^ c C ^ . 3.1. DEFINITION . Le t A b e a complex torus, and let Y c A b e a connected analytic subset . Th e smalles t subtoru s o f A containin g al l th e diiference s y — y * y > y £ ^ (i n th e sens e o f grou p structur e o n A), i s calle d th e toroidal hul l o f Y an d i s denoted b y ( Y) . We observe that fo r a n arbitrar y poin t y e Y w e have Y c y + (Y). 3.2. LEMMA . Let Y c A N be a connected compact analytic subset whose tangent subspaces at smooth points are contained in a vector subspace C m c C*. Then di m ( Y) < m . PROOF. I t i s easy to se e that ther e exis t a n Af-dimensiona l toru s A an d an m-dimensiona l subtoru s T m c A N, T m D Y suc h tha t A i s locall y isomorphic to A i n a neighborhood o f Y . I t is clear that i n a suitable neighborhood o f T i n A , an d therefor e i n sufficientl y smal l neighborhoods o f Y in A an d Y i n A, ther e exist N-m analyticall y independen t holomorphi c functions. O n th e othe r hand , fro m [5 ] an d [36 ] i t follow s tha t i n a smal l neighborhood o f Y i n A ther e exis t exactl y di m A - di m ( Y) analyticall y independent holomorphi c functions . Henc e dim ^ - di m (Y) > N-m, i.e. , dim (Y) < m . D 3.3. DEFINITION . Le t X n c A N b e a n ^-dimensiona l analyti c subse t o f an N-dimensiona l toru s A , and let Y r b e a n r-dimensiona l analyti c subse t

§3. SUBVARIETIE S OF COMPLEX TORI

29

of X . W e sa y tha t a vecto r subspac e C m c C i s tangen t t o X alon g a n analytic subset Y c X i f C m D 0^ y fo r al l y eY. 3.4. LEMMA . Let X n c A N be an analytic subset, and let C m c C ^ be a vector subspace that is tangent to X along a connected compact analytic subset Y r c X n. Then there exist an 1 SI-dimensional complex torus A , an m-dimensional complex subtorus T m c A N, T m D Y r, neighborhoods UcAN, UDY, U + (Y) A = U, UCA N, UDY, U+(Y) U,and A= an analytic subset X c TnU, such that U ~U, (Y) A ~ (Y) A = (Y) T, and X ~ X n U, and the mappings U -^ U, T ^ A, and X n U A/(Y)A, nT:T^T/(Y)T. From Lemm a 3. 4 it follow s tha t X'v = n(X n [ / ) - ft(X) ~ n

T(X),

where n(Y) = yeX'ucXf =

n(X).

f v

In particular , th e neighborhoo d X o f th e poin t y i n X 1 embed s a s a n analytic subse t i n the ( m - A:)-dimensiona l toru s B = T/{Y) T. W e put

x = n~\xf) CA, x

v

= xnu = 7c_1(^).

Then X v i s th e desire d analyti c subse t o f U and , fo r a n arbitrar y poin t z e XJJ , the tangent spac e to X v a t z ha s dimension no t exceedin g m an d is tangent t o X alon g the analyti c subse t X n n~\n(z)) =

I n (z + ( Y)A). •

30

I. THEORE M ON TANGENCIES AND GAUSS MAPS

3.6. COROLLAR Y (Theore m o n tangencie s fo r subvarietie s o f comple x tori). Let X n c A N be an analytic subset of a complex torus, and let C m c C ^ be a vector subspace that is tangent to X along a compact analytic subset Yr c X n . Then r < km> where k m is the maximal dimension of a complex subtorus C c A such that di m (X + C) < m . 3.7. REMARK . I n contras t t o th e cas e o f subvarietie s o f projectiv e space s (cf. Corollar y 1 .8) , i n Corollar y 3. 6 w e d o no t assum e tha t X i s nonde generate (a n analyti c subse t X c A i s calle d nondegenerate i f (X) = A ). However, i f dim Z < m , the n k > dim (X) > n an d Corollar y 3. 6 is trivial. Let X n c A b e an analytic subset of a complex torus, let n < m < N- 1 , and le t &>= {(x, a) e SmJi T x Gras(7V, m) \L aD@Xx], where Gras(iV , m) ~ G(N - 1 , m - 1 ) i s th e Grassman n variet y o f mdimensional vecto r subspace s i n C , L™ c C i s the vecto r subspac e cor responding t o a point a e Gra s (TV, m), an d the bar denote s closure in X x Gras(JV, m) . W e denot e b y P m'^m - • X (resp. , y m'^m - > Gras(iV , m) ) the projection ma p t o the first (resp., second) factor . 3.8. DEFINITION . Th e mappin g y m i s calle d th e mt h Gaus s map , an d its imag e X* m = y m{^m) c Gras(iV , m) i s calle d th e variet y o f tangen t m spaces to th e variety X . In particular , fo r m = n w e obtai n th e usua l Gaus s ma p y: X —- • Gras(N,n), an d fo r m = N - 1 w e get a map y N_{: ^ j / -> P ^ - 1 . 3.9. PROPOSITION . Let X n c A N be an irreducible compact analytic subset. Then there exists an analytic subtorus C c A such that (i) X + C = C; (ii) y = y'°7t\ x, where n: A -> B, B = >4/C , w ^Ae canonical holomorphic map and y : X — • Gra s (TV, «), y ; : X 7 —> Gras (N - k, n - k), X' = 7r(X ) C 5, flre ^ e Gauss maps; (iii) f/z e ma/? y : A"' —> y^X') c Gras (N - k 9 n - k) isgenericallyfinite. PROOF. Arguin g b y induction , w e assume tha t Propositio n 3. 9 i s alread y verified fo r N f < N an d prov e i t i n th e cas e di m A = N . I f th e ma p y is genericall y finite, the n i t suffice s t o pu t C = 0 , X' = X . Suppos e tha t for a genera l poin t x e X w e hav e dimy~ 1 (>'(x)) > 0 , an d le t Y b e a positive-dimensional componen t o f y~ l{y(x)). B y Lemm a 3.2 , 0 < k = dim (7) < n. Sinc e a continuou s famil y o f comple x analyti c subtor i o f A is constant, w e conclude tha t i f x i s another genera l poin t o f X an d Y i s a positive-dimensional componen t o f th e fiber y~ [{y{x)), the n ( Y) = ( Y) . We put c=

(r) , x

B = A/C, x

'=

TT(^)C5 ,

=n{Y) = n(x).

§3. SUBVARIETIE S OF COMPLEX TORI

31

Since the tangent spac e to X i s constant alon g Y n Sm X an d th e kernel o f the differentia l d x (7t\ x) coincide s wit h ® n-hx>\ x , w e se e tha t Y lie s i n a fiber of th e Gaus s ma p fo r th e subvariet y n~ (x) c C . Bu t Y span s C and dim C < n < N (otherwis e Y = X = C = A an d Propositio n 3. 9 i s obvious), so that fro m th e inductio n hypothesi s i t follow s tha t Y = C . Thus, a genera l fiber an d therefor e eac h fiber o f th e ma p n\ x coincide s with th e correspondin g fiber o f th e ma p n : A — • B; moreover , X + C = .Y an dJS T i s a locall y trivia l analyti c fiber bundl e ove r X' wit h fiber C . Furthermore, SingX = 7r" 1 (SingX / ),

where C c C ^ i s the universal coverin g of C an d y = y'°n\ x • Q 3.10. COROLLARY . Le / X " c A N be a compact complex submanifold. Then the Gauss map y: X — • Gras(iV, n) can be represented in the form y = yon, where n: X — • X' is a locally trivial analytic fiber bundle whose fiber is a complex subtorus C c A N, X 1 is a compact complex subvariety of the torus B = A/C, and the Gauss map y : X1— • Gra s (N -k 9 n -k) is finite. In particular, if X does not contain complex subtori (e.g., if A is a simple torus), then the Gauss map y is finite. PROOF. Corollar y 3.1 0 i s an immediat e consequenc e o f Theore m 3. 5 an d Proposition 3.9 . D Our result s als o allo w u s to describ e th e structur e o f Gaus s map s y m fo r arbitrary nN-\, where e(X) is the (topological) Euler-Poincare characteristic of X and Q x is the sheaf of differential forms of rank one. a ) (i) (iii) i s obvious ; an d (iii ) => (ii) follows fro m th e fac t tha t fo r m < N - 1 th e fibers o f y m (or , mor e pre cisely, their projection s t o X) ar e containe d i n th e fibers of y N_l. b) Fro m th e descriptio n o f th e ma p y N_x give n i n 3.1 3 i t immediatel y follows tha t PROOF,

degjv-, = c n ( e ; ) = c n (il lx) = {-\) ncn{X) =

\e{X)\.

In [55 , 3.1 ] i t i s shown that i f Y i s a complex manifol d an d n: Y — • F i s a finite covering o f degre e < k - 1 , the n Pi c Y = Z. T o verify b ) i t suffice s to put k = J V - 1 , Y = P W^A/X) anc * to observe that fo r J V - n - 1 > 0 w e have r k (Pic (P ( ^ / ^ ) )) > 2 . • We observ e tha t i n vie w o f Corollar y 3.1 2 assertion s (i)-(iii ) hol d i n th e case when A i s a simple torus. 3.15. PROPOSITION . Let X n c A N be an analytic submanifold. Then the canonical linear system \K X\ is base point free, and its suitable multiple defines a holomorphic mapping n: X — • X' making X a locally trivial analytic fiber bundle over a complex manifold X' ; the fiber of n is the maximal analytic subtorus C c A for which X + C = X (where X 1 embeds isomorphically in B = A/C). PROOF. I n view of the above description of a Gauss map (cf . 3.1 3) , Proposition 3.1 5 immediately follows from Corollar y 3.1 0 and from Corollar y 6.6. 3 in [30 , Chapter II].

3.16. COROLLARY . Let X n c A N be a nondegenerate complex submanifold (i.e., (X) = A). Then there exists an analytic subtorus C c A such that if n: A-> B = A/C is the projection map, then (i) n\ x: X - * X' c B is a locally trivial analytic fiber bundle with fiber C (so that X = n~ l(X'))\ (ii) the mapping n\ x is equivalent to the mapping defined by a sufficiently high multiple of the canonical class K x ; (iii) the canonical class K x, is ample; (iv) B = (X f) is an abelian variety.

34

I. THEORE M ON TANGENCIES AND GAUSS MAPS

3.17. COROLLARY . An analytic submanifold X n c A N is a variety of general type {i.e., the canonical dimension of X coincides with its dimension) if and only if the canonical class K x is ample. 3.18. REMARK . Fro m Corollar y 2. 8 i t follow s tha t fo r a nonsingula r va riety X n ^ P n ove r a n algebraicall y close d field o f characteristi c zer o th e Gauss map y i s birational, and , accordin g to Remar k 2.1 5 , the map define d by th e complet e linea r syste m \K x + {n+\)H\> wher e H i s a hyperplan e section o f X , i s a n isomorphism . However , fo r submanifold s o f comple x tori the map y an d the canonical map defined b y the complete linear syste m of canonica l divisor s ca n b e finite map s o f degre e greate r tha n one . A s a n example, it suffice s t o conside r a hyperelliptic curv e X o f genu s g > 1 em bedded in its Jacobian variety J x . In this case the Gauss map coincides with the canonica l ma p whic h clearl y ha s degre e tw o (i t i s clea r tha t th e norma l bundle JVJ /X i s ample , an d al l th e Gaus s map s y m,1 < m < g - 1 , ar e finite; cf. Propositio n 3.1 4) . I n [83 ] it i s shown that , give n the condition s o f Proposition 3.1 4 , degy ^ < \e{X)\/{N - n) . 3.19. REMARK . Th e stud y o f submanifold s o f comple x tor i wa s begu n by Hartshorn e [32 ] an d continue d b y Sommes e [84] , who reveale d th e rol e of complex subtor i using the notion o f fc-ampleness. A t the same time Uen o [93, § 10] undertook a thoroug h investigatio n o f propertie s o f th e canonica l dimension o f submanifold s o f comple x tor i (hi s result s easil y follo w fro m ours, bu t ar e state d i n differen t terms ) an d announce d i n [92 ] ou r Corol lary 3.1 7 , but his proof turne d ou t to be erroneous (cf . [93 , 10.13]). Griffith s and Harri s [29 , §4b) ] showe d tha t th e ma p y fro m ou r Corollar y 3.1 0 i s generically finite, and , o n th e basi s o f thei r result , Ra n [68 ] gave a differen t proof o f Corollar y 3.1 7 an d o f Propositio n 3.2 0 belo w i n th e cas e c = 0 . The following two results are analogs of Proposition 2.1 6 for submanifold s of comple x tori . 3.20. PROPOSITION . Let X n c A N be a complex submanifold, and let Y c X n be a complex subtorus. Then r < ["^-nlV] > where c is the maximum of the dimensions of complex subtori C c A such that X + C = X {this bound is nontrivial for c < 2n - N - 1 ) . In particular, if X is a hyper surface {i.e., n = N—l) containing a complex subtorus Y r of dimension r > n/2, then X is a locally trivial analytic bundle whose fiber is a complex torus and whose base is a hypersurface in a complex torus of smaller dimension. r

PROOF.

I t i s clea r tha t fo r a n arbitrar y poin t y e Y w e hav e Q x D

&Y = C r , wher e C r c C Hence, yx{Y) cS

Y

=

N

i s th e universa l covering o f th e toru s Y .

{a € Gras(Af, n)\L aD C

},

§3. SUBVARIETIE S OF COMPLEX TORI 3

5

and, by Corollar y 3.1 0 , r - c< di m yx{Y) < dimSY = di m (Gras (N - r , n- r)) = {n - r)(N - n) , which implies th e assertion o f the proposition. • 3.21. PROPOSITION . Let X n c A N be a complex submanifold, and let Y c X n be an analytic subset for which di m {Y) = m, where m - r = codim^ y) Y < codim^ X = N - n. Denote by d the maximal dimension r

of complex subtori D c A for which X + D ^ A. Then r < | particular, if A is a simple torus t then r < [ | ] .

^1. In

Propositio n 3.2 1 ca n b e prove d i n essentiall y th e sam e wa y a s Proposition 2.1 6 . I n the notation s correspondin g t o those of 2.1 6 w e have PROOF.

dim(y(Y)nS(M, L,

r)) > dimy(Y) - codim S{Lr) S(M, = {r-f)-{n-r) =

L , r)

2r-n-f, 1 1 (3.2 .

)

where / i s the dimension o f a general fiber of y\Y (compar e with (2.1 6.2)) . On th e othe r hand , fro m Corollar y 3. 6 i t follow s tha t dim(y(Y)nS(M,L,r)) X

T1

2

yy

Y,Xty^ 2

T X ty

(cf. [45]) . 1.2. DEFINITION . Le t / : X - » X 1 b e a morphis m o f algebrai c varieties . We sa y tha t / i s unramifie d i n th e sens e o f Johnso n (J-unramified ) wit h respect to Y c X a t a point y e Y i f the morphism d vf\a, i s quasifinite. If / i s J-unramified wit h respect to Y a t all points y e Y , the n we say that / is J-unramified wit h respec t t o Y . I f moreove r Y = X , the n th e morphis m / i s called J-unramified . 1.3. DEFINITION . I n th e notatio n o f Definitio n 1 . 2 w e sa y tha t / i s a n embedding i n th e sens e o f Johnso n (J-embedding ) wit h respec t t o Y c X 37

38

II. PROJECTION S O F ALGEBRAI C VARIETIE S

if / i s J-unramified wit h respec t t o Y an d i s one-to-one o n / " (f(Y)) . I f moreover Y = X, the n th e morphis m / i s called a J-embedding . 1.4. REMARK . I f X i s nonsingula r alon g 7 , i.e., Y n SingX = 0 an d Y C Sm X, the n / i s unramifie d wit h respec t t o Y i f and onl y i f / i s unramified a t al l point s y e Y; / i s a J-embedding wit h respec t t o Y if and onl y if / i s a closed embeddin g i n som e neighborhood o f Y in X. 1.5. PROPOSITION . Let X n c¥N be a projective algebraic variety, let Y r c Xn be a nonempty irreducible subvariety, let L N~m~l c P ^ , LnX = 0, be a linear subspace, and let n : X -» P m & e */ze projection with center in L. a) The following conditions are equivalent (i) The morphism n is J-unramified with respect to Y ; (ii) LnT'(Y,X) = 0. b) The following conditions are equivalent: (i) The morphism n is unramified at the points of Y; (ii) LnT(Y,X) = 0. c) The following conditions are equivalent: (i) The morphism n is a J-embedding with respect to Y ; (ii) LnS(Y,X) = 0. d) The following conditions are equivalent: (i) The morphism n is an isomorphic embedding', (ii) LnS(Y,X) = LnT(Y,X) = 0. PROOF. Mos t o f th e assertion s o f th e proposition ar e obvious . T o verif y a) it suffices t o us e th e fac t tha t 7rL , i s quasifinit e if f 7tL, i s finite 1 it9

Y,X,y

or equivalentl y L n T' Y X = with vertex y) . •

0 (w e recall that T'

Y,X,y

YX

i

s a projective con e

1.6. PROPOSITION , a ) In the conditions of Proposition 1 .5 , suppose that the morphism n: X n— • Pm is J-unramified with respect to an irreducible subvariety Y r c Xn , where m N — n-r). Then n is a J-embedding with respect to Y. b) Under the conditions of Proposition 1 .5 , suppose that the morphism n: X n - • Pw is unramified at all points y e Y r, where Y r c A ^ is an irreducible subvariety and m < r + n (i.e., dim L >N -n-r). Then n is an isomorphism in a neighborhood of Y. PROOF.

