Trisecant Lines and Jacobians, II

Compositio Mathematica 107: 177±186, 1997. 177

c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Trisecant lines and Jacobians, II

OLIVIER DEBARRE ? UniversiteÂ Louis Pasteur, DepartementÂ de Mathematiques,Â 7, Rue ReneÂ Descartes, 67084, Strasbourg Cedex,Â France; e-mail:[email protected]

Received 28 December 1995; accepted in ®nal form 4 June 1996

Abstract. We prove that an indecomposable principally polarized complex abelian variety X is the X

Jacobian of a smooth curve if and only if there exist points a; b; c of whose images under the

!j j X

Kummer map X 2are distinct and collinear, and such that the subgroup of generated by

b b c X a and is dense in . Mathematics Subject Classi®cations (1991): 14H42 Key words: complex abelian variety, Jacobian, trisecant, Schottky problem.

1. Introduction

X; )

Let ( be a complex principally polarized abelian variety. Symmetric repre-

sentatives of the polarization differ by translations by points of order 2, hence

the linear system j2 is independent of the choice of . It de®nes a morphism

X !j j K (X ) X (X; )

K : 2, whose image is the Kummer variety of .When is

(X )

the Jacobian of an algebraic curve, there are in®nitely many trisecants to K ,

j K (X ) i.e. lines in the projective space j2 that meet in at least 3 points. Welters conjectured in [W] that the existence of one trisecant line to the Kummer variety should characterize Jacobians among all indecomposable principally polarized abelian varieties, thereby giving one answer to the Schottky problem. The aim of this article is to improve on the results of [D], where a partial answer to this problem was given under additional hypotheses. More precisely,

our main theorem implies that an indecomposable principally polarized abelian

X; ) a; b; c X

variety ( is a Jacobian if and only if there exist points of such that

a b b c X

(i) the subgroup of X generated by and is dense in ;

(a) K (b) K (c) (ii) the points K , and are distinct and collinear. Instead of working on the intersection of a theta divisor with a translate, whose possibly complicated geometry is the source of most dif®culties in [AD], [D], [M] and [S], we perform algebraic calculations directly on a theta divisor, which has the advantage of being integral. The point is to prove that the existence of

? Partially supported by N.S.F. Grant DMS 94-00636 and the European HCM project `Algebraic Geometry in Europe', contract CHRXCT-940557.

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one trisecant line implies the existence of a one-dimensional family of such lines. Welters' criterion ([W]) then yields the conclusion.

2. The set up

X; )

Let ( be a complex indecomposable principally polarized abelian variety, let

K X !j j

be a symmetric representative of the polarization and let : 2be the

O () x 2 X

Kummer morphism. Let be a non-zero section of X .Forany ,we

+x z 7! (z x) O ( )

x X x

write x for the divisor and for the section of .If

c X K (a);K(b) K (c)

a; b and are points of , it is classical that the points and are

collinear if and only if there exist complex numbers ; and not all zero such

that

+ + = :

a a c c

b b 0

Following Welters, we consider the set

V = f 2 X j K ( + a);K( + b);K( + c) g;

a;b;c 2 are collinear

X; )

endowed with its natural scheme structure. By [W], Theorem 0.5, ( is a

a; b c V >

Jacobian if and only if there exist distinct points and such that dim a;b;c 0.

fD g n> This condition is equivalent to the existence of a sequence n 0 of constant

(") = ( )+ D(") D (") =

vector ®elds on X and of a formal curve 0 2with

D " V

n> n

0 , contained in a;b;c . This in turn is equivalent to a relation of the type

(") + (") + (") = ;

+ (") a (") b+ (") b (") c+ (") c (")

a 0

("); (") (") [["]] where and are relatively prime elements of C .

3. The case of a degenerate trisecant

In this section, we prove the following result:

X; ) THEOREM 3.1. Let ( be a complex indecomposable principally polarized

abelian variety, let be a symmetric representative of the polarization and let

X !j j u v

K : 2be the Kummer morphism. Assume that there exist points and

u 6=

of X such that 2 0 and

(u) K (v )

(i) the points K and are distinct and the line that joins them is tangent

(X ) K (u)

to K at ;

> : ru

(ii) codimX 2 2 2

r Z

X; ) Then ( is isomorphic to the Jacobian of a smooth algebraic curve.

comp4012.tex; 12/06/1997; 8:48; v.7; p.2 TRISECANT LINES AND JACOBIANS ± II 179

Note that condition (ii) in the theorem is equivalent to saying that there are no

u u X

hypersurfaces in invariant by translation by 2 ; it holds when generates .

