Rational curves and lines on the moduli space of stable bundles

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.