LetY be a smooth projective curve degenerating to a reducible curveX with two components meeting transversally at one point. We show that the moduli space of vector bundles of rank two and odd determinant on Ydegenerates to a moduli space onX which has nice properties, in particular, it has normal crossings. We also show that a nice degeneration exists when we fix the determinant. We give some conjectures concerning the degeneration of moduli space of vector bundles onY with fixed determinant and arbitrary rank.

LetX₀ be a projective curve whose singularity is one ordinary double point. We construct a birational modelG(n, d) of the moduli spaceU(n, d) of stable torsion free sheaves in the case (n, d)= 1, such that G(n, d) has normal crossing singularities and behaves well under specialization i.e. if a smooth projective curve specializes toX₀, then the moduli space of stable vector bundles of rankn and degreed onX specializes toG(n, d). This generalizes an earlier work of Gieseker in the rank two case.