Indexed on: 23 Dec '18Published on: 23 Dec '18Published in: arXiv - Mathematics - Algebraic Geometry
Let $C$ be a smooth irreducible complex projective curve of genus $g \geq 2$ and $M_1$ a moduli space of stable vector bundles over $C$. A (generalised) Picard sheaf is the direct image on $M_1$ of the tensor product of the Poincar\'e or universal bundle on $M_1\times C$ by the pullback of a vector bundle $E_0$ on $C$; when the degree of $E_0$ is sufficiently large, this sheaf is a bundle and coincides with the Fourier-Mukai transform of $E_0$. In this paper we include all results known to us and many new ones on the stability of the Picard sheaves when $M_1$ is one of the Picard variety of line bundles of degree $d$ on $C$, the moduli space of stable vector bundles of rank $n$ and degree $d$ on $C$ with $n,d$ coprime or the moduli space of stable bundles of rank $n$ and fixed determinant of degree $d$. We prove in particular that, if $E_0$ is a stable bundle of rank $n_0$ and degree $d_0$ with $nd_0 + n_0d > n_0n(2g-1)$, then the pullbacks of the Picard bundle on the moduli space of stable bundles by morphisms analogous to the Abel-Jacobi map are stable; moreover, if $nd_0 + n_0d > n_0n(n + 1)(g-1) + n_0$, then the Picard bundle itself is stable with respect to a theta divisor.