Introduction.

Let C be a smooth irreducible projective curve of genus g≥1, and for each integer d let Jd(C) be the Jacobian of C, which we view as parametrizing all line bundles on C of degree d. Denote by Lt the bundle on C corresponding to the point t∈Jd(C). Provided that d≥2g-1, the vector spaces H0(C, Lt) fit together to form the fibres of a vector bundle Pd on Jd(C), of rank d+1-g, called the degree d Picard bundle (defined by this description up tp tensoring by line bundles on Jd(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 Pd is stable with respect to the canonical principal polarization of Jd(C). Kempf [Kl] shows that this is indeed the case for the first bundle P2g-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 Pdis stable with respect to the polarization on Jd(C) defined by the theta divisor ΘC⊂Jd(C).

For g = 2, this was established by Umemura [U]. As in [K1], the proof depends on analyzing the restriction of Pd to C. We show that the restriction of Pd to both CC Jd(C) and (−C)⊂Jd(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.