I n view o f Propositio n 1 . 5 a), ou r conditio n mean s tha t f Lf)T 1 1 (Y,X) = 0. ( .6.

)

dim T f(Y, X) < codim^L 1 < r + n. ( .6.2

)

Therefore, In view of Theore m 1 . 4 o f Chapte r I , fro m (1 .6.2 ) i t follow s tha t S(Y1 9X) = T'(Y,X). ( .6.3

)

§ 1. AN EXISTENCE CRITERION FOR GOOD PROJECTIONS 3

9

In vie w o f Propositio n 1 . 5 c), assertio n a ) o f Propositio n 1 . 6 no w follow s from (1 .6.1 ) an d (1.6.3). b) According to Proposition 1 . 5 b), ou r condition mean s tha t 1 i n r ( y , i ) = 0. ( .6.4

)

Therefore, dim T\Y, X) < dim T(Y, X) < codimp* L P m be the projection with center at L . Suppose that m < In - 1 . Then a) the morphism % is J-unramified if and only if n is a J-embedding; b) the morphism n is unramified if and only if n is an embedding. 1.8. REMARK . Corollar y 1 . 7 wa s proved b y Johnso n [45 ] by mean s o f formal computation s involvin g characteristic classes under the additional assumption N < In. Fo r nonsingular varietie s Corollar y 1 . 7 wa s proved i n [26], an d the general cas e was settled i n [27 , § 5] and [97, § 2] (in these papers th e authors actuall y conside r variou s specia l case s o f Theore m 1 . 4 of Chapter I for Y = X). 1.9. LEMMA . Let I c P ^ be a nondegenerate variety, x e X, y e F N, y ^ x, z e (y, x), z # y . Then b) T

S{yiXhzD(y,TxJ;

c d

) yxXxz TxJ > where S{y 9 X) is the cone with vertex y and base X (c/ T §1 of Chapter I), d(p y is the differential of the map (py , and the bar denotes closure in the Zariski topology. PROOF,

a ) It is clea r tha t T

s(ytXhy

D S(y, X) D X. Therefore , {X) c

Ts, x) . Bu t by definition fo r a nondegenerate variet y X c P w e have (X) = P *. b) I t suffices t o conside r th e affin e case . Furthermore , w e may assum e that y coincide s with the origin. Sinc e the affine par t o f S(y , X) i s a cone, the (embedded ) tangen t space s a t the points z an d // z (ju e K* = K \ 0) coincide wit h eac h othe r an d contain th e origin . T o verify b ) it suffices t o choose JLL s o that JLLZ = x .

40 II

. PROJECTIONS O F ALGEBRAIC VARIETIE S

c) A s i n th e proo f o f b) , w e conside r th e affin e cas e an d assum e tha t y coincides wit h th e origin . The n th e restrictio n o f p\ o n th e affin e par t o f S x admit s a section a , a x

( ') = iy 9 x ' 9 ^ x) 9 x

e X, z = Ax

and "yxxxz? \®S

Vx "y,x>

,yxxxz)

-\dyxxxz(Py(@(py2y\yxx),yxxxz)> d

= (o,x,eXXz) =

yxxxz(P (

(o,e

@

CJ(X) ,yxxxz) /

xj

which implie s c) . D 1.10. PROPOSITION , a ) Let y

e Y c X c P ^ , xeX,

x

# y, z

e {y , x).

Then T S(YtX)tZD{TYty9TXtX). b) Suppose in addition that cha r A' = 0 . Then for general points y e Y, xeX, ze(y,x) we have T S{YfX)>z = (T Yy, T Xx). PROOF, a

) I t is clear tha t S(Y, X) D S(y,X), S(Y,

X) D S(x, Y).

Therefore, Ts{Y,X)9z D

YS{y,X),z> ^S(x,Y),z) 1 1 1 ' ( . 0.

)

According to statement s a ) an d b ) o f Lemm a 1 .6 , *S[y,X)tz D *S(x,Y),z

*X,x '

D

(1.10.2)

*Y,y

Assertion a ) immediatel y follow s fro m (1 .1 0.1 ) an d (1 .1 0.2) . b) Firs t o f al l we observe that i f z e {y , x) , the n i n the notatio n o f § 1 of Chapter I SYX,yxxxz \

) l (y),yxxxz9 {pi)

^y

= /©0 \S

1 \x),yxxxz/ 1 ( 03

)

\ ,yxxxz '

x

S

Yx

,yxxxz j

'

From Lemm a 1 . 9 c) i t follow s tha t

If cha r A: = 0 , the n th e ma p Y "yxxxzV '

®S Y x ,yxxxz ~*

is surjectiv e fo r a genera l poin t yxXxz e from (1 .1 0.3 ) an d (1 .1 0.4) . D

®S(Y,X),z

S

Y x

, an d assertio n b ) follow s

§ 1. AN EXISTENCE CRITERION FOR GOOD PROJECTIONS 4 1

1.11. REMARK . Th e arguments used in the proof of Lemma 1 . 9 also work in a slightly more general situation. I n particular, in Chapter IV we shall need the followin g varian t o f Lemm a 1 . 9 b): for x e SmX we have T z . x , D Tx x (w e recal l that , i n th e notatio n o f § 1 o f Chapte r I , T z s , x) i s th e tangent con e to S(y , X) a t the point z) . Similarly , i f unde r th e condition s of Proposition 1 .1 0 x, y e S m I , the n T z 5 ( y x) D T Y y , T x x . 1.12. REMARK . Robert s showe d (cf . [70] ) tha t fo r eac h prim e numbe r p > 0 ther e exist s a n irreducibl e (singular ) curv e X c F K, cha r K = p , such tha t fo r Y = X an d a genera l poin t z e P 3 = SX th e inclusio n i n statement a ) o f Propositio n 1 .1 0 i s strict . A n exampl e o f suc h a curv e i s given by the projectiv e closur e o f th e imag e o f th e affin e lin e A ^ unde r th e 2

embedding t «-• (t, f , f

) .

1.13. THEOREM . Let X n c F N be a projective variety, and let Y r c X n be an irreducible subvariety. Consider the following conditions: a) For an arbitrary linear subspace L N~m~l Q f> N the projection X —• Pm with center at L is a J-embedding with respect to Y ; b) There exists a linear subspace L c P such that the projection X— > F m with center at L is a J-embedding with respect to Y ; c) dimS(Y,X) \(r - b) + 1 , where b = dim (Y n SingZ) . Clearly , i t suffice s t o conside r th e cas e when Y i s irreducible. I t follows fro m Propositio n 1 .5c ) tha t S{Y , X) / P ^ . Le t s = dimS{Y,X) , and let z b e a general point o f S(Y, X) . I n the notation o f § 1 of Chapter I we put PROOF.

^=

Y Y Z=P X({V )-\Z)).

V , * M >Q

It follow s fro m Theore m 1 . 4 o f Chapte r I that eithe r t{Y 9 X) or s = n + r + 1 . I n th e latter cas e N>s+\ =

= S(Y, X)

n + r + 2,

n

c o d i n g X = N - n > r + 2 > \{r - b) + 1 . Thus, we may assume tha t T'(Y,X)=S(Y,X), Q It i s clear tha t LDT

xtX>

ZJ^0,

dimQ

z

= r + n-s.

V x € Q1 \ S i n g * . (2. . z1

)

§2. HARTSHORNE' S CONJECTURE ON LINEAR NORMALITY 4

Let M c and let

3

P b e a general linea r subspac e of codimension b + 1 , x' = XnM, Q' l

Y = YHM, L'

2

=Q

znM,

=LHM.

Then the variety X' c F N * is nonsingular alon g Y' , and from (2.1 .1 ) it follows tha t T\Qiz,X,) = nQ! ziX')cL\ However, it is clear that X 1 2rc+r-&+2,

i.e., codimp* Xw =TV - n > ^(r - b) + 1. D 2.2. COROLLARY . Let Y r c X n c F N, where X is nonsingular in a neighborhood of Y. Suppose that there is a point u e F N \X such that the projection n\ X— • FN~l with center at u is a closed embedding in a neighborhood of Y. Then N > n + j(r+3). 2.3. REMARK . Propositio n 1 . 6 a) implies that for N < n+r Theore m 2.1 (resp., Corollar y 2.2 ) remain s tru e i f instead o f assuming tha t n i s a J embedding wit h respec t t o Y (resp. , a n embeddin g i n a neighborhood o f Y) w e assume tha t n i s J-unramified wit h respec t t o Y (resp. , unramifie d at all points y e Y). The followin g example s sho w that th e bounds in Corollary 2. 2 and Theorem 2. 1 are sharp. 2.4. EXAMPLES . T O simplify arguments , i n the followin g example s w e assume that cha r K = 0. a) Let X 2 c P 4 b e the rational surfac e ¥ x o f degree three. Le t r 1 c X2 be the minimal section , s o that Y i s an exceptional curv e o f the first kind. Then 7 c P 4 i s a projective line , an d the embedding X P 4 i s defined by th e complete linea r syste m \Y + 2F\, wher e F c X i s a fiber of the

44

II. PROJECTION S OF ALGEBRAIC VARIETIES

ruled surfac e ¥ x. Sinc e F c P i s a projectiv e line , th e tangen t plan e a t an arbitrar y poin t o f X contain s th e fiber passin g throug h thi s poin t an d therefore intersect s Y . Fro m Propositio n 1 .1 0 b) i t follow s tha t dimS{Y ,X) = r + n = n + ±(r+l) =

3.

Hence, 4

S(Y,X) = T(Y,X)^F

,

and b y Propositio n 1 . 5 d) ther e exist s a projectio n n: X - > P 3 whic h i s a n isomorphic embeddin g i n a neighborhoo d o f Y (i n a suitabl e coordinat e system n{X) c P 3 i s defined b y the equation u Qu23 = u xu\ ) . I n this exampl e N = 4 = n + ±(r + 3), i.e., the inequalit y i n Corollar y 2. 2 turns int o equality . b) Let X 6 = (7(4 , 1 ) c P 9 (th e Plucker embedding), and let Y = P 3 b e the linear subspace of lines passing through a fixed point of P 4 . Then for genera l points y eY, x e X, th e line T Y n T x x parametrize s lines in P passin g through a fixed point an d intersectin g a fixed line. Fro m Propositio n 1 .1 0b ) it follow s tha t dimS{Y, X) = dim T{Y, X) = 8 = r + n - 1

= n + \{r + 1 ) = N - 1 ,

so that agai n th e inequality i n Corollar y 2. 2 turns int o equality . To sho w tha t th e boun d give n i n Theore m 2. 1 i s als o shar p fo r b > -1 , it suffice s t o conside r th e con e with verte x P ove r on e o f the varieties con structed i n Example s 2.4 . Let X n c F N b e a nonsingula r variety , an d le t D m (resp. , R m) b e th e double poin t (resp. , ramification ) locu s wit h respec t t o a genera l projectio n fN __ + p m 9 n 0, then Y rnRm ^ 0 for an arbitrary subvariety Y r c X n . Suppos e tha t Y r n D m = 0 (resp. , Y r n R m = 0 ) . The n fo r a general linea r subspac e L N~m~l c P ^ w e hav e S(Y , I ) n L = 0 (resp. , T ( 7 , X ) n L = 0 ) , an d therefore d i m 5 ( F , ^ ) < m (resp. , d i m r ( F , X ) < m) (compar e wit h Propositio n 1 .5b),d)) . O n th e othe r hand , fro m Theo rem 2. 1 i t follow s tha t PROOF.

s = dimS(Y,X)>n +

±{r+ 1 )

§2. HARTSHORNE' S CONJECTUR E ON LINEAR NORMALITY

45

(cf. (2.1 .5)) , and b y Theorem 1 . 4 o f Chapte r I eithe r T(Y,X) =

S(Y,X)

or dimT(Y,X) =

r + n,

so that fo r r > 0 dimT{Y,X)>n + ±(r+l). Thus, under our assumptions, m> n + j(r+l), i.e. , r 2(r a - n) (resp. , r > ma x {1, 2( m - n)}) w e hav e Y r n D m ^ 0 (resp., r r n i ? m ^ 0 ) . D 2.6. REMARK . Corollar y 2. 5 i s nontrivia l onl y i f m < \(in - 1 ) . I t i s clear that for this corollary to hold it suffices to require only the nonsingularity of X alon g Y (w e considered the case of a nonsingular X i n order to avoi d introducing the definition o f ramification locu s and double point locus in th e general situation; cf., e.g. , [39], [45], [48]). Example s 2.4 show that the bound in Corollar y 2. 5 is sharp. 2.7. REMARK . I f m = W , the n i t i s eas y t o deduc e fro m Corollar y 2. 5 that the linear system \R n\ generate d by the ramification divisor s R L, wher e L run s through th e se t of genera l (N - n - l)-dimensiona l linea r subspace s of F N, i s ampl e o n X . Thi s resul t wa s alread y prove d i n Propositio n 2.1 2 of Chapte r I . Examples 2. 4 sho w that th e boun d give n i n Theore m 2. 1 i s sharp . How ever, i n on e importan t specia l case , viz. , whe n Y = X, thi s boun d (whic h takes th e for m n < \{2N + b - 2) , wher e b = di m (SingX)), ca n b e im proved somewhat . Thi s i s du e t o th e fac t tha t fo r Y = X th e subvariet y Qz involve d i n th e proo f o f Theore m 2. 1 ca n b e replace d b y th e subvariet y Yz = p x (cp~\z)), wher e di m Y z = dim Qz + \. 2.8. THEOREM . Let X ncPN be a nondegenerate variety, b = dim (Sing X). Suppose that there exists a point u €FN\X such that the projection n : X —• P ^ - 1 with center at u is a J-embedding. Then n < \(2N + b) - 1 {i.e., codimpAr X > j(n - b + 3)) . PROOF. A S w

e alread y pointe d out , th e proo f i s paralle l t o th e proo f o f Theorem 2.1 . I n th e notatio n o f Theore m 2.1 , from Propositio n 1 .1 0 a) i t follows tha t LDTX x

1 (2.8. z=px((p-\z)).

VxeY z\SingX, Y

)

Let M N~b~l c¥ N b e a general linear subspac e o f codimensio n b + 1 , an d let

x' = xnM9 Y

z

= YznM, L'

1

= LnM.

Then th e variet y X' c p^"^" i s nonsingular , an d fro m (2.8.1 ) i t follow s that

T'iY^X^^TiY^X^cL'.

