Deformations of the Generalised Picard Bundle
نویسندگان
چکیده
Let X be a nonsingular algebraic curve of genus g ≥ 3, and let Mξ denote 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 are coprime. We assume that if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3, and suppose further that n0, d0 are integers such that n0 ≥ 1 and nd0 + n0d > nn0(2g − 2). Let E be a semistable vector bundle over X of rank n0 and degree d0. The generalised Picard bundle Wξ(E) is by definition the vector bundle over Mξ defined by the direct image pMξ∗(Uξ ⊗ p ∗ XE) where Uξ is a universal vector bundle over X ×Mξ. We obtain an inversion formula allowing us to recover E from Wξ(E) and show that the space of infinitesimal deformations of Wξ(E) is isomorphic to H (X, End(E)). This construction gives a locally complete family of vector bundles over Mξ parametrised by the moduli space M(n0, d0) of stable bundles of rank n0 and degree d0 over X . If (n0, d0) = 1 and Wξ(E) is stable for all E ∈ M(n0, d0), the construction determines an isomorphism from M(n0, d0) to a connected component M of a moduli space of stable sheaves over Mξ. This applies in particular when n0 = 1, in which case M 0 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 .
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