Moduli spaces of vector bundles on curves

In this work, we generalize Bertrams work on rank two vector bundles on a smooth irreducible projective curve to an irreducible singular curve C with only one node as singularity. We resolve the indeterminancy of the rational map L : P(Ext 1 C(L,OC)) . SUC(2, L) de.ned by L([0.OC . E . L . 0]) = E in the case degL = 3, 4 via a sequence of three blow-ups with smooth centers. Here SUC(2, L) is a compacti.cation of the moduli space SUC(2, L) of semi-stable vector bundles of rank 2 and determinant L on C. An additional part is on the base locus of the generalized theta divisor r on SUC(r, L) for a smooth curve C. Among our results, we show, using results of Raynaud, that the base locus is always non-empty when r = 2g and L = OC.