Raynaud’s vector bundles and base points of the generalized Theta divisor

Abstract

We study base points of the generalized Θ-divisor on the moduli space of vector bundles on a smooth algebraic curve X of genus g defined over an algebraically closed field. To do so, we use the derived categories Db(Pic0(X)) and Db(Jac(X)) and the equivalence between them given by the Fourier–Mukai transform \({\rm FM}_\mathcal {P}\) coming from the Poincaré bundle. The vector bundles P m on the curve X defined by Raynaud play a central role in this description. Indeed, we show that E is a base point of the generalized Θ-divisor, if and only if there exists a non-trivial homomorphism Prk(E)g+1 → E.