u;v

Proof. As explained in Section 2, it is enough to prove that the scheme u;

fD g n>

has positive dimension at 0: we look for a sequence n 0 of constant vector

D 6= ("); (") (")

®elds on X with 1 0 and relatively prime elements and of

"]]

C[[ such that

P (z; ") = (") + (")

+D(")= uD(")= u+D(")= uD(")=

u 2 2 2 2

+ (")

+D(")= vD(")=

v 2 2 (3.2)

D (")= D " n vanishes, with n>0 . This is nothing but equation (1.4) from [D]. It

follows from loc. cit. that we may assume

(")= + " ; (")= ; (")=":

1n 1

n>0

P (z; ")= P " P O ( ) n >

> n n X

Write n 0,where is a section of 2 for each 0. =

One has P0 0and

P = + D D + :

u u u u u v v 1 1 u 1 1 (3.3)

As explained in loc.cit., hypothesis (i) in the theorem is equivalent to the van-

P D K (D ) K (u)

ishing of 1 for a suitable 1 such that 1 is tangent at to the line that

K (u) K (v ) P

joins and , and a suitable 1. In general, note that n depends only

;::: ; D ;::: ;D P n

on 1 n and 1 . Knowing that 1 vanishes, we need to construct

fD g X f g

n> n n>

a sequence n 0 of constant vector ®elds on and a sequence 1 of

P n

complex numbers such that n vanishes for all positive integers .

It is convenient to set X

R (z; ")=P(z+ D(");")= R (z )" : 2n

n>0

We begin by proving a few identities. Note that

R (;")= (") +" :

u u v

uD(") uD(") v D (")

(3.4)

Modulo u ,weget

R (;") +" ;

u v

D(") v D (")

u (3.5)

(") + " R (;") ;

u u u+v

uD(") v D (")

232 2u (3.6)

R (;") (") :

uv uv v

vD(") uvD(") 2 2 (3.7)

comp4012.tex; 12/06/1997; 8:48; v.7; p.3

180 OLIVIER DEBARRE

P u

Since 1 and its translate by 2 both vanish, formula (3.3) yields, modulo u ,

D ;

u u v v

1 0 (3.8)

+ : D

u uv u+v 3u 1 2 2 0 (3.9)

The following result is the main technical step of the proof.

LEMMA 3.10. Modulo u , one has

(")R (;") : + R(;") + "R (;")

v u u v uv u u+v 3u 2 20

Proof. By (3.5), (3.6) and (3.7), the left-hand side of the expression in the

lemma is congruent modulo u to

(") + " (")

u u v v u v

D (") v D (")

u 3 3

+ (") + "

u u v u+v u v

D (") uv D (")

3 u 2 2

" : +" (")

uv u u+v v u+v u

v D (") v D (") 2 2 2u 2

All terms cancel out but the second and the ®fth. Since (3.8) and (3.9) yield

+ 2

u v uv u u+v

v 3 2 2 0, the sum vanishes.

n > ;::: ;

We proceed by induction: let be an integer 2 and assume that 1 n 1

D ;::: ;D P = = P =

and 1 n 1have been constructed so that 1 10. We want

D P X

n n

to ®nd a complex number n and a tangent vector such that vanishes on . P

By [D], Lemma 1.8, it is enough to show that the restriction of n to the scheme

\ ;::: ; D ;::: ;D

u n n

u (which depends only on 1 1and 1 1)vanishes.

R = = R =

Our induction hypothesis can be rewritten as 1 n 10and

P = R R \

n n u u

n . Therefore, we need to prove that vanishes on the scheme . +

n 1

(") " "

Since 1 modulo , the identity of the lemma taken modulo yields

R +(R ) ;

v n u n u u

3 20

v 6= u u

modulo u ;since is integral and ,weget

+(R ) R :

u n u u

n 3 20(3.11)

F L X R F

LEMMA 3.12. If is a section of an ample line bundle on such that n

\ r R F

u n ru

vanishes on the scheme u , then, for any integer , the section 2

vanishes on the scheme u .