II. PROJECTION S OF ALGEBRAIC VARIETIES

46

However, i t i s obvious that X 1 dimSX' > dim5(1^, X 1 ) = dim y j + d i m *' + 1 = ((2/ i + 1 - s) - (b + 1 ) ) + (n - b - 1 ) + 1 = 3n - s - 2b ,

i.e., 3« < 2s + 6 - 1 < 2N + 6 - 3 . • 2.9. REMARK . I n [97 ] w e gave a proo f o f Theore m 2. 8 base d o n Theo rem 1 . 7 o f Chapte r I . In th e cas e when K = C, b = - 1, Lazarsfeld showe d that Theorem 2. 8 can be derived directly from th e connectedness theore m of Fulton an d Hanse n (cf . [27 , § 7]). 2.10. REMARK . Fo r n > 1 i n the statement o f Theorem 2. 8 it suffice s t o assume that there exists a point u eP N\X suc h that the projection n : X -> FN~l wit h cente r a t u i s J-unramified. I n fact , fro m Propositio n 1 . 5 a) an d Theorem 1 . 4 of Chapter I it follows that if n i s J-unramified, the n T'X ^ F N and eithe rT V > dimSX = In + 1 , n < {{N - 1 ) < \{2N + b) - 1 , o r SX = T rX 3 u an d n i s a J-embedding, s o that i n both case s the condition s of Theore m 2. 8 are satisfied . 2.11. COROLLARY . If a nondegenerate nonsingular variety X n c F N can be isomorphically projected to a projective space P m , m < N, then n < | ( m - l ) . Ifn>\, then it suffices to require the existence of an unramified projection X-+Fm. 2.12. REMARK . Fo r m = N - I th e boun d give n i n Corollar y 2.1 1 i s sharp, bu t fo r m < N - 1 thi s boun d ca n b e improve d (cf . [1 00 ] o r Corol lary 2.1 6 i n Chapte r V) . 2.13. REMARK . Th e varietie s fo r whic h th e inequalit y i n Theore m 2. 8 or Corollar y 2.1 1 turn s int o equalit y wil l b e classifie d i n Chapte r I V (cf . Chapter IV , Theorem s 1 . 4 an d 4.7) . W e observ e tha t fro m th e proo f o f Theorem 2. 8 i t follow s tha t i f n = 5(27 V + b) - 1 , the n fo r a general poin t z e SX w e have di m (Y z n Sing^T) = b , i.e. , Y z contain s a componen t o f Sing*. 2.14. THEOREM . Let I n c P \ b = dim (Sing*), n > \{2N + b - 1 ) . Then X is not the projection of a variety of the same dimension and degree, nontrivially embedded in a projective space of larger dimension. PROOF.

Suppos e th e converse . The n ther e exis t a variet y

X' c P * ' , d i m *

' = d i m * = n, deg*

' = d e g *1 , (2. 1 4.

and a linear subspac e L c P " \ dimL

= N' -N-1

,

)

§2. HARTSHORNE' S CONJECTURE ON LINEAR NORMALITY 4

7

such tha t X' i s nondegenerate an d i f n: P — • F N i s the projectio n wit h center in L , the n n(X f) = X. Fro m ou r assumption s o n the dimension an d degree of X' i t follow s tha t L f l l , = 0 . We may assume that N f = N + 1 . I n fact , i f A r' > N + 1 , then w e pick a general linear subspac e l / c L , dimZ and denot e by n : P — • P th

/ = iV'-JV- 2 e projection wit h cente r i n L . Pu t

f

X" = n'{X ), L"

= n'{L),

and le t n" ': P^" 1 "1 — • P ^ b e th e projectio n wit h cente r L " (her e L " i s a point i n P ^ + 1 ) . The n i t i s clear that L" $ X" an d n"(X") = n(X') =X, dim X" = dim X' = n, degX" = degJ*r ' = d e g X Thus, w e ma y assum e tha t N* = N + I an d L i s a poin t i n P^ b' - di m (Sing^f 7). Sinc e for L £ X' , deg^'= (^(X

,

+1

. Le t

):^W)-degX,

from (2.1 4.1 ) i t follow s tha t7 T is a finite birational map , s o that 7r(Sing^') c SingX , b' 1 < b. (2. 4.2

)

From th e conditio n o f th e theorem an d inequalit y (2.1 4.2 ) i t follow s tha t 2A T + 6 -1 2(iV+l A. , dimX = n >r = — ^

) + 6 -3 ^ 2N' + b' , > -1

,~* . (2. 1 4.3

An

)

In view of Theorem 2. 8 and Propositio n 1 . 5 c), fro m (2.1 4.3 ) i t follows tha t 1 SX' = P N' (2. 4.4 and therefore L e SX r. Let (p : S^i— » SX , /?j

)

: S%> — • X

be the canonica l projections . Pu t

^L=/?i((^)"1(i)). It is easy to se e tha t 1 n(DL) cSingX. (2. 4.5

)

On th e othe r hand , fro m (2.1 4.4 ) i t follow s tha t dimZ^ = di m ( ( / ) - 1 ( L ) ) >2n + l-N' =

1 2n-N. (2. 4.6

)

By our assumption , 2n-N>b 1 + (N-n-\). (2. 4.7

)

48

II. PROJECTION S OF ALGEBRAIC VARIETIES

Hence, from (2.1 4.5 ) an d (2.1 4.6 ) i t follow s tha t b = dim (SingZ) > di m (n(D L)) = dimDL >2n-N, which contradict s (2.1 4.7 ) fo r N > n + 1 . Fo r N = n w e hav e X = F N, and th e assertio n o f the theorem i s obvious. • 2.15. COROLLARY . For n > |(JV — 1 ) a nonsingular variety X n c P ^ cannot be obtained by projecting a variety of the same dimension and degree, nontrivially embedded in a projective space of larger dimension. 2.16. DEFINITION . A variety X n c P ^ i s called linearly norma l i f the lin ear system of hyperplane sections of X i s complete, which means that the restriction map H°(F N ,^{1 )) -*H°(X s surjective (i f X i s non9ffix(l)) i degenerate, this condition i s equivalent to H°(F N, ^ ( 1 ) ) ^ H°(X, ^ ( 1 ) ) ) Thus, a variety X c P i s not linearly norma l i f an d onl y i f there exist s a nondegenerate variety X' cF N , N f > N, an d a projection n : F N — • P^ such tha t n\ x,: X' ^ X . Th e followin g corollar y immediatel y follow s fro m Corollary 2.1 5 . 2.17. COROLLARY . For n > \{N - \) any nonsingular variety X n c P ^ is linearly normal. The simplest case when Corollary 2.1 7 can be applied and yields a nontrivial result i s the cas e of threefolds i n P ; until no w it wa s unknown whethe r or not the y ar e linearly normal . 2.18. REMARK . Corollar y 2.1 7 was stated as a conjecture b y R. Hartshorn e in 1 97 3 (cf . [33,4.2]) . 2.19. REMARK . A variety X n c P ^ i s called projectively normal i f al l its Veronese embeddings v k(X), k > 1 , are linearly normal (or , in other words, if th e restriction map s H°(F N, ffj^ik)) - • H°(X , 0 x{k)) ar e surjectiv e fo r all k > 1 ) . Ra o [69 ] constructed threefold s i n P 5 tha t ar e no t projectivel y normal. W e already observe d tha t al l suc h varieties ar e linearly normal .

10.1090/mmono/127/04

CHAPTER II I

Varieties o f Small Codimensio n Corresponding to Orbits of Algebraic Groups §1. Orbit s of algebraic groups, null-forms, an d secant varietie s 1.1. Le t K b e a n algebraicall y close d field, char # = 0 , an d le t G b e a linea r algebrai c grou p ove r K actin g o n a vector spac e V = K N+l. Le t v e V b e a vector fo r whic h Gv i s a punctured con e i n V , an d le t Xn = Gv/K* - > P * be the correspondin g projectiv e variety . Let H b e the stabilizer of v . B y Corollary 2 from [67 , n ° 4 ], H contain s a maximal unipotent subgrou p of G (i n our case this simply reflects th e fac t that eac h paraboli c subgrou p contain s a Borel subgroup; cf. [7]) . I n particu lar, H contain s th e unipoten t radica l o f G , an d withou t los s o f generalit y we may assume tha t th e group G i s reductive (thi s is a classical theore m o f E. Cartan ; cf . [1 2]) . Moreover , sinc e w e ar e intereste d onl y i n th e variet y X correspondin g t o th e orbi t Gv , we may assum e tha t G i s a semisimpl e group (perhap s in this situation i t would be more natura l t o conside r group s with one-dimensiona l center , bu t th e notio n o f semisimpl e grou p i s univer sally accepted, and it seems inconvenient to use notation i n which everythin g should be tensore d b y GLj) . Fixing a Bore l subgrou p B correspondin g t o a maxima l unipoten t sub group contained i n H w e can represen t v i n the for m V = V j + \-V

r,

where for b e B bvt = A-(6)vp /

= 1 , .. . , r ,

Az i s th e highest weigh t o f th e restrictio n o f actio n o f G o n a n invarian t subspace V. c V , an d v t i s a highes t weigh t vecto r (primitiv e element ) i n Vt. I t i s clear tha t

i=\ 49

50 III

. VARIETIE S OF SMALL CODIMENSION CORRESPONDING TO GROUPS

and without los s of generality w e may assum e tha t

1=1

Since A " is a projective variety , Gv consist s of two orbits, viz., Gv an d 0 , and therefor e al l A ( ar e collinea r (cf . [1 7] , [85]). O n th e othe r hand , sinc e Gv i s a cone , al l A ;. li e i n a n affin e hyperplan e (cf . [1 7]) . Thu s w e ma y assume tha t r = 1 an d Gv i s the orbi t o f th e highes t weigh t vecto r o f a n irreducible representatio n o f a semisimpl e grou p G (varietie s o f suc h typ e were considere d i n [95 ] an d wer e calle d HV-varieties). I n particular , fro m this it follows tha t th e variety X = Gv/K* c P N i s rational (cf . [72 ] and 1 . 3 below) an d i s defined i n F N b y quadratic equation s (cf . [57]) . Let A b e th e highest weigh t o f ou r representation , le t v A b e th e corre sponding highest weight vector, and le t g b e the Lie algebra of the group G. It i s clear that th e variety o f tangent s TX

=\JTX.x xex

corresponds t o the affine con e 1 1 1 GQVAC V. ( . .

)

Furthermore, i f P A c G i s th e paraboli c subgrou p stabilizin g th e lin e Kv A (or, whic h i s th e same , th e poin t x A e X correspondin g t o v A), the n th e stabilizer o f gv A i n G coincide s wit h P A (thi s follow s fro m Corollar y 2. 8 in Chapte r I) . Similarly, let N A c V* b e the subspace o f points corresponding t o hyper planes passin g throug h QV A (i.e. , the "normal " subspace) . The n th e variet y corresponding t o the cone GN A coincide s with th e dua l variety X* c (P^) * (here we consider the contragredient representatio n o f G i n V*) . Moreover , the stabilize r o f N A coincide s wit h tha t o f gv A. I t i s well known tha t fro m this it follows that the varieties TX an d X* (a s well as X ) ar e rational an d arithmetically Cohen-Macaula y (cf . [47]) . We proceed with finding out which of the varieties corresponding to orbit s of highest weigh t vector s ar e complet e intersections . I f X i s a complet e intersection, then , accordin g t o Propositio n 2.1 0 o f Chapte r I , ** = ^ _ , ( p ( ^

/ x

(-i))),

where J^N , X i s the normal bundl e an d

is a finite birationa l morphism . Since , a s w e have alread y observed , i n ou r case th e variet y X* i s normal , fro m thi s i t follow s tha t X* c (P N)* i s a nonsingula r hypersurface . O n th e othe r hand , X* ~ F(J^ FN/X(-1 )) i sa

§ 1. ORBIT S OF ALGEBRAIC GROUPS, NULL-FORMS, AND SECANT VARIETIES 5 1

projective bundl e ove r X wit h fibe r p ^ " " " 1 . Hence ,T V - n - 1 < 0 , i.e. , either X = P " o r X i s a hypersurface. I n th e latte r cas e it i s clear tha t X is a quadric. Summin g up , we obtain th e following result . 1.2. THEOREM . Let X n = Gv/K* c (V\0)/K* = F N be the projective variety corresponding to an orbit Gv c V of an irreducible representation of an algebraic group G which is a punctured cone. Then X = Gx , where x is the point of P corresponding to a highest weight vector v . Furthermore, X, TX , and X* are rational arithmetically Cohen-Macaulay varieties. The variety X is defined in F N by quadratic equations. Moreover, either X = Pn {i.e., G acts transitively on V \ 0 ) or X is a nonsingular quadric, or X is not a complete intersection. By analogy with [95] , we call projective varietie s satisfyin g th e condition s of Theorem 1 . 2 HV-varieties. 1.3. W e turn to secant varieties. I n the above notation, let M be the lowest weight o f th e representatio n o f G i n V , an d le t v M b e th e correspondin g weight vector. Althoug h the orbit Gv A doe s not necessarily contain all weight vectors, we have VMeGvA,

since M = w 0(A), where w 0 i s th e involutio n i n th e Wey l grou p W o f th e grou p G trans forming th e positiv e Wey l chambe r t o th e negativ e on e (cf . [9 , Chapte r VI, § 1, n ° 6, Corollar y 3]), and we may assume that w 0 i s contained i n the nor malizer o f a maxima l toru s o f G. Le t x A (resp. , x M) b e th e poin t i n X corresponding t o v A (resp. , v M), an d le t P A (resp. , P M) b e th e stabilize r of x A (resp. , JC M) . Conside r the orbi t o f the point x Ax x Me X x X unde r the natural actio n o f G o n X x X. I t i s clear that th e stabilize r of x A x x M coincides with jP A nP M . Sinc e P A contain s the "upper" and P M th e "lower " Borel subgroup o f G , w e have dim(P A .P M ) = dim G (cf. [7 , Chapter IV, Theorem 1 4.1 ]) . Hence , dim (G • {x A x x M)) = di m G - di m {P A n P M) = (di m G - di m PA) + (di m PA - di m (P A nP M)) = d i m X + ( d i m ( P A - P M ) - d i m P M ) = 2dim X = 2n (similar computation show s that di m (B U/(BU n P A)) = n , so that B M • xA = X an d fro m [72 ] it follow s tha t X i s a rationa l variety) . Hence , th e orbi t G • (x A x x M) i s dense i n X x X an d 1 1 SX = G(x A,xM). ( .3.

