Vector Bundles on Curves with Many Components

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. 1991 Mathematics Subject Classification : 14H60, 14F05.