Vector bundles on curves with many components

Abstract

We construct the moduli spaces M(n) of semistable parabolic sheaves of rank n and fixed Euler characteristic on a disjoint union X of integral projective curves with parabolic structures over Cartier divisors on X. In the case where X is non-singular, M is a normal projective variety. Suppose that X is the desingularisation of a reducible reduced curve Y with at most ordinary double points as singularities. We show that, for a suitable choice of parabolic structure, M(n) is the normalisation of the moduli space of torsion-free sheaves of rank n and fixed Euler characteristic on Y, and it is a desingularisation if semistability coincides with stability. We find explicit descriptions of M(n) for small n in some cases.