P = R O ( ) R F

n X n

Proof. Recall that n is a section of 2 ,sothat is a section

L( ) I

of 2 . The Koszul complex yields an exact sequence

!L!L( ) L( ) !L( ) I : !

u \

0 u 20

comp4012.tex; 12/06/1997; 8:48; v.7; p.4 TRISECANT LINES AND JACOBIANS ± II 181

H (X; L)= B C

Because L is ample, one has 0, and there exist sections and

R F = B + C (R ) ) F B (

n u u n u u u u

such that . It follows that 2u 2 2 3 mod . F

Multiplying (3.11) by 2u ,weget

R F + B ( ):

u u u u u u

n 2 3 2 3 0 mod

u 6=

u u

Since 2 0and is integral, 3u is not a zero divisor modulo and we

R F ( ( ; )) R F

n u u n u

get 2u 0 mod . By a similar reasoning, 2 0

( ( ; )) 2

mod u .

R \ r

n u+ u u

(3.13) The lemma implies that 2ru vanishes on for all . Because R

of hypothesis (ii), it follows that n vanishes on each primary component of codi-

\ R

u n mension 2 of u ; since this scheme has no embedded components,

vanishes on it. This concludes the proof of the theorem. 2

R R

v n v

It follows from (3.5) that n , hence also its image by the involution

x 7! x \

u , vanish on the scheme u . Hypothesis (ii) in Theorem 3.1 can

therefore be relaxed to

> ( \ \ ) :

X ru v v

codim u 2 2 2 r Z

4. The case of a non-degenerate trisecant In this section, we prove, under an extra hypothesis, that the existence of a non- degenerate trisecant line implies the existence of a degenerate trisecant of the type

studied in Section 3.

X; )

THEOREM 4.1. Let ( be an indecomposable principally polarized abelian

K X !

variety, let be a symmetric representative of the polarization and let :

j j a; b c X 2 be the Kummer morphism. Assume that there exist points and of

such that

(a);K(b) K (c)

(i) the points K and are distinct and collinear;

> :

+qb+rc

(ii) codim pa 2 2

p;q ;r Z

+q +r =

p 0

X; ) Then ( is isomorphic to the Jacobian of a smooth algebraic curve.

Note that condition (ii) in the theorem is equivalent to saying that there are no

a b b c a b

hypersurfaces in invariant by translation by and ; it holds when

c X and b together generate .

comp4012.tex; 12/06/1997; 8:48; v.7; p.5

182 OLIVIER DEBARRE V

Proof. Instead of proving that a;b;c has positive dimension at 0, we will proceed

as follows. As explained in Section 2, condition (i) translates into the existence of

nonzero complex numbers ; and such that

+ + = :

a a c c

b 0 (4.2)

x X P

+b+c

For any in , we will write for a . Our ®rst aim is to show that

P \ a vanishes on the scheme b .

LEMMA 4.3. One has

c c

P ( ): + P

a 0 mod

b b

a 2

a + c) (a b)

Proof. Equation (4.2) and its translates by ( and yield, modulo

a ,

+ ;

c c

b 0

+ ;

a+c c

+b+c ab+c

2 a 0

+ :

b b ab+c abc

2a 0

It follows that, still modulo a

c c

(P + P )

b ab

b b

a 2

( )+ ( )

a+c c c c

bc a+b+c ab+c abc

2 a

( + ) :

c a+c c

bc a+b+c ab+c

a 2 0

b 6= a

a b Since is integral and , the section is not a zero divisor modulo

and the lemma follows. 2

X P F

LEMMA 4.4. If F is a section of an ample line bundle on such that

\ s P F

(ab)

vanishes on the scheme , then, for any integer , the section s

\ a

also vanishes on the scheme b .

a b P F

Proof.Since and play the same role, it is enough to prove that a

\ B a

vanishes on b . As in the proof of Lemma 3.12, there exist sections and

c c

C P F = B + C P F B ( )

a a

ab ab

such that b.Then mod .Using

b ab a 2

Lemma 4.3, we get

+ B P F ( ):

ab ab ab b b

a 2 2 0 mod

a b 6= a

b Since 2 and is irreducible, we can divide out by 2a ,andthe

lemma is proved. 2

comp4012.tex; 12/06/1997; 8:48; v.7; p.6

TRISECANT LINES AND JACOBIANS ± II 183

X P F

LEMMA 4.5. If F is a section of an ample line bundle on ,then vanishes

\ P F \

a a c

on the scheme b if and only if vanishes on the scheme .

F B ( )

c a

+b+c b Proof. As in the proof of Lemma 3.12, write a mod .