)

52 III

. VARIETIE S OF SMALL CODIMENSION CORRESPONDING TO GROUP S

Let U cV be the plane spanned by the vectors v A an d v M , let 9 1 be the cone of null-forms i n V (i.e. , 9 t i s the subset in V o n which all G-invarian t polynomials vanish), and let Z c P ^ b e the projective variety correspondin g to th e con e 91 . It i s clear that I c Z . Conside r th e actio n o f th e maxima l torus T C B o n U . Le t

where a A , a M ^ 0 . The n Tv c U and ther e ar e two possibilities: eithe r A + M ^ O, dimTv

= 2,

A + M = 0 , dimTv

= 1.

or In th e firs t cas e GVDTVBO,

and, therefore , GU = Gv1 cfJt 9 SXcZ ( .3.2

)

(cf. [1 7] , [85]). I n the secon d cas e 1 GU = ~G(Kvj', ( .3.3

)

examples sho w that i n thi s case SX ma y o r ma y no t li e in Z . We observe that fro m (1 .3.2 ) an d (1 .3.3 ) i t follow s tha t i f

then Gz = SX. The involutio n w Q fo r simpl e Li e groups i s described i n the tables in [9]. In particular , w 0 = - 1 (an d therefore A + M = 0 fo r all representations) i f and only if G i s a simple group of one of the following types: A x, B r {r > 2), C r ( r > 2 ) , £> 2 / (/>2), E 7, E %9 F 4 , G 2. From (1 .3.2) , (1 .3.3) , and (1 .3.4 ) i t follow s tha t i f SX = F N, the n eithe r 9t = V , i.e. , / G [F] = # , wher e I G[V] i s the algebr a o f polynomial s o n V invariant with respect to the action of G , o r Gv i s a hypersurface i n V and , since WDGV, di

m 9 1 > di m Gv

,

7G[K] = K[F] , wher e F generate s the ideal of 9 1 in K[V] . Al l representa tions for which the algebra of invariants has such a form hav e been classifie d (cf. [46] , [85 (addendum)]) , viz., if G i s a simple group, then I G[V] - K i f and onl y if G = SL r , Sp

2r ,

A 2 (SL 2r+1 ) ( r > l ) , Spin 1 1 0 ; ( .3.5

)

§1. ORBIT S OF ALGEBRAIC GROUPS, NULL-FORMS, AND SECANT VARIETIES 5

3

IG[V] = K[F] i f an d onl y if G=SOr(r/4), A

2

(SL 2 r ), S

2

Aj(Sp 6 ), S 3(SL2), Spin,(

(SL 2r) ( r > 1 ) , A

3

(SLr) ( r = 6 , 7 , 8) ,

r = 7 , 9 , 1 1 , 1 2 , 1 4 ) , E 6, E

G 2, (1.3.6) where A denote s th e exterio r powe r o f th e representatio n an d A 0 i s th e "principal part" of the decomposition of A into irreducible summands. Fro m the lists (1.3.5) and (1.3.6) it is easy to select representations for which SX = FN . I t turns out that i f SX = P ^ fo r a variety X n c P ^ correspondin g to a representation o f a simple group G , the n there are the following possibilities: X = Fn, X = Q ncFn+\ X

= v

7,

l 3{F )cP\

I = G(4,l)6cP9, I = G(5,2)9cP19, X = S

1 0

cP1 5,

;r = s15 cP 3 1, I = C 6 C P 1 3 , I = £ 2 7 C P 5 5 , where Q n i s a nonsingular quadric, ^ ( P 1 ) i s a rational cubic curve, S 1 0 an d S ar e the spino r varieties parametrizing linea r subspace s o n a nonsingula r eight- an d ten-dimensiona l quadric , respectivel y (cf . [1 1 ] , [35], [87]), C 6 i s the variet y correspondin g t o th e orbi t o f th e highes t weigh t vecto r o f th e representation A 0 (Sp6 ), an d E 21 i s the variet y correspondin g t o th e orbi t of the highest weight vector of the standard representatio n o f the group E 7. Summing up, we obtain th e followin g result . 1.4. THEOREM . If X = Gx A = Gx M, then SX = G(x A, x M). If A + M ^ 0, then SX c Z and, in particular, SX ^ F N, if the representation of G in V has at least one nontrivial invariant. If SX = P , then I G[V] = K[F] (here F is a form which may belong to K). Furthermore, if G is a simple group, then X is one of the following nine varieties: F n , Q n , ^ 3 (P 1 ), G(4, 1 ) , G(5,2) , S 1 0 , S 1 5 , C 6 , E 12 . 1.5. B y Theorem 1 . 2 w e may assum e tha t G = G { x • • • x Gd , V

= V x ® • • • 0 V d, d

> 1,

where G ( (i = 1 , .. . , d) i s a simple group and the representation G —> Aut V is a tensor product of nontrivial irreducible representations G i -* Aut V ( wit h highest weight s A . an d highes t weigh t vector s v. G V %,. I t i s clea r tha t th e highest weight A o f the representation G -> Aut V i s equal to A j H h A^ and th e correspondin g highes t weigh t vector ca n b e represente d i n th e for m v = v Y • • -®v d.

Let X t c V N' = F(Vt) b e the projectiv e variet y correspondin g t o th e orbi t Givi, an d le t n. = dimXt (i = 1 , .. . , d). The n i t i s clear tha t th e variet y l " c P = F(V) correspondin g t o th e orbi t Gv i s projectivel y isomorphi c to the Segr e embedding o f (X { x . . • x X d)n^"'+rid i n P W + 1 )-W+ 1 )- 1 .

54 III

. VARIETIE S OF SMALL CODIMENSION CORRESPONDING TO GROUP S

In what follows we shall need information abou t secant varieties of projective varieties correspondin g t o orbits of highest weigh t vectors of irreducibl e representations o f semisimpl e (bu t no t simple ) algebrai c groups . I n view o f 1.5, i t suffice s t o prove th e following genera l theorem i n which w e no longe r assume that varieties correspon d t o group actions . 1.6. THEOREM . Let {X n c P* } = {X^ x .. . x X n/ c pW + I )-W + 1 )" 1 } 5 where Xf ' c P^' , 0 < n { < • • • < n d , d > 2 (the Segre embedding). Then dimSX = 2/ i + 1 , except in the case when d = 2 , X x = P* 1 , X 2 = P" 2 ; in this latter case dimSX = 2n - 1 . Furthermore, SX = F N if and only if {Xn C P*} = {P 1 x P"" 1 c P 2 "" 1 } or {X n c F N} = {P 1 x V n~l c P 2 r t + 1 }, where n>2 and V n~l c¥ n is a hypersurface (we observe that this last case includes the variety P 1 x P 1 x P 1 c P 7 ). Arguin g b y induction , i t i s eas y t o reduc e everythin g t o th e cas e d = 2 . Le t x = (Xj , x 2), x = (x[ , x 2) b e a genera l pai r o f point s o f th e variety X = XX x I 2 c P ^ ' xP^ 2 = P , PROOF.

and le t z b e a general point o f the chord (x , x) . I f dimSX < 2n , then 1 1 d i m 7 z > 1 , ( .6.

)

where, in th e notatio n o f § 1 of Chapte r I an d Theore m 2. 8 o f Chapte r II , Y

z = {PxU9x l(z)) =

1( .6.2

)

ze(x 1 9y)}. ( .6.3

)

{xeX\3yeX,y^x, ze(x,y)}.

On th e othe r hand , i t i s clear tha t

YzCYz, where Y2 = bMt^iz)) =

{xe¥\3ye¥, y^x,

But i t i s well known (cf. , e.g. , [33] , [38]) that i f x[ ^ x {, x' Yz =1 (x x ,x[)x (x 2,x2) ( .6.4

2

^ x 2, the n )

is a nonsingular two-dimensiona l quadri c an d YzcYznX= {(y

{,

y 2) \ y x e (x x, x[) n X x , y

2

e (x 2, x 2) n X 2 ) . 1( .6.5 )

Suppose that at least one of the varieties X {, X 2, sa y ^ , is not a projectiv e space. The n fro m (1 .6.5 ) i t follow s tha t Y z consist s o f a finite numbe r o f points an d a finite number o f line s fro m on e o f th e tw o familie s o f line s o n Yz . Bu t from (1 .6.2) , (1.6.3), and (1 .6.4 ) it follows that, along with each lin e / C Y z c Y z fro m on e famil y o f line s o n Y z , Y z contain s a line fro m th e other family , viz. , th e line intersecting / a t th e poin t / n (p F){ (y/~ x(z)). I n view of (1 .6.1 ) w e come to a contradiction. Thus , if dimSX < 2n + 1, the n Xn = P" 1 x P" 2 i s a Segr e variety.

§2. //K-VARIETIE S OF SMALL CODIMENSION

55

Suppose no w that SX = P . The n N = (N l + l)>~{N d + l)-l Au t V is an irreducible representation of a semisimple, but not simple, group G such that N < 2n + 1 , then X is of one of the following types: a) X n = P 1 x P"" 1 c P 2 "" 1 ( w > 2) ; b) X n = P 1 x Q n~l c P 2n+l, where (»>2); c) I 4 c P 2 x P 2 c P 8 ; d) Z 5 = P 2 x P 3 c P n .

Q n~l is a nonsingular quadric

Furthermore, in cases a) and b) SX = P and PROOF.

in cases c) and d) SX ^ P .

Propositio n 2. 1 immediatel y follow s fro m Corollar y 1 .7 . •

2.2. PROPOSITION . / / X n = Gx A c P ^ = V(V), where G -> Au t V is an irreducible representation of a simple group G such that N < 2n + 1 , then dim V < dimG except in the following cases: a) r k G = 2 , di m G = di m V, G —• Aut K w £/*e adjoint representation-, b) G = SL 2 , G -> Aut K w f/*e adjoint representation, X = v 2(¥l) = Ql c P 2 is a conic, c) G = SL2 , J T = ?; 3(P1 ) c P 3 w a rational cubic curve. PROOF. Sinc e the parabolic subgroup P A stabilizin g the point x A contain s a Borel subgroup B A c G, w e have

n = dimA r = d i m G - d i m P A < d i m G - d i m £ A = ± ( d i m G - r k G ) ,

56 III

. VARIETIES OF SMALL CODIMENSION CORRESPONDING TO GROUP S

i.e., d i m1 G > 2n + rkG. (2.2.

)

From (2.2.1 ) i t follows tha t fo r N < In + 1 dimG > In + rkG > dim V + (rkG - 2) . (2.2.2

)

The inequalit y (2.2.2 ) show s tha t i f r k G > 2, the n di m V < dim G. Th e proof o f Proposition 2. 2 is completed b y a direct check . • All irreducible representation s G —• Au t V o f simpl e algebrai c group s G for whic h di m V < dim G wer e classified i n [2] and [20]. Usin g tables fro m these paper s (whic h fo r ou r purpose s shoul d b e complemente d b y addin g the adjoint representation s o f the groups SL 3 , Sp 4 , and G 2 an d the second and third symmetric powers of the standard representation o f SL 2) we select those representations fo r whichT V < In + 1 . First, w e describe thos e varieties X n c P ^ fo r which SX ^F N. Specia l attention wil l be devoted t o the case o f Severi varieties, which i s importan t for wha t follows . 2.3. DEFINITION . A nonsingular nondegenerat e (no t necessarily homoge neous) variet y X n c P ^ i s calle d a Sever i variet y i f n = |(J V - 2 ) an d

SX^FN. We recal l tha t fro m Corollar y 2.1 1 in Chapte r I I it follow s tha t fo r n > | (T V - 2 ) w e have SX = ¥N, s o that, fo r a fixed TV , Severi varietie s hav e maximal dimension amon g the varieties that can be isomorphically projecte d to a projective spac e of smaller dimension . A complete classificatio n o f Severi varietie s wil l b e given i n Chapte r IV , and her e w e restrict ourselve s t o classifying homogeneous Severi varieties ( a posteriori all Severi varieties tur n out to be homogeneous). First, we consider the case when the group G i s not simple. 2.4. THEOREM . If X n cf N is a projective variety corresponding to the orbit of the highest weight vector of an irreducible representation of a simple group G in a vector space V N'+l, where N < In + 1 and SX / P ^ , then either G is a simple group or X is a Segre variety of the form P* 1 x P"2 c p*i n2+*i+"2 j where n x = 2, 2 < n 2 < 3 . In the last case, X is a Severi variety if and only if n x = n 2 = 2 ,T V = n xn2 + n { + n 2 = 8 ; for this Severi variety, V can be identified with the space M 3 of 3 x ^-matrices over the field K , X corresponds to the cone of matrices of rank less than or equal to one, and SX is the cubic hypersurface corresponding to the cone of degenerate matrices defined by the equation de t M = 0 , M e M 3. Furthermore, for the above variety X = SingSX, (SX)* ~X, X* ~SX. PROOF. Th e first assertion o f th e theorem follow s fro m Propositio n 2.1. Furthermore, it is clear that the variety P" 1 x P"2 correspond s to the standard representation o f the group SL „ +1 x SLn +1 i n the space of matrices of order (nx +1) x (n2 + 1), and the orbit of a highest weight vector consists of matrices

§2. tf K-VARIETIE S OF SMALL CODIMENSION 2

57

2

of ran k one . I n particular, fo r th e Sever i variety P x P w e have V^M„ I

SLiXSL3[V]

=

K[det].

By Theorem 1 .4 , i n this cas e

sxcz~(m\o)/K*. Q

Hence, SX = Z i s a cubi c hypersurfac e i n P (correspondin g t o th e con e of matrice s o f ran k les s tha n o r equa l t o 2) , X = SingSX , an d fro m th e structure o f orbit s o f th e representatio n o f G i n V an d th e contragredien t representation o f G i n V* i t immediatel y follow s tha t ther e exis t natura l G-isomorphisms (SX)*~X, X*~SX. U 2.5. No w le t G b e a simpl e group . Propositio n 2. 2 an d a n analysi s o f the result s o f [41 ] an d table s fro m [2 ] an d [20 ] yiel d th e followin g lis t o f irreducible representations for which dimS X < N X is defined by the linear system of quadrics containing the spinor variety S 1 0 c P 1 5 c P 1 6 parametrizing four-dimensional linear subspaces from one family on the eightdimensional quadric. The variety X is obtained from the Veronese variety v2(F16) c P 1 52 by projecting it from (v 2(S1 0)) = P 12 5 . 5) Let X = S 1 0 c P 1 5 . The projection n: X — > P 1 0 with center in a linear subspace P 4 c X is a birational isomorphism; moreover, if H is a singular hyperplane section, then n\ X\H is an isomorphism. The inverse map a\ P 10 —- > X is defined by the linear system of quadrics containing the Grassmann variety (7(4 , 1 ) c P 9 C P10 . The variety X is obtained from the Veronese variety v 2(F1 0) c P 65 by projecting it from {v 2(G(4,.l))) = P4 9 . 3.9. REMARK . I n Theorem 3. 8 w e describe d projection s o f those HVvarieties that will be discussed in Chapters IV and VI and analyzed the structure of maps n\ H an d a = n~l. Som e other Z/F-varietie s for which H ma y contain mor e tha n tw o orbit s o f th e grou p H A an d a ha s a more comple x structure also present geometric interest. A direct application of Theorem 3. 3 shows that fo r al l d > 1 th e projection o f th e Veronese variety v d(Pn) wit h center in a subspace (^(P"" 1 )) i s an isomorphism invers e to v d ; the Grassmann variet y G(m, k) ca n b e birationally projecte d ont o p (/c+ 1 ^ m ~ /c) ? and the fundamenta l subse t A = n(H) o f the ma p a coincide s wit h th e Segr e variety p fc x f

m-k-\ p

(*+l)(w-A:)-l c

v

(k+\){m-k) .

the spinor variety S k parametrizin g the /c-dimensiona l linear subspaces fro m one family o n a nonsingular 2fc-dimensiona l quadri c Q c P 2 + 1 (S k cor responds to the orbit of the highest weight vector of the spinor representatio n k(k+l)

of the group D k+l = Spin2A:+2) ca n be birationally projected ont o P 2 , and the fundamenta l subse t A = n(H) o f th e ma p a coincide s wit h th e Grass *(*+!) t

k(k+l)

mann variet y G(k, 1) c P 2 c P 2 , etc . Nonsingula r hyperplan e sections o f Severi varietie s (example s A 0 ), A 2 ), C) , an d F ) from § 2) ca n also be interpreted i n this way (fo r suc h a variety, A i s a hyperplane sectio n of the variety A fo r th e correspondin g Sever i variety). 3.10. REMARK . I n th e case of the Segr e variety P 2 x P 2 an d Grassman nians th e existenc e o f a birational projectio n n: X —•* F n wa s proved , by different methods , o f course , in th e classica l papers [79 ] and [81].

10.1090/mmono/127/05

CHAPTER I V

Severi Varietie s §1. Reductio n to the nonsingular case 1.1. Th e goa l o f thi s chapte r i s t o giv e a classificatio n o f extremal vari eties wit h smal l secan t varieties , i.e. , varietie s fo r whic h th e inequalit y i n Theorem 2. 8 o f Chapte r I I turn s int o equality . I n othe r words , w e classif y nondegenerate varietie s = 2N * b - 1 , 6 = di m (Sing*) 1 1 1 ( . . ) 3\ that can be J-isomorphically projecte d to P ^ " 1 . B y Proposition 1 . 5 of Chapter II , th e las t conditio n hold s i f an d onl y i f SX ^ P ^ (i n vie w o f Re mark 2.1 0 i n Chapte r II , fo r n > 1 thi s conditio n ca n b e replace d b y th e condition T'X ^ ¥ N which , accordin g t o Propositio n 1 . 5 o f Chapte r II , ensures th e existenc e o f a J-unramifie d projectio n o f X t o P ^ - 1 ) . Fro m Theorem 2. 8 o f Chapte r I I it follow s that , unde r thes e assumptions , X*cP*,n

dimS* = N - 1

u 2

= 3n ~2 +l'
= T sx z fo r al l point s z e {x , z ) \ x . Therefore , fro m Proposi tion 1 . 9 a) o f Chapte r I I i t follow s tha t th e hyperplan e T sx z i s tangent t o X a t al l points o f y z n S m I, wher e

Z'E\J

C

Applying th e argument s use d i n th e proo f o f Theore m 2. 8 o f Chapte r I I t o the subvariet y y z (o r applyin g Theore m 1 . 7 o f Chapte r I t o th e subvariet y n{yz) c n(X) c F n~l an d the hyperplane n{T sx^) c P ^ - 1 , wher e n i s the projection wit h cente r i n a general point o f T sx z \ SX), w e see that d i m | ^ = di m Y z =

n + b + l

= 2n-N +

2.

