A note on the Picard bundle over a moduli space of vector bundles

Let ℳ( n , d ) be a coprime moduli space of stable vector bundles of rank n ≥ 2 and degree d over a complex irreducible smooth projective curve X of genus g ≥ 2 and ℳ ξ ⊂ ℳ( n , d ) a fixed determinant moduli space. Assuming that the degree d is sufficiently large, denote by the vector bundle over X ×ℳ( n , d ) defined by the kernel of the evaluation map H 0 ( X , E ) → E x , where E ∈ℳ( n , d ) and x ∈ X . We prove that and its restriction ξ to X × ℳ ξ are stable. The space of all infinitesimal deformations of over X ×ℳ( n , d ) is proved to be of dimension 3 g and that of ξ over X × ℳ ξ of dimension 2 g , assuming that g ≥ 3 and if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)