On the Picard bundle

Abstract

Fix a holomorphic line bundle ξ over a compact connected Riemann surface X of genus g, with g≥2, and also fix an integer r such that degree(ξ)>r(2g-1). Let M_ξ(r) denote the moduli space of stable vector bundles over X of rank r and determinant ξ. The Fourier-Mukai transform, with respect to a Poincare line bundle on X× J(X), of any F ∈ M_ξ(r) is a stable vector bundle on J(X). This gives an injective map of M_ξ(r) in a moduli space associated to J(X). If g=2, then M_ξ(r) becomes a Lagrangian subscheme.