Thus, Y z consist s of component s o f J ^ , and w e may assum e tha t 1 Yz,DY2 ( .4.5

)

for z e (x , z ) . Fro m (1 .4.5 ) i t follow s tha t

c = p2((px x (p)~\y' x (x, z») c i n n is a one-dimensiona l subvariet y o f th e plan e I I = (x , y\ z) . Withou t los s of generalit y w e may assume tha t (x,z)HCl =

(x, 1 z)nX = x, ( .4.6

)

where C i s a one-dimensional component o f C passin g through y . I n fact , otherwise SX - S(x , X) an d therefor e dim Y z = In + 1 - dimSX = In + 1 - dim5(x , X) = n , so tha t Y z = X an d fro m (1 .4.2 ) i t follow s tha t X c T sx z , contrar y t o the assumptio n tha t X i s nondegenerate . Fro m (1 .4.6 ) i t follow s tha t a general line in n passin g through x intersect s C 1 onl y at x , and therefor e Cl = (x,y) (cf.,e.g. , [64 , 5.1 1 ]) . Thus, for a general, and therefore fo r each point y e X , w e have (x , y) c X, i.e. , X i s a con e wit h verte x x . Sinc e x i s a n arbitrar y poin t o f E, from thi s i t follow s tha t X i s a cone wit h verte x H = F b, an d sinc e X i s irreducible, 6 < w - 1 . Let M N° c P ^ b e a genera l linea r subspace , M n SingJ f = 0 , N Q = J V - & - 1 , I 0 = I n M , rc 0 = di m X0 = / 2 - f e - l > 0 . B y Bertini' s theore m (cf. [28 , Vol. I, Chapter I , §1 ] , [34, Chapter II , 8.1 8]) , X Q i s irreducible an d nonsingular. Furthermore ,

SXncSXHM^M

74

IV. SEVER I VARIETIE S

and n0 = n-b-l= ( ^

1 —

J- b -1

= - ^—

J- =

-(N 0 - 2) ,

i.e., X 0 i s a Severi variety. The convers e i s a n immediat e consequenc e o f th e fac t that , a s i t i s eas y to see , SX i s the con e ove r SX 0 wit h vertex P = Sin g X, an d sinc e n 0 =

f(^o-2),

, 2(7V + bt 0 + Z>+l) + 6 -3 2N n = n 0 + 6 + 1 = -i— 2 ^ = 1 —

n

.•

§2. Quadric s on Severi varieties 2.1. PROPOSITION . Let X n c P ^ , n = §(JV-2) , be a Severi variety, and let z be a general point of SX . Set L = 7 ^ z , P L = {ueSX \ Tsx u = L} . r/zen X n P L = Y z = / ^ ( p - 1 ^ ) ) i s a nonsingular ^-dimensional quadric in the projective space P L = P I + . PROOF. Accordin g to C . Segre's reflexivity theore m (cf . [49] , [50]), P L i s a linear subspac e o f P ^ . Let XL = {xzX\T XxcL}.

Then X L i s a close d subvariet y o f X , !T(Jf L, X) c L , an d sinc e X i s nondegenerate, 5(^T L, X) qL L. Hence , Theore m 1 . 4 o f Chapte r I show s that dimSX = y - + 1 > dimS(jr L , X ) = dimX L + n + 1 , so that 1 1 d i m X L < ^ . (2. .

)

On th e othe r hand , fo r a genera l poin t u e P L, fro m Propositio n 1 . 9 o f Chapter I I i t follow s tha t T(Yu,X)cTSXtU^L9 Y

l u=Pl( L = r z. (2. .5

)

From (2.1 .4 ) i t follow s tha t degr z > 2 . Fro m th e trisecan t lemm a (cf . [39, 2.5 ] an d als o [34 , Chapte r IV , §3 ] and [64 , §7B] ) i t follow s tha t fo r a general pair o f point s x , y € X 1 (x,y)nX = {x,y}. (2. .6

)

Since z i s a general point o f SX , fro m (2.1 .6 ) i t follow s tha t de g 7Z = 2 . It remain s t o verif y tha t th e quadri c Y z i s nonsingular . Suppos e tha t this i s not so . The n Y z i s a quadrati c con e wit h verte x a t a point y e Y z. Therefore, SX — S(Y Z, X) i s also a con e wit h verte x y , an d fo r a genera l point u e SX w e have y e P M, M = T s;r M . I n vie w o f (2.1 .5) , from thi s it follows tha t 1 ye f | Y u. (2. .7 ) uesx From (2.1 .2 ) an d (2.1 .7 ) i t follow s tha t T x c f | 7 ^ u , i.e., S X i s a cone with vertex T Y „ an d

which is impossible sinc e dimP M = - + 1 < n = dimTx y . This contradiction show s that Y z i s a nonsingular quadric . • 2.2. LEMMA . Let X n cF N, n = §(iV-2), be a Severi variety, and let z be a general point of SX. Then, in the notation of § 1 of Chapter I, the morphism tpz = (p Yz: SY x - * 5(1^ ,X) = SX, Y z= p x{(p'\z)), is birational PROOF.

Fro m Propositio n 1 . 9 a) o f Chapte r I I i t follow s tha t T(Y2,X)CT^Z,

and sinc e (S(YZ,X)) =

(X) = F\

Theorem 1 . 4 o f Chapte r I shows that dimS{Yz, X) = dimS Y x

= dimSX = N - 1

=^ + 1 .

Hence, S(Y z, X) = SX an d th e morphis m cp z is generically finite, i.e. , fo r a general point u e SX w e have card(7 z n Y u) < oo . T o prove Lemm a 2. 2

76 IV

. SEVER I VARIETIE S

it suffice s t o verify tha t fo r a general point u e SX th e quadrics Y z an d Y u (cf. Propositio n 2.1 ) intersec t transversel y a t a uniqu e poin t [Y z D Yu ^ 0 since u e S{Y z, X) = SX). Let P z (resp. , P u) b e the ( § + l)-dimensional linea r subspace spanned by the quadric Y z (resp. , Y u). Suppos e that Y znYu3 x, y, an d let / = (x , y ) (in th e cas e whe n y = x , i.e. , 7 Z an d Y u ar e tangen t a t x , / i s thei r common tangen t line) . The n /cPznPMcSing(5^), since, by Propositio n 2.1 , the tangen t spac e t o SX a t a n arbitrar y poin t o f Pz n P u contain s bot h T sx z an d T sx u ^ T sx z . Varyin g u e SX , w e see that a genera l poin t y e Y z lie s o n a lin e / c P z n (Sing^ f \ Jf) , fro m which i t follow s tha t z e P z c SingSX , contrar y t o th e choic e o f z . Thi s contradiction complete s the proo f o f Lemm a 2.2 . • 2.3. LEMMA . Let X n c¥ N, n = \(N - 2) , be a Severi variety. Then the quasiprojective variety SX \ X has a nonsingular normalization. PROOF. Le t v e Sin g (SX \ X), an d le t Y' v b e a n irreducibl e componen t of Y v =p {( : S x — • SX. Fro m Propo sition 2. 1 i t follow s tha t v\ SX - * SX i s th e normalizatio n morphism . Lemma 2. 3 shows tha t Si1 n g l e ^ ( X ) . (2.4. ) Suppose tha t SingS X ^ X , le t v b e a general point o f Sing(5 X \ X), an d let v e v~ l(v). I n view of (2.4.1 ) an d the Serre normality criterio n (cf. , e.g., [30, Chapter IV 2 , (5.8.6)]), dim (Sing(5Z \ X)) = dimSX - 1

= ^ .

Hence, d i m p " 1 ^ ) = f an d ] )) = ( P i U f e ' V ) ] ) = [Y 2l (2.4.2

)

By Proposition 2. 1 and Lemm a 2.2 , Y z i s a nonsingular quadri c an d l. (2.4.3

(Y*)x =

)

Hence, all components of Y^ = Px{(j>~X(v)) ar e also ^-dimensiona l quadrics . From (2.4.2 ) an d (2.4.3 ) i t follows that fo r a general point v e Sing SX \ X , the fiber cp~ (v) i s connected , i.e. , th e morphis m v i s one-to-one . Sinc e for x £ X , th e fiber 0 \/ueS(x,X)\X, which contradict s Theorem-2.4f) . D 3.2. PROPOSITION . Let X n cW N, n = §( N - 2) , be a Severi variety, and let Zj , z 2 e SX \ X . Then either z x e P z , z 2 e P z , and Y z = Y z , p = p , or Y 9 n 7 is a linear subspace (we use the notation from the Z Z

\

2Z

\Z

2

proof of Lemma 2.2) . PROOF. A S we have already observed , S(YzrX) =

1 SX,

= 1 ,2 ,

and therefore 7 n T , ^ 0 . Suppos e that Y 7 n Y 7 i s not a linear subspace . Then fro m Theore m 2. 4 it follow s tha t S(Yz nY z) = Z Z

\

2Z

Pz nP \Z

tx

z

.

2

But for an arbitrary point z e S(Y Z C\Y z )\X, fro m Theore m 2.4b) i t follow s that YZl=Yz = Y Z2. D

§ 3. DIMENSIO N OF SEVERI VARIETIES 8 1

Let l " c P , N = \{n - 2) , be a Severi variety , let x b e an arbitrary point of X, an d le t

y, = rv Y

2=

Y ZI, z

l,z2es(x,x)\x

be two quadrics fo r whic h (Y l • Y 2) = x (cf . Lemma 2.2) . Put Ci = Y^TXx =

Yi^TYnX, i=l,2.

Then C i i s an (f-l)-dimensiona l con e with vertex x i n the f-dimensiona l projective spac e T Y x whos e bas e i s a nonsingula r ( f - 2)-dimensiona l quadric (i = 1 ,2) . I t is clear tha t fo r n > 2, S ^ , C2) £ X (here , as in § 1 of Chapter I, S(C l, C 2) is the join of cones C x an d C 2 ). 3.3. PROPOSITION .

a) dim5(C 1 ? C 2) = / i - 2 ; b ) L ^ o 2 , Z G S(Cj, C 2) \ X, Y z ^ 7 Z. (/ = 1 , 2). Then Y z n Y) w a /wea r subspace of dimension [ J ] ; c) fo r n > 2 w e /zave rc = 0 (mo d 4) ; d) For n > 4 we have n = 0 (mo d 8). a ) Let Q. be the base of the con e C . (i.e. , Q ( i s the intersectio n of C . with a general hyperplan e in T Y x , / = 1 , 2). The n PROOF,

S{C19C2) = S(x,S{Q

lfQ2))

and (Q 1 )n«2 2 > = 0. Hence, dim5(C 1 , C 2) = d i m S ^, G 2) + 1 = 2 (^ - 2 ) + 2 = * - 2. Assertion a ) is proved. b), c ) Let z b e a general point o f 5(Cj , C 2) \ X. B y Proposition 3.2, YznYt = P°', 1 /= ,2

.

Since C x an d C 2 ar e cones with vertex x , a t > 0 ( / = 1 , 2) an d

P a '3x, 1

= 1,2, P

ai

nPQ2 = C1nC2 = ^0^2 = x, zeS(V

a

\Fa2).

Furthermore, if z e S(Cl, C 2) \ X, then 7 z , = 7 z i f and onl y if Ql+£ 2 z' G 5(PQ l , P° 2) = P1 * \ X. (3.3.

)

By a) and (3.3.1), varying z 6 ^ ( C j, C 2)\X w e obtain an ( ( « - 2 ) - ( a 1 + a 2 ) ) dimensional famil y o f quadrics passin g throug h x an d intersectin g Y j an d Y2 alon g linear subspace s of positive dimension . We already know (cf. Propositio n 3.1 ) that there is an ^-dimensiona l fam ily of quadrics Y u passin g throug h x an d parametrize d b y a quadric Y x . Furthermore, the ( § - l)-dimensiona l subfamil y o f quadrics Y u intersectin g

82 IV

. SEVER I VARIETIE S

Yj alon g a positive-dimensional linea r subspac e i s parametrized b y th e subcone with verte x (T sx z ) * i n Y x . Fro m thi s i t follow s tha t th e dimensio n of the family o f quadrics Y u passin g through x an d intersecting Y x an d Y 2 along positive-dimensional linea r subspace s i s equal to \ - 2 (th e base Y n of this family i s the intersection o f two ( ^ - l)-dimensiona l subcone s i n Y x with vertices (T sx z ) * an d (T sx z )*,sotha t Y l2 i s an ( J - 2)-dimensiona l quadric). Thus , (n - 2 ) - (a { + a 2) = \ - 2 , i.e. , a

i + a 2 = = 2 " (3.3.2

)

On th e othe r hand , i t i s wel l know n (cf. , e.g. , [37 , Chapte r XIII , §4] , [28, Chapter VI , §1 ] ) that th e maximal dimensio n o f a linear subspac e lyin g on a nonsingular \ -dimensional quadri c Y- ( / = 1 , 2 ) i s equal t o [ f ] . Hence ,

a. < [J] , i

= l , 2. (3.3.3

)

Combining (3.3.2 ) an d (3.3.3) , we see that a

i

=a

rni n 2 = U J= 4

which simultaneously proves b) and c) (under specialization o f z th e dimension of Y z n 7 . coul d onl y jump). d) Let z E 5(Cj, C 2)\X . Fro m b), c), and Proposition 3. 1 it follows tha t the set of quadric s Y u passin g through x an d intersectin g Y z alon g a linear subspace o f dimensio n | i s parametrized b y the con e with vertex (T sx z )* in Y x whos e base is a nonsingular ( f - 2)-dimensiona l quadric . Fo r n > 4 this con e i s irreducible, an d therefor e al l J -dimensional linea r subspace s o f the for m Y z n Y u belon g to on e an d th e sam e famil y o f linea r subspace s o n Yz . It is well known (cf . [37 , Chapter XIII, §4], [28, Chapter VI, §1]) that th e dimension o f th e intersectio n o f tw o \ -dimensional linea r subspace s fro m one famil y o n Y x ha s th e sam e parit y a s | (w e recal l tha t o n th e nonsin gular even-dimensiona l quadri c Y ZJ ther e exis t tw o irreducibl e familie s o f ^-dimensional linea r subspaces) . O n th e othe r hand , i n th e notatio n use d i n the proof o f assertion s b) an d c) , pQl n P° 2 = Y x n Y 2 = x. Hence, for n > 4

0 = J (mo d 2). This completes th e proof o f assertio n d) . D 3.4. REMARK . Assertion s c) and d ) of Proposition 3. 3 were independentl y proved b y Fujit a an d Robert s (cf . Proposition s 5. 2 an d 5. 4 i n [25]) , wh o used the techniques of computations with Chern classes . Thei r approach wa s developed by Roberts (unpublished) an d Tango [89] (cf. Remar k 4.1 1 below).

§3. DIMENSIO N OF SEVERI VARIETIES 8

3

3.5. COROLLARY . If in the conditions of Proposition 3. 2 Y z # Y z , then Y9 n y is either a point or a linear subspace of dimension ~ . z z

\

2H

In th e proo f o f Propositio n 3. 3 w e showe d tha t b y varyin g z i n S(Cl, C 2) \I w e obtain a family o f \ -dimensional linea r subspace s o n th e j -dimensional quadri c Y x. Th e bas e o f thi s famil y i s a nonsingula r (f - 2)-dimensiona l quadric . Hence , fo r n > 4 al l linea r subspace s o f th e form Y z n Y x, z e S(C X, C 2) \ X belon g t o on e an d th e sam e irreducibl e family o f ^-dimensiona l linea r subspace s o n Y j passin g throug h x (cf . th e proof o f Propositio n 3.3d)) . W e denot e thi s famil y b y 9" an d th e othe r family b y 9 1 . n.

