Secant varieties and Hirschowitz bound on vector bundles over a curve

For a vector bundle V of rank n over a curve X and for each integer r in the range 1 ≤ r ≤ n ? 1, the Segre invariant s _r is defined by generalizing the minimal self-intersection number of the sections on a ruled surface. In this paper we generalize Lange and Narasimhan’s results on rank 2 bundles which related the invariant s ₁ to the secant varieties of the curve inside certain extension spaces. For any n and r, we find a way to get information on the invariant s _r from the secant varieties of certain subvariety of a scroll over X. Using this geometric picture, we obtain a new proof of the Hirschowitz bound on s _r .     

Moduli of rank 4 symplectic vector bundles over a curve of genus 2

Let X be a complex projective curve which is smooth and irreducibleof genus 2. The moduli space ₂ of semistable symplectic vectorbundles of rank 4 over X is a variety of dimension 10. Afterassembling some results on vector bundles of rank 2 and odddegree over X, we construct a generically finite cover of ₂by a family of 5-dimensional projective spaces, and outlinesome applications.