Secant varieties and Hirschowitz bound on vector bundles over a curve

TL;DR

This paper explores the relationship between secant varieties and invariants of vector bundles over curves, providing new geometric insights and a novel proof of the Hirschowitz bound.

Contribution

It generalizes previous results on rank 2 bundles to higher ranks, connecting secant varieties to the invariants s_r of vector bundles.

Findings

Established a method to derive s_r from secant varieties of subvarieties of scrolls.

Provided a new geometric proof of the Hirschowitz bound on s_r.

Extended Lange and Narasimhan's results from rank 2 to arbitrary rank n.

Abstract

For a vector bundle V over a curve X of rank n and for each integer r in the range 1 \le r \le 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.