3.6. LEMMA . Let n > 4 , and let P 04 be an arbitrary linear subspace on Y x passing through x and belonging to the family 9 . Then for some z e S(C X ,C 2)\X we have Y zf\Yx=P0. PROOF. W e argue by induction. Le t z e S(P 0 ,C 2)\X. The n Y z n P0 i s a linea r subspac e o f positiv e dimension . I t i s clea r tha t i t suffice s t o prov e the following assertion . Le t z e S(C X ,C 2)\X, Y

zHP0

= P a3x, 0

0. Hence , ther e exist s a linea r subspace o n Y z tha t passe s throug h P a an d intersect s Y 2 a t a poin t y 1 e Y2 \ X z . Thi s point y' satisfie s al l the abov e conditions . Let ue(y,y')\XcS(Cl,C2)\X, and le t a e¥ 9 b e a n arbitrar y point . B y constructio n (y,a)cP0cX, (y\a)cY

zcX,

and therefor e , zx P

z,x'>

which i s impossibl e sinc e Y u is a nonsingular |-dimensiona l quadri c an d dimPz x = J + 1. Therefore , S(P ZX, P zx.) c X an d th e fiber o f th e map (4.2.1 ) ove r th e poin t correspondin g t o P z x n Y z i s a linea r subspac e of positiv e dimension , whic h i s als o impossibl e sinc e otherwis e th e variet y Alz° c P 1 5 woul d contai n a n exceptiona l divisor , contrar y t o a theorem o f Barth (cf . [6] , [33]). Thus , in case c) the map (4.2.1 ) i s an isomorphism, an d from th e fac t tha t PicSz-Z

§4. CLASSIFICATIO N THEOREMS

87

(cf. [1 1 ] , [35], [87]; by virtue o f Theore m 5 from [95 ] this als o follows fro m the result s o f § 2 of Chapte r III , an d i n th e cas e when dimS z = 1 0 on e ca n apply a Barth-type theore m [54] , [65], [60]) i t follow s tha t th e embeddin g

4° - P 1 5 c P 1 6 corresponds t o the spino r representation . In cas e b) dimJ9z = 4 , S

z

= P 3.

From Remark 3.1 1 it follows tha t the fibers o f the morphism (4.2.1 ) ar e projective line s an d th e preimag e o f a n arbitrar y projectiv e lin e fro m S z (cor responding t o a poin t fro m Y z) i s a nonsingula r two-dimensiona l quadric . Furthermore, t o eac h point o f S z ther e correspond s a four-dimensional lin ear subspac e o n X mappin g t o a lin e o n B z, an d thu s w e obtai n a ma p Sz x P 1 - » B z. Thus , i n cas e b), B z i s projectively isomorphi c t o th e Segr e embedding o f th e variety P 1 x P 3 i n P 7 c P 8 . In cas e a), l dimBz = 1 , S z= F . By Remark 3.8 , each line on the quadric Y z i s cut by a plane, and we obtain a surjectio n B z - + S z ]}S z . Arguin g a s in cas e c) , we see that B z = P 1 UP 1 and H z consist s of tw o irreducible component s intersectin g alon g Y z . • 4.3. REMARK . Sinc e TSXtZriSX =

T{Y

z,X),

we see that

Hz = T SXiZnx= \J{T

x§ynX).

yer2 Furthermore,

*>.,"* = ( J C

w=

| J P« , C

u

= YunTYmty.

uesyx\x ye¥ icx Hence, T x 0 X i s a con e wit h verte x y , an d fro m Propositio n 3. 1 an d Corollary 3. 5 i t follow s tha t j . , _ __ (n , \ n 2>n xn 2 w e hav e d — 2. Th e cas e n = 2 (B z = 0 ) i s deal t wit h i n a simila r wa y (cf . Remar k 4.4) ; thi s cas e was first studied b y Sever i (cf . [82] , [62], [15]). Summin g up , we obtain th e following result . 4.5. THEOREM . If X n cF N, n = |(JV—2), is a Severi variety, then n = 2, 4, 8 , or 1 6 and X is the image of F n under the rational map a : F n — • P^ defined by the linear system of quadrics passing through a subvariety A c

P""1 c P\ where a) for n b) for n c) for n d) for n

=2 A = 4^ =8 4 =16 J

= 0; 4 = P 1 UP1 is a union of two skew lines; = P 1 x P 3 i s f/*e &#n? var/^ y / w P 7 ; = 5 1 0 w //ze spinor variety parametrizing one of the two families of four-dimensional linear subspaces on the nonsingular quadric in F (cf § 2 of Chapter III). In other words, the Severi variety X is obtained from the Veronese vann(n+3)

ety v 2(F ) c P 2 (n = 2 , 4 , 8 , 1 6 ) by projecting it from the linear span (v2(A)) of the image of the subvariety ^ c P " under the Veronese embedding v2. 4.6. REMARK . Fro m Remark 4.3 and the arguments given before the statement of Theorem 4.5 it immediately follows that the linear system of quadrics cut in a general linear subspace F n~l c T x x by the linear system of quadric s passing through X an d definin g a rational ma p p"" 1 --» • (?? c P

f+1

§4. CLASSIFICATIO N THEOREMS

89

(where Q 2 = Y x i s a nonsingular quadric ) i s the second fundamental form in th e sens e of [29 ] and th e subvariet y A c¥ n~l i s the fundamenta l subse t of this form . Theorem 4.5 shows that in each of the dimensions 2, 4, 8, 1 6 there exists at most on e Sever i variety. T o complet e th e classificatio n o f Sever i varieties i t remains to verify that the necessary conditions formulated i n Theorem 4.5 are also sufficient, i.e. , the varieties X n describe d in Theorem 4.5 are nonsingular and ca n be isomorphically projecte d t o P 2 However, i n Chapte r II I w e already constructe d fou r example s o f Sever i varieties (th e first thre e o f them , viz . th e Veronese , Segre , an d Grassman n varieties, ar e classical ; cf . Remar k 1 .3 , [33] , [38]) . Moreover , usin g meth ods fro m representatio n theory , i n § 3 of Chapte r II I w e studie d th e map s nz an d a z i n thes e example s an d describe d geometri c propertie s an d com puted invariant s o f th e correspondin g varieties . Thus , Theore m 4. 5 yield s the followin g basi c result . 4.7. THEOREM . Over an algebraically closed fieldofcharacteristic zero each Severi variety is projectively equivalent to one of the following four projective varieties: a) u 2 ( P 2 ) c P 5 (Veronesesurface); b) P 2 x P 2 c P 8 {Segre variety); c) G(5 , l) 8 c P 1 4 (Grassmann variety); d) E 1 6 c P 26 (Cartan variety); All these varieties are homogeneous, rational, and are defined by quadratic equations. Furthermore, a Severi variety X corresponds to the orbit of the highest weight vector of an irreducible representation of a semisimple group G in a vector space V with highest weight A , where : a) G = SL 3 , A = 2^ ; b) G = SL 3 xSL 3 , A = ^ 1 ©?? 1 ; c) G = SL 6 , A = cp 2 ; d) G = E 6, A = ^ (here

1 6 , the n n = 2 m (m > 1 ) o r

90

IV. SEVER I VARIETIE S

n — 3 • 2 (r a > 5) . Mor e intriguin g i s th e "explanation " base d o n th e following resul t (independentl y discovere d b y Roberts). 4.8. THEOREM . Let 2 1 be a composition algebra over the field K, and let Z be the Jordan algebra ofHermitian (3x3 ) -matrices over 2 1 ( a matrix A is called Hermitian if A 1 = A, where l denotes transposition and the bar denotes the involution in 21 ) , so that dim^ S = 3(dim^2H - 1 ) (cf [1 0] , [44], [76]). Let X c P(3 ) = P be the projective variety corresponding to the cone {A e 3 | vkA < 1 } . Then X is a Severi variety and SX is the hypersurface corresponding to the cone {A e 3 | det A = 0 } . Conversely, each Severi variety is obtained in this way. PROOF. B y Jacobson' s theore m (cf . [43] , [44 , Chapte r IV , n°3]) , ther e exist exactl y fou r compositio n algebras—on e i n eac h o f th e dimension s 1 , 2, 4 , 8 , viz. th e algebra s 2l 0 = K, 2l j = K[t]/(t 2 + 1 ) , 2l 2—the algebr a o f quaternions over K , an d 2l 3—the Cayley algebra over K . Fo r these algebras

Nt = dimP^. ) = 3 - 2'" + 2, n

t

= dimX. = 2 M = 2dim2l / ,

where 3 - an d X t ar e th e Jorda n algebr a an d th e projectiv e variet y corre sponding to the algebr a %L. ( 0 < i < 3). It i s clea r tha t th e surfac e X 0 coincide s wit h th e Verones e surface . Th e algebra 3 j i s identifie d wit h th e algebr a o f ( 3 x 3)-matrices ove r th e field K, an d the variety X { i s identified wit h the Segr e variety P 2 x P 2 (cf . The orem 2. 4 i n Chapte r III) . Sinc e th e field K i s algebraicall y closed , 2l 2 i s isomorphic t o th e algebr a o f ( 2 x 2) -matrices ove r K . Fro m thi s i t i s eas y to deduc e tha t X 2 i s projectivel y equivalen t t o G{5, 1 ) (cf . Chapte r III , 2.5, A 3 )). Finally , from FreudenthaF s result s [23] it follows tha t the variety X3 i s isomorphic t o E (cf . Chapte r III , 2.5 , E)) . D It i s eas y t o se e that Theore m 4. 8 ca n b e restate d a s follow s (judgin g b y [56], a simila r resul t (fo r complexification s o f rea l divisio n algebras ) wa s proved b y T . Banchoff) . 4.9. THEOREM . X is a Severi variety if and only if X is a "Veronese surface" over one of the algebras 21 . ( 0 < / < 3) , i.e., X is the image of the "projective plane" P 2(2l.) = (2l z3 \ 0)/2l * (where 21 * is the set of invertible elements of the algebra 21 . ) with respect to the map (x0 : JCJ : x2) H- * (• • •: x lxm :•••) > 0 < / < r a < 2

.

4.10. REMARK . W e do not kno w if there exist s som e intrinsi c connectio n between compositio n algebra s (o r som e othe r clas s o f algebras ) an d Sever i varieties, o r i f thi s i s a n accidenta l coincidence . I n an y case , th e classifica tion o f Sever i varieties give n i n Theorem s 4.7-4. 9 allow s one to giv e a new,

§5. VARIETIE S OF CODEGREE THREE

91

unexpected proo f o f th e well-know n Jacobso n theore m o n th e structur e o f composition algebra s (cf. , e.g. , [43] , [44, Chapte r IV , n° 3 ]). I n Chapte r V I (cf. Remar k 5.1 0 an d Theore m 5.1 1 there ) w e shal l se e tha t extrema l vari eties with small secant varieties also correspond to matrix Jordan algebras (or to Verones e varietie s ove r compositio n algebras) . Furthermore , al l varietie s except E c P correspon d t o specia l Jorda n algebras , an d th e variet y E corresponds to the exceptional algebra of Hermitian (3x3 ) -matrices over the Cayley numbers. 4.11. REMARK . I n th e cas e o f surface s Theore m 4. 7 wa s first prove d b y Severi [82 ] (th e proo f o f Sever i i s reproduce d i n [62] , an d th e pape r [1 5 ] is devote d t o finding ou t whic h part s o f thi s proo f wor k i n th e cas e whe n char# > 0) . Griffith s an d Harris , wh o apparentl y didn' t kno w abou t Sev en's paper, proved a local version of his result (cf. [29 , 6c)]). Scorz a [77], [78] classified (possibl y singular ) threefold s an d fourfold s wit h smal l secant vari eties. However , thes e results o f Scorz a were forgotten, an d i n 1 97 9 Griffith s and Harri s prove d tha t eac h four-dimensiona l Sever i variet y (o r a Zarisk i open subset of such a variety) ha s the same second fundamental for m a s the Segre variety F 2 x P 2 c F 8 (cf . [29 , (5.62)]) . I n th e sam e paper , Griffith s and Harri s conjectured tha t up to projective equivalenc e P 2 x P2 i s the only four-dimensional Sever i variety . Base d o n th e author' s results , Fujit a an d Roberts [25 ] proved thi s conjecture, an d Fujit a [24 ] gave a modern proo f o f Scorza's resul t fo r nonsingula r threefolds . Tang o [89 ] showe d tha t i f ther e exists a Sever i variet y X n, n > 1 6 , the n n > 9 6 an d eithe r n = 2 m o r n = 3-2 m , wher e m i s a natural number . Theore m 4. 7 wa s first proved i n [99] (cf. als o [56]).

§5. Varieties of codegree three From Theore m 2. 4 an d Remar k 2. 5 i t follow s tha t th e dua l variety o f a n arbitrary Sever i variet y i s a cubi c hypersurface . Thi s propert y i s share d b y isomorphic projections o f Severi varieties. Thi s observation indicate s that i n the contex t o f th e presen t chapte r i t i s relevant t o give a classificatio n o f al l nonsingular varieties whose dual varieties have degree three. Classification o f varietie s o f smal l degree has bee n a popular topi c sinc e A. Wei l [1 04 ] classifie d al l projectiv e varietie s o f degre e thre e i n 1 95 7 (th e classification o f varieties of degree one and two is trivial). Late r Swinnerton Dyer [86 ] succeede d i n classifyin g varietie s o f degre e four , an d i n a serie s of paper s Ionesc u classifie d smoot h projectiv e varietie s u p t o degre e eight . Several papers ar e devote d t o low-dimensional varietie s o f smal l degre e an d to varieties whose degree is not to o big with respec t t o codimension . Du e t o efforts o f Hartshorne , Barth , Va n d e Ven, and Ran , i t was discovered tha t i f the dimension of a nonsingular variety i s sufficiently larg e with respect t o it s degree, then th e variety i s a complete intersection .

92

IV. SEVER I VARIETIE S

However, her e w e are mor e intereste d i n class, which i s anothe r classical invariant o f projectiv e varietie s whos e rol e i n enumerativ e geometr y i s no t less than tha t o f th e degree . Traditionally , th e clas s of a nonsingular variet y Xn c P N i s define d a s th e numbe r o f singula r divisor s i n a generi c penci l of hyperplan e section s o f X . Thus , i f th e dua l variet y X * i s a hypersur face, the n th e clas s of X i s equal t o the degre e o f X* . Varietie s fo r whic h codimJF* > 1 hav e clas s zero , bu t fo r u s i t i s mor e convenien t t o us e th e notion o f codegree. 5.1. DEFINITION . Th e numbe r d* = degX* i s calle d th e codegre e o f X in P ^ an d i s denoted b y codegX . Thus, codegre e i s equal to clas s provided tha t X* i s a hypersurface . It is clear that the only varieties of codegree one are linear subspaces F n c P^ an d th e onl y varieties o f codegre e two ar e quadric s Q n c¥ N . We notic e tha t i f X n c P M c P* , the n th e dua l variet y o f X i n P * i s the con e ove r th e dua l variet y o f X i n P ^ wit h verte x (P M)* = p A r ~ M " 1 . Hence, the codegree of X i n P ^ i s equal to the codegre e of X i n P , and in the classificatio n o f varieties of a given codegree it suffice s t o consider th e case when X i s nondegenerate, i.e., (X) = P , where (X) i s the linear spa n of X. Classification o f nonsingular varieties of small codegree is apparently mor e difficult tha n tha t o f varieties o f smal l degree, e.g., because i n th e latter cas e one ca n procee d b y inductio n o n dimensio n b y takin g hyperplan e sections , while i n th e first cas e ther e i s n o suc h possibility . Furthermore , th e flavor of th e proble m fo r codegre e i s quit e different . A n importan t par t o f th e problem i s to characterize th e structure o f singularitie s o f hypersurfaces o f a given degre e whos e dua l varieties ar e nonsingular . Whil e ther e alway s exis t varieties of a given degree and arbitrary dimension (e.g. , hypersurfaces), ther e are reasons to expect that, if we denote by n{d) th e smallest natura l numbe r (or oo ) suc h that for eac h nonsingular variety X wit h code g X = d w e have d i m Z < n(d) , the n n(d) < oo fo r d > 2 (o f course , n(2) = oo) . However , the numbe r n(d) i s no t small ; i n th e presen t sectio n w e sho w tha t alread y /i(3) = 1 6 . There ar e man y paper s devote d t o surface s o f smal l clas s (codegree ) an d some papers devoted to threefolds (cf . [53] ; a survey and bibliography can be found i n [91 ]) , but, i n general, varieties of small codegree remain completel y unexplored. I n th e presen t sectio n w e mak e th e first ste p i n thi s directio n and giv e a complet e classificatio n o f nonsingula r nondegenerat e varietie s o f codegree three (it turns out that up to projective equivalenc e there are exactly ten suc h varieties). 5.2. THEOREM . Let X n cF N be a nonsingular irreducible nondegenerate projective variety of codegree three over an algebraically closed field K of characteristic zero. Then there are the following possibilities:

