Rational curves and lines on the moduli space of stable bundles

TL;DR

This paper characterizes rational curves on the moduli space of stable bundles over a curve, showing they are generalized Hecke curves, and analyzes the structure and coverage of lines within this space.

Contribution

It proves that all rational curves are generalized Hecke curves and provides a detailed study of lines, including their coverage and structure depending on rank and degree.

Findings

Rational curves on the moduli space are generalized Hecke curves.

The moduli space is covered by lines when (r, d)=r.

For (r,d)<r, lines form a closed subvariety with specific irreducible components.

Abstract

Fix a smooth projetive curve $\mathcal {C}$ of genus $g\geq 2$ and a line bundle $\mathcal{L}$ on $\mathcal{C}$ of degree $d$. Let $M:= \mathcal{SU}_{\mathcal{C}}(r, \mathcal{L})$ be the moduli space of stable vector bundles on $\mathcal{C}$ of rank $r$ and with fixed determinant $\mathcal{L}$. We prove that any rational curve on $M$ is a generalized Hecke curve. Furthermore, we study the lines on $M$, and prove that $M$ is covered by the lines when $(r, d)=r$; for the case $(r,d)<r$, the lines fill up a closed subvariety of $M$, and we determine the number of its irreducible components and the dimension of each irreducible component when $g>(r,d)+1$. Finally, we prove that there are no $(1,0)$-stable (resp., $(0,1)$-stable) bundles for $g=2$, $r=2$ and $d$ is odd as an application.