We get

B F

c c c

b a+b+c

F = P F ( );

+b+c b b b

a mod

a 6= b 2

whereweused(4.2).Since a is irreducible and , the lemma is proved. s

We combine the last two lemmas to get, for all integers r and ,

P F ( ; ) a

0 mod b

=) P F ( ; ) c

0mod a

( ; ) =) P F

a c

(ac)

r 0mod

=) P F ( ; )

(ac)

r 0mod

( ; ) : =) P F

(ac)+s(ab)

r 0 mod

P \

+r (ac)+s(ab)

It follows in particular that a vanishes on for all s

integers r and . As in (3.13), hypothesis (ii) in the theorem shows that

P \ a

vanishes on the scheme b (4.6)

a b

P \ P \

c c a

(hence also on b ,and on ).

X u = a b v = u a c

Let u be any point of such that 2 ,andset . Translating

(u b) \

v u u

(4.6) by , we get that v vanishes on . As explained in [D], D this is equivalent to the existence of a complex number 1 and a tangent vector 1

to X such that

+ D D + = :

u u u u u v v

1 u 1 1 0 (4.7)

(u) K (v ) K (X ) K (u) In other words, the line that joins K and is tangent to at . Note that we cannot apply Theorem 3.1 directly, since hypothesis (ii) may not be

satis®ed. However, we will still follow the same method, i.e. we will show that the

V V (a b)

u;u;v

scheme a;b; (which is a translate of ) has positive dimension at ,

b;c

but we will need to prove at the same time that a; has positive dimension at

a c)

the point ( .

n > V

Let be an integer 1. As in Section 3, the scheme a;b; contains a +

n 1

"]=" (a b)

scheme isomorphic to C[ and concentrated at if and only if

;::: ; D ;::: ;D n

one can ®nd complex numbers 1 n and tangent vectors 1 such

R ;::: ;R D

that 1 n, de®ned in Section 3, vanish ( 1 and 1 are the same as in (4.7),

comp4012.tex; 12/06/1997; 8:48; v.7; p.7

184 OLIVIER DEBARRE

R V

b;c

and 1 is the left-hand side of that equation). Similarly, the scheme a; con- +

n 1

"]=" (a c)

tains a scheme isomorphic to C[ and concentrated at if and only

0 0 0 0

;::: ; D ;::: ;D

if there exist complex numbers and tangent vectors such n

1 n 1

0 0 ;::: ;R that R vanish.

1 n

n > ;::: ; ;

We proceed as in Section 3: let be an integer 2 and assume that 1 n 1

0 0 0 0

;::: ; D ;::: ;D ;D ;::: ;D

and n have been constructed so that

1n 11 111

0 0

R ;::: ;R ; R ;::: ;R X

n vanish on . As in the proof of theorem 3.1, it

1 11n 1

R \

n a

is enough to show that the restriction of to the scheme b (which depends

;::: ; D ;::: ;D R \

n a c only on n and ), and the restriction of to

1 11 1n

0 0 0 0

;::: ; D ;::: ;D

(which depends only on and ) both vanish.

n 1 n 11 1

LEMMA 4.8. One has

n n 0 n n 0

( ) R R ( ) D D ): (

n c n a c a b

b mod

n n

Proof. Formula (3.4) translates into

R (;")= (") +" ;

a c

D(") aD(") a+b+cD (")

b (4.9)

0 0

R (;")= (") 0 0 +" 0 :

a c

D(") aD(") a+b+cD (")

c (4.10)

t 0 t

For 0 ,let be the property ª t whenever 0

0 s

D ( ") D( ") " P (s) s,º or equivalently (mod ). Assume holds; using (4.9),

(4.10) and (4.2), we get

R (; ") R (; ")

s+ s 0

s 1

( ;" ) : ( ) D D

a a c s

b mod (4.11)

s+ s s 0

0 1

Assume ;then and vanish modulo ,weget s 0

(s + ) P ( ) P (n)

and P 1 holds. Since 1 is empty, this proves that holds, hence also

= n 2

formula (4.11) for s .

F X R F

LEMMA 4.12. Let be a section of an ample line bundle on .Then n

\ R F a

vanishes on the scheme b if and only if vanishes on the scheme

\ c

a .

R F B

n a Proof. As in the proof of Lemma 3.12, we can write b (mod ).

Multiplying the congruence of Lemma 4.8 by F ,weget

n n 0

( ) B R F

b b

n n 0

): F ( ) D D (

a c n a

b mod

a 6= b a Since is irreducible and , one can divide out by b . This proves the Lem-

ma. 2

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TRISECANT LINES AND JACOBIANS ± II 185

R \ r

a c

+r (ac)

By Lemma 3.12, a vanishes on for all integers ;by

R \ r

n a

+r (ac)

Lemma 4.12, this implies that a vanishes on for all , and by

R r s

+r (ac)+s(ab)

Lemma 3.12 again, so does a for all and . Hypothesis (ii) in

R \

n a the theorem implies, as in (3.13), that vanishes on b , which concludes

the proof of the theorem. 2

5. Complements In this short Section, we will indicate how to combine the techniques used here with those of [D] to get better results in the degenerate case when the theta divisor is not too singular. We will use the following lemma, inspired by Proposition 2.6

in [D].

