00 1 Deformations of the Picard Bundle

Let X be a nonsingular algebraic curve of genus g ≥ 3, and M ξ the moduli space of stable vector bundles of rank n ≥ 2 and degree d with fixed determinant ξ over X such that n and d coprime and d > n(2g − 2). We assume that if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3. Let W ξ (L) denote the vector bundle over M ξ defined as the direct image π * (U ξ ⊗ p * 1 L) where U ξ is a universal vector bundle over X × M ξ and L a line bundle over X of degree zero. The space of infinitesimal deformations of W ξ (L) is proved to be isomorphic to H 1 (X, O X). This construction gives a complete family of vector bundles over M ξ parametrized by Pic 0 (X) such that W ξ (L) is the vector bundle corresponding to L ∈ Pic 0 (X). The connected component of the moduli space of stable sheaves with the same Hilbert polynomial as W ξ (O) over M ξ containing W ξ (O) is in fact isomorphic to Pic 0 (X) as a polarised variety.