On the stability of the direct image of a generic vector bundle
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On the stability of the direct image of a generic vector bundle Arnaud BEAUVILLE Introduction We discuss in this note the following conjecture: Conjecture .? Let pi : X? ? X be a finite morphism between smooth projective curves, and L a generic vector bundle on X? . The vector bundle pi?L is stable if g(X) ≥ 2 , semi-stable if g(X) = 1 . I do not have a strong motivation towards the conjecture, except that it is a rather natural statement. As we will see below, the crucial case is when L is a line bundle; the (easy) case when pi is a double covering was used in [B] to control the theta divisor on the moduli space of rank 2 vector bundles on X . One may hope that a proof of the conjecture would lead to a better understanding of the theta linear system in arbitrary rank. We have only partial results in the direction of the conjecture: we will show that stability holds with respect to sub-bundles of small degree ( 1), for small values of ?(L) ( 2), or when pi is etale ( 3). 1. General remarks (1.1) It is of course sufficient to prove the conjecture for one vector bundle with the same rank and degree as L .
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- l? pi??
- line bundle
- f? pi?l
- effective divisor
- instance ?
- bundles
- divisor ??
On the stability of the direct image of a generic vector bundle
Arnaud BEAUVILLE
Introduction We discuss in this note the following conjecture: 0 Conjecture.−Letπ: X→Xbe a finite morphism between smooth projective 0 curves, andLa generic vector bundle onX. The vector bundleπ∗Lis stable if g(X)≥2, semi-stable ifg(X) = 1 . I do not have a strong motivation towards the conjecture, except that it is a rather natural statement. As we will see below, the crucial case is whenL isa line bundle; the (easy) case whenπis a double covering was used in [B] to control the theta divisor on the moduli space of rank 2 vector bundles onX .One may hope that a proof of the conjecture would lead to a better understanding of the theta linear system in arbitrary rank. We have only partial results in the direction of the conjecture: we will show that stability holds with respect to sub-bundles of small degree (§1), for small values of χ(L) (§2), or whenπ(elate´is§3).
1. General remarks (1.1) It is of course sufficient to prove the conjecture foronevector bundle with 0 000 the same rank and degree asL .Letπ: X→vecoleta´eanbeXrgeefoedirgn 00 0 rk L, andM ageneral line bundle onX ofdegree degL .ThenπsameM has ∗ 00 rank and degree asL ;so our conjecture holds if it holds for line bundles onX 0 w.r.t. the coveringπ◦π. Therefore it is enough to prove the conjecture in the case L isa line bundle. (1.2) From now on we suppose thatL isa line bundle. We denote byrthe degree of the coveringπ, so thatπ∗L isa rankrvector bundle; we denote byg 0 0 the genus ofX andbygX .the genus of The assertion depends only of course on the degree ofL ,and actually on the degree ofL (mod.r), whereris the degree of the coveringπ: this is because the ∗ (semi-) stability ofπ∗L isequivalent to that ofπ∗(L⊗πany line bundleM) forM ∼ −1∗ on X. Moreover, the duality isomorphismπ∗(KX⊗= KL )X⊗(π∗L) implies 0 that the conjecture is true forχ(L) =nif and only if it is true forχ(L) =−n. (1.3) The weaker conclusion of the Conjecture in the casegis due to the= 1 fact that there are no stable bundles of rankrand degreedon an elliptic curve if 1
