Stability and restrictions of Picard bundles, with an application to the normal bundles of elliptic curves

Summary

Introduction.

Let C be a smooth irreducible projective curve of genus g≥1, and for each integer d let J_d(C) be the Jacobian of C, which we view as parametrizing all line bundles on C of degree d. Denote by L_t the bundle on C corresponding to the point t∈J_d(C). Provided that d≥2g-1, the vector spaces H⁰(C, L_t) fit together to form the fibres of a vector bundle P_d on J_d(C), of rank d+1-g, called the degree d Picard bundle (defined by this description up tp tensoring by line bundles on J_d(C)). These bundles have been the focus of considerable study in recent years, notably by Kempf and Mukai ([K1], [K2], [K3], [M]). To better understand their geometry, it is natural to ask whether P_d is stable with respect to the canonical principal polarization of J_d(C). Kempf [Kl] shows that this is indeed the case for the first bundle P_2g-1. The main purpose of this note is to complete Kempf's result by proving the following

Theorem.For every d≥2g, the Picard bundle P_dis stable with respect to the polarization on J_d(C) defined by the theta divisor Θ_C⊂J_d(C).

For g = 2, this was established by Umemura [U]. As in [K1], the proof depends on analyzing the restriction of P_d to C. We show that the restriction of P_d to both CC J_d(C) and (−C)⊂J_d(C) are stable; either of these statements implies the result. In the hope that the techniques involved may find other uses in the future, we give rather different arguments for the stability of each of these restrictions.