X; )

LEMMA 5.1. Let ( be a principally polarized abelian variety and let be a X

representative of the polarization. Let x be a non-torsion element of and assume

Z \ Z rx

that is a component of x such that is contained in . Assume 2

red r Z

) > Z (Z \

that codimX Sing 3.Then is reduced.

Z + rx \ r Z + A

Proof.Since red is contained in x for all integers ,sois red , x

where A is the neutral component of the closed subgroup generated by .It

+ A = Z Z A A

follows that Zred red, hence red contains a translate of that satis®es

(A \ ) > Z

codimA Sing 2. If is not reduced, it is contained in the singular

\ A D =D x

locus of x , hence so is . By the Jacobian criterion, the ,for

D 2 T A O ( ) x

0 , de®ne a section of A which is regular outside of the

0 0

\ > A

closed subset A Sing . Since this subset has codimension 2in ,the

O x ( ) x line bundle A is trivial. This means that is in the kernel of the

0 0 0

X ) ! (A ) A restriction homomorphism Pic ( Pic , hence so is . The composed

! (A)

homomorphism A Pic is therefore zero. Since it is the morphism associated

A A =

with the restriction of the polarization to , this implies 0, which contradicts

Z 2 the fact that x is not torsion. Hence is reduced.

In the case of a degenerate trisecant, this lemma allows us to prove the following

improvement on Theorem 2.2 of [D].

X; ) THEOREM 5.2. Let ( be a complex indecomposable principally polarized

abelian variety, let be a symmetric representative of the polarization and let

X !j j u v K : 2be the Kummer morphism. Assume that there exist points and

of X such that

(u) K (v )

(i) the points K and are distinct and the line that joins them is tangent

(X ) K (u) to K at ;

(ii) the point u is not torsion;

\ > ru

(iii) codimX Sing 3. 2 r Z 2

comp4012.tex; 12/06/1997; 8:48; v.7; p.9

186 OLIVIER DEBARRE

X; ) Then ( is isomorphic to the Jacobian of a smooth non-hyperelliptic algebraic

curve.

Note that by [BD], condition (i) implies codimX Sing 4. On the other hand,

(X; ) >

the indecomposability of implies codim X Sing 3 ([EL]): if hypothesis X (iii) fails, there is a component of Sing of codimension 3 in , invariant by the

abelian subvariety generated by u.

Proof. We keep the notation of the proof of Theorem 3.1. The point is to

R \ Z

u u

show that n vanishes on the scheme .Let be a primary component

\ Z

u u+ ru

of u ; it has codimension 2. If is not contained in , 2

red r Z 2

R Z

Lemma 3.12 implies that n vanishes on . Otherwise, Lemma 5.1 implies that

Z R \

is reduced. On page 9 of [D], it is proved that vanishes on u .Since

Z R Z R n

is reduced, it follows that n vanishes on . Hence vanishes on all primary

\ 2

u components of u , which proves the theorem.

This approach does not seem to work in the non-degenerate case.

References [AD] Arbarello, E. and De Concini, C.: Another proof of a conjecture of S.P. Novikov on periods of abelian integrals on Riemann surfaces, Duke Math. J. 54 (1984), 163±178. [BD] Beauville, A. and Debarre, O.: Une relation entre deux approches du problemeÁ de Schottky, Inv. Math. 86 (1986), 195±207. [D] Debarre, O.: Trisecant Lines And Jacobians, J. Alg. Geom. 1 (1992), 5±14. [EL] Ein, L. and Lazarsfeld, R.: Singularities of theta divisors, and the birational geometry of irregular varieties, J. Amer. Math. Soc. 10 (1997), 243±258. [M] Marini, G.: A Geometrical Proof of Shiota's Theorem on a Conjecture of S.P. Novikov, Comp. Math., to appear. [S] Shiota, T.: Characterization of Jacobian varieties in terms of soliton equations, Invent. Math. 83 (1986), 333±382. [W] Welters, G.: A criterion for Jacobi varieties, Ann. of Math. (2) 120 (1984), 497±504.

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