Torelli type theorems for moduli spaces of vector bundles on curves

 A lot is known about moduli spaces of semistable vector bundles on smooth projective curves of genus g>=2 if the rank r and the degree d of the bundles are coprime, (r,d)=1, for example there is a Torelli type theorem for theses spaces. In this talk we will look at the first examples where the condition (r,d)=1 is not satisfied, that is we look at moduli spaces of semistable vector bundles of rank two and degree zero (more exactly with trivial determinant) on smooth projective curves of genus 2. It turns out that these spaces have a connection to a lot of classical algebraic geometry. They are for example related to quadratic line complexes and Kummer surfaces. Using this classical geometry we will discuss in which sense these spaces satisfy a (birational) Torelli type theorem. If time permits we will also discuss some ideas for the case of genus g>=3.