Projective Poincaré and Picard bundles for moduli spaces of vector bundles over nodal curves

Abstract

Let UL′s(n,d) be the moduli space of stable vector bundles of rank n with determinant L where L is a fixed line bundle of degree d over a nodal curve Y. We prove that the projective Poincaré bundle on Y×UL′s(n,d) and the projective Picard bundle on UL′s(n,d) are stable for suitable polarisations. For a nonsingular point x∈Y, we show that the restriction of the projective Poincaré bundle to {x}×UL′s(n,d) is stable for any polarisation. We prove that for arithmetic genus g≥3 and for g=n=2,d odd, the Picard group of the moduli space UL′(n,d) of semistable vector bundles of rank n with determinant L of degree d is isomorphic to Z.