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

TL;DR

This paper investigates genus zero stable maps of degree three within the moduli space of stable rank two vector bundles on a genus g≥3 curve, revealing a structure with two irreducible components.

Contribution

It generalizes previous work by describing the stable map space in a specific subvariety of the moduli space, identifying its two irreducible components.

Findings

The stable map space has two irreducible components.

One component parameterizes smooth rational cubic curves.

The other component parameterizes unions of lines and smooth conics.

Abstract

Let $X$ be a smooth projective curve with genus $g\geq3$. 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.