Rational Curves on Moduli Spaces of Vector Bundles

TL;DR

This paper classifies rational curves in moduli spaces of vector bundles on curves, identifying unobstructed components, constructing obstructed ones, and providing bounds on rational connectedness.

Contribution

It completely describes the components of rational curves in moduli spaces of vector bundles, including unobstructed and obstructed components, and bounds on rational connectedness.

Findings

Number of unobstructed components equals gcd(r, deg L)

Existence of obstructed components for certain degrees

Upper bound on degree of rational connectedness

Abstract

We completely describe the components of expected dimension of the Hilbert Scheme of rational curves of fixed degree $k$ in the moduli space ${\rm SU}_{C}(r,L)$ of semistable vector bundles of rank $r$ and determinant $L$ on a curve $C$. We show that for every $k \geq 1$ there are ${\rm gcd}(r, \deg L)$ unobstructed components. In addition, if $k$ is divisible by $r_1(r-r_1)(g-1)$ for $1\le r_1\le r-1$, there is an additional obstructed component of the expected dimension for each such $r_1$. We construct families of obstructed components and show that their generic point is not the generic vector bundle of given rank and determinant. Finally, we also obtain an upper bound on the degree of rational connectedness of ${\rm SU}_{C}(r,L)$ which is linear in the dimension.