Non-simple semistable vector bundles over a curve (Q1319316)

Let \(X\) be a nonsingular algebraic curve over \(\mathbb{C}\) of genus \(g\geq 2\). Let \(S(3,0)\) be the set of indecomposable semistable vector bundles over \(X\) of rank 3 and degree 0. The nonsimple vector bundles in \(S(3,0)\) must have their algebra of endomorphisms isomorphic to \(\mathbb{C}[t]/(t^ 2)\), \(\mathbb{C}[t]/(t^ 3)\) or \(\mathbb{C}[r,s]/(r,s)^ 2\). We shall prove that if we fix the algebra of endomorphisms then there exists a moduli space \(M_ i\) 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 \(M_ i\). Mainly we use the universal extensions of families of vector bundles over \(X\) and we prove that under a suitable equivalence relation these extensions define the required moduli spaces.