Stable maps of genus zero in the space of stable vector bundles on a curve

Abstract

Let $X$ be a smooth projective curve with genus $g\ge 3$. Let $\mathcal{N}$ be the moduli space of stable rank two vector bundles on $X$ with a fixed determinant ${\mathcal{O}}_X(-x)$ for $x\in X$. In this paper, as a generalization of Kiem and Castravet's works, we study the stable maps in $\mathcal{N}$ with genus $0$ and degree $3$. Let $P$ be a natural closed subvariety of $\mathcal{N}$ which parametrizes stable vector bundles with a fixed subbundle $L^{-1}(-x)$ for a line bundle $L$ on $X$. We describe the stable map space ${\mathbf{M}}_0(P,3)$. It turns out that the space ${\mathbf{M}}_0(P,3)$ consists of two irreducible components. One of them parameterizes smooth rational cubic curves and the other parameterizes the union of line and smooth conics.