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Proc. Indian Acad. Sci. (Math. Sci.), Vol. 111, No. 3, August 2001, pp. 263269. Printed in India

Stability of Picard bundle over moduli space of stable vectorbundles of rank two over a curve

INDRANIL BISWAS and TOMAS L GOMEZ

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road,Mumbai 400 005, IndiaE-mail: indranil@math.tifr.res.in; tomas@math.tifr.res.in

MS received 14 September 2000

Abstract. Answering a question of [BV] it is proved that the Picard bundle on themoduli space of stable vector bundles of rank two, on a Riemann surface of genus atleast three, with fixed determinant of odd degree is stable.

Keywords. Picard bundle; Hecke lines.

0. Introduction

Let X be a compact connected Riemann surface of genus g, with g 3. Let be a

holomorphic line bundle over X of odd degree d, with d 4g 3. Let Mdenote the

moduli space of stable vector bundles E over X of rank two and2

E = . Take a

universal vector bundle E on X M. Let p : X M Mbe the projection. The

vector bundle P := pEonMis called thePicard bundle forM. In [BV] it was proved

that the Picard bundle Pis simple, and a question was asked whether it is stable. In [BHM]

a differential geometric criterion for the stability ofPwas given. But there is no evidence

for this criterion to be valid.

In Theorem 3.1 we prove that the Picard bundle PoverMis stable.

1. Preliminaries

In this section we prove some lemmas that will be needed.

A vector bundle E of rank two and degree d is called superstable if for every subline

bundle L ofE the inequality

deg(L) d /2. LetE M

be a vector bundle and0 = s H0(E) a nontrivial section. Then scannot simultaneously

vanish at all the chosen points {x1, . . . , xm}.

Proof. Ifs vanishes at all chosen pointsx1, . . . , xm, thens : O Efactors as

s : O E(D) E,

whereD is the divisorD = x1 + + xm. Since deg E(D) = d 2m

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Stability of Picard bundle 265

0 0 0 Ex / l Wx Cx 0

0 E W

f0 Cx 0 (1)

W (x) === W (x) 0 0

Here Cx is the skyscraper sheaf at x with stalkC. Instead off0 : W Cx we may

consider an arbitrary nontrivial homomorphism

f Hom(W, Cx ) = Hom(Wx , Cx ) = Wx

and defineEfas the kernel

0 Ef W f Cx 0 . (2)

This way we obtain a family of vector bundles parametrized by the projective line P(Wx ),

withEf0= E. More precisely, there is a short exact sequence on X P(Wx ),

0 E XW f OxP(Wx ) (1) 0 ,where X : X P(W

x ) Xis the projection to X. It has the property that if f W

x

and we restrict the exact sequence to the subvariety X [f] = X ofX P(Wx ), then a

sequence isomorphic to (2) is obtained.

For everyf Wx , the vector bundle Ef is stable. Indeed, ifL is a subline bundle of

Ef, then by composition with the homomorphismEf Win (2) it is a subline bundle

ofW. The stability condition forWsays that deg(L) < (d+ 1)/2. Sincedis odd this is

equivalent to

deg(L) d 1

2 d /2.

We need the following lemma for the proof of the theorem.

Lemma3.2. There is a nonempty open set ofM such that ifE is a vector bundle corre-

sponding to a point of that open set, then E has the following four properties:

(i) E is superstable;

(ii) Fis locally free atE;

(iii) FE PE is an injection;

(iv) Letxibeone ofthefixed pointsandl anylineon Exi . LetP = PE, xi , lbe the associated

Hecke line. Then Fis locally free at all points of the image of : P M.

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268 Indranil Biswas and Tomas L Gomez

Proof. The subsetUofMwhere property (i) is satisfied is open and nonempty by Lemma

1.1. LetU Ube the subset where also property (ii) is satisfied and U U the subset

where furthermore property (iii) is satisfied. Clearly,U is a nonempty open subset ofM.

Let S Mdenote the subvariety where F is not locally free. SinceF is reflexive,codim(S) 3. Let Si be the union of the images of all Hecke lines PE, xi , l , whenE runsthrough all points inSand l runs through all lines ofEx . Then

codimSi 3 1 1 = 1 .Finally consider the union

S :=m

i=1

Si .

Since this is a union of a finite number of subvarieties, we still have codimS 1. Conse-quently,U := U

(M\

S) is nonempty and open. By construction, any vector bundle

Ecorresponding to a point inU satisfies conditions (i) to (iv). This finishes the proof ofthe lemma.

Continuing the proof of Theorem 3.1, fix a vector bundle Esatisfying the four properties

in the above lemma. Let v FE be a nonzero vector in the fiber, and lets be its image in

the fiber PE = H0(E). It is still nonzero because of property (iii).

From the fixed set of chosen points {x1, . . . , xm}, pick one of them xi such that the

sections does not vanish at xi . The existence of such a point is ensured by Lemma 1.2.

Let l Exi be a line such that s (xi ) / l. Consider the Hecke lineP = PE, xi , l defined

with this data.

Note that Fis a vector bundle because Fis locally free on all points of the image of

P inM(property (iv)), andF Pis injective as a sheaf homomorphism because

both Fand Pare vector bundles and property (iii).

The Proposition 2.2 says that Phas a canonical subbundle Vwith

OP(k)N1 = V

P= OP(k)

N1 OP(k 1). (6)

We can think ofv and s as vectors in the fibers of Fand Pat [f0]. Sinces (xi ) / l ,

Proposition 2.2 also gives thats / V = VE . Consequently,

F V . (7)

By Grothendiecks theorem

F = OP(b1) OP(br ) .

Since F Pis injective, (6) implies that bi kfor alli , and (7) implies that for

somei (sayi = 1),b1 k 1.Fix a polarization L on M. Let be the degree of L. The Corollary 2.3 says that

> 0. Now,

1

deg(F)

rank(F)=

deg(F)

rank(F) k

1

r< k

1

N=

deg(P)

rank(P)=

1

deg(P)

rank(P)

and hence the Picard bundle Pis stable. This completes the proof of the theorem.

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Stability of Picard bundle 269

References

[BV] Balaji V and Vishwanath P R, Deformations of Picard sheaves and moduli of pairs,DukeMath. J. 76 (1994) 773792

[BHM] Brambila-Paz L, Hidalgo-Sols L and Mucino-Raymondo J, On restrictions of the Picardbundle. Complex geometry of groups (Olmue, 1998) 4956; Contemp. Math. 240; Am.Math. Soc. (Providence, RI) (1999)

[Gr] Grothendieck A, Sur la classification des fibres holomorphes sur la sphere de Riemann.Am. J. Math.79 (1957) 121138

[NR] Narasimhan M S andRamanan S, Moduliof vectorbundleson a compact Riemann surface.Ann. Math. 89 (1969) 1951