Remarks on minimal rational curves on moduli spaces of stable bundles

TL;DR

This paper investigates minimal rational curves on the moduli space of stable rank 2 bundles with fixed determinant over a smooth projective curve, revealing the existence of non-Hecke minimal rational curves in specific cases.

Contribution

It demonstrates the existence of minimal rational curves that are not Hecke curves on the moduli space for genus 3 and even degree, extending previous results by Xiaotao Sun.

Findings

Existence of non-Hecke minimal rational curves for genus 3 and even degree

Minimal rational curves pass through any point in the moduli space

Complements prior work by Xiaotao Sun on Hecke curves

Abstract

Let M be the moduli space of stable bundles of rank 2 and with fixed determinant \mathcal{L} of degree d on a smooth projective curve C of genus g>= 2. When g=3 and d is even, we prove, for any point [W]\in M, there is a minimal rational curve passing through [W], which is not a Hecke curve. This complements a theorem of Xiaotao Sun.