§5. VARIETIE S OF CODEGREE THREE

93

0. n = 3, X = P1 x P 2 c P 5 (X is a Segre variety); 1. n = 2, X = ¥ x c P 4 (F 1 is a bundle with fiber P1 ove r P 1 embedded in P 4 so that its fibers and the minimal section s are projective lines am/ ( 5 ) = - 1 ) ; II. X is a Severi variety. More precisely, in this case there are the following possibilities: (1) II.l . n = 2 , X = v 2(¥2) c P 5 (A T w the Veronese surface); (2) II.2 . « = 4 , I = P 2 x P 2 c P 8 {X is the Segre variety); (3) II.3 . n = 8, X = G(5, 1 ) c P 1 4 ( X w J/ze Grassmann variety); (4) II.4 . n = 1 6 , I = £ c P 2 6 (X corresponds to the orbit of the highest weight vector of the nontrivial representation of the group E6 having the smallest possible dimension); II'. X is an isomorphic projection of one of the Severi varieties X n c P ^ + 2 described in II to P ^ + 1 , n = 2' , 1 < / < 4 (a s /« II, /zer e we obtain four cases Il'.l-Il'.4) . 5.3. REMARK . I n al l th e abov e case s X i s th e imag e o f P n unde r th e rational ma p define d b y the linear syste m o f quadrics in P n passin g throug h B, wher e B ha s the following form : 0. B = p°U P 1 ; I. B = P°; (1) II.l, II.1. B = 0 (2) II.2, II.2. B = vl U F 1 ; (3) II.3 , Il'.3 . B = P 1 x P 3 ; (4) II.4 , II 7.4. B = iS 1 0 , the spino r variet y i n P 1 5 correspondin g to the orbit o f the highest weight vector o f the spino r represen tation o f th e group Spin 9 (o r Spin 1 0 ). 5

In cas e 0 w e hav e X* ~ X ~ P 1 x P 2 ; i n cas e I , X* i s th e projectio n of P 1 x P 2 fro m a poin t o f P 5 \ P 1 x P 2 . I f X i s a Sever i variety , the n by (2.5.4 ) X* ~ SX, an d i n cas e II ' th e variet y X* i s obtaine d fro m th e corresponding Sever i variety b y intersecting i t wit h a general hyperplane . According to Theorems 4. 8 and 4.9 , all Severi varieties can be interprete d as "matrice s o f ran k 1 " i n th e spac e o f Hermitia n ( 3 x 3)-matrices (o r a s "Veronese surfaces" ) ove r on e o f th e fou r standar d compositio n algebras ; X* i s define d b y th e equatio n de t = 0 an d therefor e ha s degre e three . I n case II ; , X* i s define d b y th e sam e equatio n i n th e subspac e o f matrice s with vanishing trace . Variety I i s a hyperplane sectio n o f variet y 0 ; thes e varieties hav e degre e three. Varietie s II.l an d Il'. l hav e degree four, varietie s II.2 and Il'. 2 hav e degree six, varieties II.3 and Il'. 3 hav e degree 1 4 , and varieties II.4 and Il'. 4 have degree 7 8 (cf . Chapte r III , Proposition 2.1 0) . The remaining part o f this sectio n i s devoted t o a proof o f Theore m 5.2 .

94

IV. SEVER I VARIETIE S

5.4. LEMMA . In the conditions of the theorem, let Z = SingX* . Then SXcX*. PROOF. Fo r a e l w e hav e mult Q X* > 2. Hence , i f a , / ? e E , a ^ / ? , then the line (a , P) intersect s X* wit h multiplicity a t least 4, and therefor e this line lie s in X* . Thu s SlcX*. • 5.5. REMARK . Sinc e codimX * < degX * - 1 = 2 , ther e ar e tw o possi bilities: codimZ * = 1 an d codimX * = 2 . Th e secon d cas e i s eas y t o investigate sinc e classificatio n o f varietie s o f degre e 3 and codimensio n 2 is fairly simple : al l nondegenerate varieties with such invariants ar e cones over sections o f th e Segr e variety P 1 x P 2 c P 5 b y linea r subspace s o f P 5 (cf . [104]), and sinc e X i s nondegenerate an d codim X > 1 (becaus e otherwis e codeg* = d e g * - ( d e g * - l ) d i m * ^ 3) , X i s th e Segr e threefol d 0 . On e can avoi d referenc e t o [1 04 ] by considerin g a general hyperplan e sectio n Y of a variety X wit h codimX * = 2 . I t i s eas y t o se e tha t Y* c p ^ " 1 ^ i s obtained b y projectin g X* fro m a general poin t o f P^* . Sinc e Y* i s a hypersurface, i t suffices t o classify varietie s X o f codegree 3 for whic h X* i s a hypersurface an d to find out which of them ar e smoothl y extendible , i.e. , ar e hyperplane section s of nonsingula r varietie s (cf . Corollarie s 5. 7 an d 5.1 0) . Thus, i n wha t follow s w e may assum e tha t X* i s a hypersurface .

5.6. LEMMA . Either Z = P ^ - 2 or the hypersurface X* is normal. Suppos e tha t X* i s not normal . The n fro m th e Serr e normalit y criterion i t follows tha t there exists a component I 0 c Z suc h that di m X0 = dimX* - 1 = i V - 2 , an d Lemm a 5. 4 show s tha t SI. Q c X* . Le t A b e a genera l plan e i n P** , an d pu t X*' = X* nA, l! Q = Z Q n A. The n X*' is a n irreducibl e plan e cubic , l! 0 i s a unio n o f degS 0 distinc t point s an d Sl!0 c X*' . Hence , degZ 0 = 1 an d Z 0 = p^"" 2 . Furthermore , I = Z 0 sinc e otherwise fro m Lemm a 5. 4 it would follo w tha t X* contain s the hyperplan e spanned b y Z 0 an d a point fro m X \ I 0 . • PROOF.

5.7. COROLLARY . Let X c P be a nonsingular projective variety such that codeg Z = 3 , eodimX * = 2, and let Y n~x c P ^ _ 1 be a general hyperplane section of X . Then code g Y = 3, codi m 7* = 1 , and Sin g Y* = ¥ N~~3. I t i s clea r tha t Y* i s obtaine d b y projectin g X* c F N* fro m a general point £ e P N*. Sinc e SX* = FN* (t o prove this it suffices t o consider the section o f X* b y a general three-dimensional linea r subspace A c P^*) , the finite ma p X* — > Y* i s no t a n isomorphism . Hence , th e variet y Y* cannot be normal, and from Lemm a 5. 6 it follows that Sin g Y* = FN~ . • Since X* i s a hypersurface , a e SmX* i f an d onl y i f th e hyperplan e section L af) X ha s a uniqu e nondegenerat e quadrati c singula r poin t (cf . [16]). Fo r x e X w e denot e b y I, x c & x th e se t o f hyperplane s / ? e 3? x for whic h the singularity o f the hyperplane sectio n L Bf\X a t the point x i s PROOF.

§5. VARIETIE S OF CODEGREE THREE

95

not a nondegenerate quadrati c singula r point (w e identify & x an d X ^ wit h their image s in P^ * unde r the morphism n) . The n X D {\JxeX^x^ anc * t * le points from X \ (U^e* ^c) correspon d to hyperplane section s having severa l nondegenerate quadrati c singula r points. I n the case when X* i s normal we have X = \J xexi:x. Let x G X b e a poin t fo r whic h I, x ^ &> x. Takin g N - n point s a 0 , .. . , OLN_n_x i n genera l positio n i n & x an d denoting by A a, a e ^ x , the quadrati c ter m i n th e Taylo r expansio n o f th e equatio n o f th e hyper plane section L af)X i n some system of local coordinates in a neighborhood of th e poin t x i n X , w e se e tha t X ^ c & x i s define d b y th e equatio n det \tnA„ H Vtjj „ *A„ 1 = 0, an d so for J V > n + 2, X v i s a hypersurface o f degree at most n i n & x = p N~n~~l. Sinc e X c P^ * i s defined b y quadratic equation s (b y vanishing o f the partial derivative s o f the equatio n of X*) , fo r all x fo r which l x i s distinct fro m & x th e subvariety X x c & x is a hypersurface o f degree at most two. For N > n + 2 w e will distinguish betwee n the following tw o main cases : I. For all x € X for which & x dimc5^-(/c-l ) = 3 , and fro m th e theorem o n tangencie s i t follows tha t n > 4. Since d i m ^ - dim Z = dim E - n + 3 > 1 , we have dimi ? > n - 2 , an d arguing as in the case k = N - 3 w e see that £ c L = ( S ) * = P 3 . Hence , the case di m is = n - 1 i s impossible , an d fo r d i m £ = n - 2 w e have n = 5 , N = 9 , an d £ = L = P 3 . Moreover , fro m th e abov e i t follow s tha t i n th e last case l! i s a hypersurface i n £ = P 5 an d di m Yp = 3 fo r a general poin t j8el'. Let x b e a general point of X , an d let Y x = \JpYp , where / ? run s through the se t of genera l points o f L x . B y the theorem o n tangencies, di m Yx > 4 . On the other hand , i f di m Y x = 5 , the n Y x = X s o that fo r a general poin t y e l ther e exists a hyperplane (/?) * tha t is tangent to X a t x an d y . Fro m the Terracini lemma it follows that X ca n be isomorphically projected to P , which was already show n to be impossible ( 8 = In - 2) . Thus , di m Y x = 4 , and Bertini' s theore m yield s the inclusion Y x c ((£*)* ) = (r £ , L 3 ). Le t y b e a general point o f Y x . A dimension coun t show s that y lie s on a onedimensional family o f Y« . Hence , E^n E = ^ xn^ = P 1 , and a dimension count show s that, varying y e Y x , we thus obtain a general line in the plan e Z v . Fro m th e theore m o n tangencie s i t follow s tha t di m MVYV = 4 , wher e •* 7

7

y run s throug h th e se t o f genera l point s o f th e lin e Z v n Z.,. Thus , U F •* .

coincides with both Y an d Y , so that Y = Y v an d y\

y I

where y run s throug h th e set of general points o f Y x.

r7

7

98

IV. SEVER I VARIETIE S

Since for a general point y e Y x w e have d i m ^ n Z ) = l , dim(X x , Z y)* = 5 an d Y x c (2^ , X y)* = P * D L 3 . Le t x b e anothe r genera l poin t o f X. The n Y x, c P ^ D L3 an d di m ( p j, P*, ) < 1 0 - 3 = 7 . Sinc e Y x n l y is nonempt y (thes e subvarietie s intersec t wit h eac h othe r o n L) , w e hav e dim(Yx n Y^ ) > 8 - 5 = 3 . Thus , the linear subspace (P* , V5X,) i s tangent t o X alon g the subvariety Y xV\Yx>, which contradicts the theorem on tangencies since d i m ^ n Y x,) > 3 > 2 > dim(P * , P*,) - dim* . Thus, the cas e k = N - 4 i s also impossible. It remain s t o conside r th e cas e k = N - 2 . Sinc e £ = n{^ E), w e hav e d i m ^ g = d i m £ + (N - H - 1 ) > dim Z = N - 2 , an d s o d i m £ = n - 1 . Arguing a s in th e cas e k = N - 3 , w e se e tha t i?"" 1 c L = (X) * = P 1 an d therefore n < 2. Sinc e for / ? e l! w e have di m Y fi>(N-2)-(k-l)=l, X i s a surfac e an d E = L i s a lin e o n X . No w th e lemm a follow s fro m Proposition 3 from [96] . D 5.10. COROLLARY . Let X n c¥ N be a nondegenerate nonsingular variety such that codeg X = 3 , codimZ * = 2 . Then n = 3 , N = 5 , and X = P1 x P 2 c P 5 is a Segre variety. Fro m Corollar y 5. 7 i t follow s tha t a genera l hyperplan e sectio n Y o f th e variety X satisfie s th e condition s o f Lemm a 5.9 ; so Y = ¥ { . Th e corollary now follows from well-know n results on extension of projective varieties (i n thi s cas e i t i s eas y t o verif y directl y tha t th e standar d morphism s Fj - • P 1 an d ¥ { - > P 2 define d b y th e linea r system s \F\ an d \s + F\ , respectively, exten d t o X an d defin e a n isomorphis m I ^ P ^ P 2 ) . D A different proo f o f th e corollar y i s given i n Remar k 5.5 . 5.11. REMARK . A close analysis of our proof o f Lemma 5. 9 shows that w e actually use d onl y th e fac t tha t S = Sin g X* i s a linear subspace . Thus , ou r method allow s us to giv e a classification o f al l varieties havin g thi s propert y (the lis t o f suc h varietie s include s al l rationa l scroll s ¥ e (e > 0) o f degre e e+2 embedde d in P ^ (N < e+3) b y means of a very ample linear subsystem of the linear system \s + (e+l)F\, wher e s ~ P 1 , {s 2)x = -e i s the minima l section, an d F ~ P i s a fiber). Similarly , usin g our technique s i t shoul d b e possible to classif y al l varieties fo r whic h 5 1 c X* (cf . Lemm a 5.4) . To prov e th e theore m i t remain s t o conside r cas e II . W e begi n wit h giv ing a lowe r boun d fo r th e dimensio n o f I whic h hold s unde r ver y genera l assumptions. PROOF.

5.12. LEMMA . Let X n c P be a nonsingular projective algebraic variety such that di m X* ~ N - 1 , and let L = SingX* . Then either X is a quadric and 2 = 0 , or di m Z > n - 1 . PROOF. Sinc e X* i s a hypersurface, a e X* \ I i f an d onl y i f th e hyper plane section ( a f f l l ha s a unique nondegenerat e quadrati c point . I f X i s

§5. VARIETIE S OF CODEGREE THREE

99

a hypersurface o f degree d , the n the Gauss map n : X - > X* i s finite, and it is clear that fo r d > 2, dim Z = n -1 (fo r d = 2, X i s a quadric, X * ~ ^ , 1 = 0 ). Suppose now that rc < JV-2. W e have already shown that in this case there exists a n irreducible subvariet y S? x c & x suc h tha t di m S?x = d i m ^ . 1 = i V - 2 , p( dimX 0 + \ - (N ~ n-\) =

?r

and, a s w e hav e jus t shown , (i? a)* D T z a , i.e. , d i m r z a < N - 1 dim (R)*. Sinc e dimS 0 = N - f - 2 < di m r 2 a , from thi s i t follow s tha t dim (R a) < f + 1 , wher e equalit y hold s i f an d onl y i f di m r L a = dimZ 0 , i.e., a i s a nonsingular poin t o f Z 0 . We observe that R a i s not a linear subspace of F N . In fact, whe n fi run s through the set of general points of 2 0 an d y run s through the set of genera l points of (a , ft) , x = p (n~l (y)) b y definition run s through the set of general points of R a , and from th e Terracini lemma it follows that (x)* D T z ^ . I f Ra wer e a linear subspac e o f P ^ , fro m th e Bertin i theore m i t woul d follo w that Z 0 c (R a)* — (R a)* 5 whic h contradict s th e nondegenerac y o f X 0 . Thus, R a ^ (R a), an d sinc e dimi? a > § an d dim(i? a ) < f + 1 , w e se e that R a i s an irreducibl e hypersurfac e i n (RJ = P ? + 1 , 7r _1 (5^Z0) = 3° R , and a e S m Z0 . Sinc e thi s i s tru e fo r a n arbitrar y poin t a e Z 0 , E 0 i s a nonsingular variety . D 5.15. LEMMA . In case II, X n is a Severi variety\ and in case II' , X n is an isomorphic projection of a Severi variety X n to P 2 + I (we use the notation of Lemma 5.1 4) .

