Small rational curves on the moduli space of stable bundles

TL;DR

This paper investigates small rational curves on the moduli space of stable bundles over a smooth projective curve, providing classifications and codimension estimates, especially for rank 3 bundles.

Contribution

It classifies all small rational curves on the moduli space for rank 3 bundles and estimates the codimension of their locus.

Findings

All small rational curves are classified for r=3.

Codimension of the locus of small rational curves is estimated.

Provides new insights into the geometry of the moduli space.

Abstract

For a smooth projective curve C with genus g >= 2 and a degree 1 line bundle L on C, let M := SU_{C}(r;L) be the moduli space of stable vector bundles of rank r and with the fixed determinant L. In this paper, we study the small rational curves on M and estimate the codimension of the locus of the small rational curves. In particular, we determine all small rational curves when r = 3.