Deformations of the generalised Picard bundle

Let X   be a nonsingular algebraic curve of genus g⩾3g⩾3, and let MξMξ denote the moduli space of stable vector bundles of rank n⩾2n⩾2 and degree d   with fixed determinant ξξ over X such that n and d   are coprime. We assume that if g=3g=3 then n⩾4n⩾4 and if g=4g=4 then n⩾3n⩾3, and suppose further that n0n0, d0d0 are integers such that n0⩾1n0⩾1 and nd0+n0d>nn0(2g-2)nd0+n0d>nn0(2g-2). Let E be a semistable vector bundle over X   of rank n0n0 and degree d0d0. The generalised Picard bundle Wξ(E)Wξ(E) is by definition the vector bundle over MξMξ defined by the direct image pMξ*(Uξ⊗pX*E) where UξUξ is a universal vector bundle over X×MξX×Mξ. We obtain an inversion formula allowing us to recover E   from Wξ(E)Wξ(E) and show that the space of infinitesimal deformations of Wξ(E)Wξ(E) is isomorphic to H1(X,End(E))H1(X,End(E)). This construction gives a locally complete family of vector bundles over MξMξ parametrised by the moduli space M(n0,d0)M(n0,d0) of stable bundles of rank n0n0 and degree d0d0 over X  . If (n0,d0)=1(n0,d0)=1 and Wξ(E)Wξ(E) is stable for all E∈M(n0,d0)E∈M(n0,d0), the construction determines an isomorphism from M(n0,d0)M(n0,d0) to a connected component M0M0 of a moduli space of stable sheaves over MξMξ. This applies in particular when n0=1n0=1, in which case M0M0 is isomorphic to the Jacobian J of X as a polarised variety. The paper as a whole is a generalisation of results of Kempf and Mukai on Picard bundles over J, and is also related to a paper of Tyurin on the geometry of moduli of vector bundles.