§5. VARIETIE S OF CODEGREE THREE

101

PROOF. I n case II, Z 0 i s a Severi variety by definition (cf . Definitio n 1 .2) . From the description o f the structure of Severi varieties given in Remark 2. 5 it follow s tha t X = (SZ 0)* i s also a Severi variety. Furthermore , SX = Z j , Z = Z 0 . Thi s proves Lemma 5.1 5 i n case II. It remain s t o conside r cas e II' . A s in th e proof o f Lemm a 5.9 , le t 5? = \JX Zx c &* x , where x run s throug h th e se t o f genera l point s o f X . Fro m Lemma 5.1 4 i t follow s tha t fo r a general poin t x e X w e have Z 0 n ^ = lx , s o tha t n{^) = Z 0 . Sinc e d i m ^ = n + (N - n - 2 ) = T V - 2 an d dimZ 0 = n - 1 , fo r eac h point Q G I 0 w e have

dim Y a > (N - 2 ) - (n - 1 ) =T V - n - 1 (here, as above, Y a denote s th e varieties n~ {(a) an d p(n~ l(a)) whic h ar e naturally isomorphi c t o eac h other) . Bu t accordin g t o th e theore m o n tan gencies di m Y a < N - n - 1 . Hence , di m Y a = N-n-l = % for al l a G Z0. Let a e Z 0 , y G Y a b e genera l points . The n th e hyperplan e (y)* i s tangent t o X* alon g & , and fro m Lemm a 5.1 4 an d th e Terracin i lemm a it follow s tha t (y)* i s tangen t t o Z 0 alon g th e quadri c Z 0 n ^ y = Z ^ 3 a. I n particular , th e linea r subspac e (Y a)* c P^ * i s tangen t t o Z 0 a t th e point a , an d therefor e (Y a) c {{T z j Q )*)* + 1 . I f / ? G ly i s anothe r poin t for whic h (a , /? ) £ Z , the n i t i s clea r tha t Y anYp = y e R a an d th e intersection i s transvers e (s o tha t i n particula r (Y*) x = 1 ) . Fo r a genera l point a e l 0 w e have Y a c R a , where i? a i s the variety introduce d i n th e proof o f Lemma 5.1 4 . Sinc e dim7 a = \ an d R a i s an irreducible nonlinea r hypersurface i n (RJ = ((r z J * ) f +1 , w e conclude tha t 7 Q = i? a . Let x b e a general point of X . The n the hyperplane (x) * i s tangent to Z 0 along the subvariety Z x c Z 0 , i.e. , T(Z x , Z 0) c (x)* . Fro m Theorem 1 . 4 i n Chapter I it follows that dim5(Z ;c , Z 0) = ( f - 1 ) + (n - 1 ) + 1 = \n - 1 an d a general point { e S(Z X, Z 0) lie s on a finite number o f secant s joining point s of Z x wit h point s o f Z 0 . W e hav e £ G & = 5Z y fo r a suitabl e genera l point y G X, an d b y the above, the intersection Z v n Z„ = ^ n ^v reduce s ^_ y


r (dim X = n , h°{X, @ x{\)) = N + I) an d to projec t i t isomorphicall y t o P r ; th e imag e o f X unde r thi s projectio n i s r

0

a nonsingula r variet y X c F fo r whic h dim^ T = n an d h (X, 0 , be a Scorza variety. Suppose that n = 0 (mo d 3), so that M(n , 5) = f(k 0), k 0 = | , and let u e S °~ lX and z e SX be generic points. Then s k _ x = N - 1 and

{?1 v

=\-(

n

+ 1- \) =

N L

-

From (1 .4.2 ) i t follow s tha t T{Yu,X)cLu = 1 Ts^Xu. (2.2.

)

Since X i s a nondegenerate variety, fro m (2.2.1 ) an d Theore m 1 . 4 o f Chap ter I it follow s tha t dimS(Y u , X) = dim Y u + n + 1 = In + 1 - 6 = s. (2.2.2

)

Equality (2.2.2 ) mean s tha t S(YU,X) =

SX (2.2.3

(similarly, on e ca n sho w tha t S(Y Z, S k°~2X) = S k°~[X). Fro follows tha t YzCiYu* 0 . (2.2.4

) m (2.2.3 ) i t )

In vie w o f Corollar y 1 .5 , Y dz c P^ +1 = SY z i s a nonsingular quadric . Sinc e Lu ~jf> P z , L u D P z i s a hyperplane i n P z tha t i s tangent t o Y z a t al l point s of Y znYu. Fro m thi s and (2.2.4 ) i t immediatel y follow s tha t (Y z-Yu)=l. Lemma 2. 2 i s proved . We retur n t o th e cas e 3=1 . I n thi s cas e (2.2.1 ) mean s tha t L u-X = rYu + Eu, wher e r > 2, Y u 3), ( 5 = 0 , fee a Scorz a vartery . 77zen f/ze algebraic system of divisors H u = (L u • X)FN , L u = T skQ-\x u , u e does not have fundamental points on X . PROOF O F LEMM A

point. The n

2.4. Suppos e the converse, and let y b e a fundamenta l

^n T£

5 •kn-i ^>~1A',iis

u€Sk0-\x

and therefor e S ° l X i s a con e wit h verte x y . Hence , Lemm a 2. 4 i s a consequence o f th e following result . 2.5. LEMMA . Let X n c F M{n'S), S S °~ lX is not a cone.

= 8{X) >0,bea Scorza

variety. Then

PROOF. W

e prove the lemma by induction. I f n = 23 , the n X i s a Severi variety an d Lemm a 2. 5 follow s fro m result s o f Chapte r IV . Usin g Theo rem 1 .4 , suppos e tha t fo r a generi c poin t u e S k°~lX th e variet y S k°~2Yu k— 1/

is not a cone an d th e variety S ° X i s a cone wit h verte x v . Le t u b e a generic point o f the lin e (v , u) . The n u eS° X, L u, = T sk0-ijCtU, = T ^ o - i ^ = L M, and henc e Y ^ = Y" M (cf . (1 .4.4)) . Fo r a generic poin t x e Y u, w e conside r the curv e < / ' ^ T V x (v , u)°)) C in the plan e U = (v,x,u)cS k°-lYu, (2.5.2

S ^\ (2.5. 1

) )

where (v , u)° denote s th e se t o f generi c point s o f th e lin e (v , u) . Sinc e C sk _ 2-sk-\~2 ( ^- f Qrm ula (1 .5.4 ) i n Chapte r V) , we may assum e tha t

(v, u) n s*0 - 2 y u = (v, u) n s*°~\jf = v. (2.5.3

)

128

VI. SCORZ A VARIETIE S

From (2.5.1 ) an d (2.5.3 ) i t follows tha t (v , u) n C v x u = v, an d sinc e u i s a generi c poin t o f II , C v x u consist s o f severa l line s passin g throug h v . Hence, fo r a generi c poin t w e S °~ 2YU w e hav e (v , w) c S °~ 2Yu, i.e. , contrary t o ou r assumption , S k°~2Yu i s a cone with vertex v . This contradiction prove s Lemma 2. 5 and therefore als o Lemmas 2. 4 an d 2.3. From Lemm a 2. 3 it follow s tha t L U - X = 2Y U, (2.5.4

)

so that , i n particular , Y u i s a n ampl e diviso r o n X . No w i t i s eas y t o prove Theore m 2. 1 b y inductio n o n n usin g Theore m 1 .4 , Corollar y 1 .6 , the classica l resul t o f Sever i fo r n = 2 (cf . [82 , n ° 8 ] or Theore m 4. 7 a) i n Chapter IV) , and th e well-know n theore m t o th e effec t tha t i f a nonsingula r variety X n , n > 2, contain s a n ampl e diviso r Y ~ F n~l, the n X ~ P " (cf . [63]). However , w e give a direct proo f o f Theore m 2.1. From (2.5.4 ) i t follow s tha t th e image o f th e rationa l ma p Sn~lX — > Pic°X, u

^ cl(Y

u

- Y UQ)

(where u 0 i s a fixed generic point of S n~ X) i s contained in the set of points of orde r tw o o n th e Picar d variety . Sinc e S n~lX i s a n irreducibl e variety , from thi s it follows tha t fo r generi c points u , u e S n~lX w e have Y u ~ Y u> (where ~ denote s linear equivalence) . Consider th e complet e linea r syste m o f divisor s %* — | YJ o n th e variet y X. W e observe that a general diviso r H e %? has the for m H=YU, ueS

n l

~ X. (2.5.5

)

In fact , le t x Qy .. . , xn_l e H b e a generi c collectio n o f n points , an d le t u e {x Q, .. . , xn_x) b e a generic point . The n i t i s clea r tha t u i s a generi c point o f S n~xX, an d fro m Propositio n 1 .4b ) o f Chapte r V it follows tha t L

u = (T x^,...,TXxJ. (2.5.6

)

On th e othe r hand , sinc e X i s a n extrema l variety , X i s linearl y normal , and therefor e th e diviso r 2H ~ 2Y U i s cu t b y a hyperplan e L H c P , i.e., 2H = L H X . Hence , T(H , X)cL H, s o that fro m (2.5.6 ) i t follows tha t LH = Lu, 2H

= LH-X =

LU.X = 2Y U,

and H = Y u a s stated i n (2.5.5) . From Lemm a 2. 4 it follow s tha t th e linear syste m 2) .

Thus,

qVc 2 ", c;nc

u

M 2

Dy,

From th e abov e consideration s i t follow s tha t H u is a connected diviso r whose component s pairwis e intersec t wit h eac h othe r alon g cycle s o f codi mension tw o i n X lyin g in Sin g Hu an d

(CT + c2" • Y2) = i\ + /* = (L, • rz v = (*« • *,)*• (3- 1 -2) Let £ , F b e a n arbitrar y pai r o f irreducibl e component s o f H u . The n £ H F c S i n g i / M = YL i.e. , r1 ( £ n F , I ) c L „ . (3. .3

)

On th e othe r hand , fro m th e nondegenerac y o f X i t follow s tha t S(EnF,X)£L 1 u. (3. .4

)

In view o f (3.1 .3 ) an d (3.1 .4) , Theorem 1 . 4 fro m Chapte r I implies tha t dimS(Ef)F, X)

= &\m{Er\F) + n + 1 = In - 1 =&imSX.

Hence S(E nF , X) = SX, and therefore , fo r a generic point z e SX, 0 < car d (Y z • (£ 1 n F)) < oo (3. .5

)

(compare wit h th e proo f o f Lemm a 2.2) . Fro m (3.1 .1 ) , (3.1 .2) , an d (3.1 .5 ) it follow s tha t u Hu = Cul+CZ, (C" Y u. (3. .6 ) rC 2)1 x=

132

VI. SCORZ A VARIETIES

Since s = In -1, th e variety X ca n be isomorphically projected t o P " From Barth' s theore m [6 ] it follow s tha t hl(X9 0 X) = \h\X 9 C

l

) = 0, H

(X, @*

X)

0, be a Scorza variety. Suppose that n = 0 (mo d S) {so that, in accordance with Proposition 1 .2 , n - k 0S), let a, b be natural numbers such that a + b = k 0, and let v e S aX and w € S bX be generic points. Then (Y V*YW)X=- 1 . 4.2 . Firs t w e sho w tha t Y vnYw^0. T suffices t o verify tha t fo r a generic point w e S X we hav e PROOF O F LEMM A

1 S(Yw,Sa~lX)=SaX. (4.2.

o d o thi s i t

)

We verify equalit y (4.2.1 ) b y induction o n a . Fo r a = 1 , (4.2.1 ) reduce s t o formula (2.2.3 ) prove d i n Lemm a 2.2 . Assumin g (4.2.1 ) , we show tha t S(Yw,,SaX)=Sa+lX for a generic point w' G S ~ lX. Le t x b e a generic point o f X , and let w be a generic point o f (w f, x) c S X. The n fro m (4.2.1 ) , Theorem 1 .4 , an d

136 VI

. SCORZ A VARIETIES

Lemma 2. 2 it follow s tha t S(YW,, S aX) = S (Y w., S(Y

w

a l

, S

- X))

= S (S(Y W,, 7 J , S*" 1 *) = S(SY w , S a~lX) /

(4-2-2)

i x

= 5 ( ^ , 5 ( ^ , 5 " - ^ ) =5(y = S (S{Y W , 5fl"1^T), x) =

fll>5

fl

jr)

5(5 f l X, X ) = S* +1 X

as required . In view o f Propositio n 1 . 2 c) and formul a (1 .5.4 ) fro m Chapte r V , sa =s a_x+n+ l-ad = sa_l+kQS+l-aS = sa_x+bS+l =s a__l+dimYw + l 9 and fro m (4.2.1 ) i t follow s tha t fo r generi c point s v e S aX, w e S X th e varieties Y v an d Y w intersec t a t finitely many points . To prove Lemm a 4. 2 it remains t o verify tha t th e morphis m < >:

f ^Y w,tSaX -"

*

%$is birational . Thi s i s agai n prove d b y inductio n o n a , and , i n vie w o f th e chain of equalities (4.2.2), it suffices t o show that in the commutative diagra m of rationa l map s (4.2.3)

I 4 c

k

a+l v

e

*YtlliSaX *

**

A

the generi c fibers of th e morphis m a t th e botto m ar e connected . Bu t thi s i s really so , sinc e otherwis e fo r a generi c poin t v e S a+lX th e intersectio n (Yy/ • Y w)x woul d consis t o f severa l distinc t J-dimensiona l quadric s o f th e form Y z, z e SX (cf . Corollar y 1 .5) , wher e by Lemm a 2. 2 1

{Y-Y)Y = ^ z v'Y

i

contrary t o the induction assumption , accordin g t o which

(Yv-(Yv.-YJx)Yi=(Yv.YJx =

l. D

v

We need several results valid for an arbitrary Scorza variety X n c P ^ " ' ^ , 6 = 8{X) > 0, suc h tha t n = 0 (mo d S) (recal l tha t b y Lemm a 2. 2 unde r these assumption s S °~ lX i s a hypersurface i n P ^ ) . 4.3. LEMMA . Let X n c F M{n'S), 6 = 8{X) , be a Scorza variety. Suppose that n = 0 (mo d S) . Let 0 < a < k Q - 1 , and let v be a generic point of SaX. Then degS* 0 " 1 * = k 0 + 1 , mult,, Sk°~]X = k Q-a. e argue by induction . Suppos e tha t Lemm a 4. 3 holds fo r k fQ < k0 . Fo r a = k Q-1 th e assertion of the lemma is obvious. Le t 0 < a < kQ- 1 , PROOF. W

§4. SCORZ A1 VARIETIE S WIT H< 5 = 4 3

7

FIGURE 6. 1

b = k Q- a, an d le t v b e a generic point o f S aX an d w a generic point o f SbX. B y Lemma 4.2 ,

YvnYw=xeX By Theorem 1 .4 , Y v an d Y assumption i t follow s tha t (x,v)nSa~lX =

w

ar e Scorz a varieties , an d fro m th e inductio n

{x,v'}, {x,w)nS

b l

~X =

{x,w'}, 1 (4.3.

where v an d w' ar e generi c point s o f th e varietie s S a~xX an d S ~ respectively. Pu t u = {v,w)n(v' 9 w) e S k°~lX (4.3.2

) l

X, )

(cf. Figur e 6.1 ) . Then u i s a generic point o f S k°~l X , an d therefor e mult,, Sk°~lX + mult^ S k°~lX + 1 < d , (4.3.3

)

where rf = d e g 5 V 1 J f . We claim tha t (4.3.3 ) i s actually a n equality , i.e. , mult,, Sk°~lX + mult^ S k°~lX + 1 = rf.

(4.3.4

)

To sho w this it suffice s t o verify tha t (v, w)r\S k°~lX =

vuwuu.

Suppose tha t thi s is not so , and le t u € {(v , w) n S k °' l X) \(v\Jw\Ju): (4.3.5 Set u = (v , w ) n (x, u ).

)

138 VI

. SCORZ A VARIETIE S

Then u" € S °~ X, an d fro m th e genericity assumption s i t follow s tha t (JC, u")