Hecke correspondence, stable maps, and the Kirwan desingularization

Abstract

We prove that the moduli space ${\stackrel{̲}{M}}_{0,0}(N,2)$ of stable maps of degree $2$ to the moduli space $N$ of rank $2$ stable bundles of fixed odd determinant $O_X(-x)$ over a smooth projective curve $X$ of genus $g\ge 3$ has two irreducible components that intersect transversely. One of them is Kirwan's partial desingularization ${\tilde{M}}_X$ of the moduli space $M_X$ of rank $2$ semistable bundles with determinant isomorphic to $O_X(y-x)$ for some $y\in X$. The other component is the partial desingularization of the geometric invariant theory (GIT) quotient $\mathbb{P}\mathrm{Hom}(\mathrm{sl}(2){}^{\vee },W)//\mathrm{PGL}(2)$ for a vector bundle $W=R^1{\pi }_{*}{L}^{-2}(-x)$ of rank $g$ over the Jacobian of $X$. We also show that the Hilbert scheme $H$, the Chow scheme $C$ of conics in $N$, and ${\stackrel{̲}{M}}_{0,0}(N,2)$ are related by explicit contractions