Non-simple semistable vector bundles over a curve

Abstract

Let X be a non-singular algebraic curve over C of genus g > 2 . Let S(3, 0) be the set of indecomposable semistable vector bundles over X of rank 3 and degree 0. In [21 we prove that non-simple vector bundles in S(3, 0) must have algebra of endomorphisms isomorphic to C [t]/(t2), C[t]/(t 3) o r C [r, s]/(r, s) 2. In this paper we prove that if we fix the algebra of endomorphisms (and make a further subdivision in the case C[t]/(t2)), then there exists a moduli space Mi for such vector bundles in S(3, 0). For some i, M i is a fine moduli space, i.e. there exists a universal family parametrized by Mg (see Theorems 6 and 9). Mainly we use the universal extensions constructed in [3] and we prove that under a suitable equivalence relation these extensions define the required moduli spaces. The equivalence relations are defined by considering the C*, C* x C, C* x C* and the GL(2, C)-action. In Sect. 1 we recall from [2] the conditions for a vector bundle to have a particular algebra of endomorphisms. In the following sections we construct the moduli spaces My