Duke Mathematical Journal

Hecke correspondence, stable maps, and the Kirwan desingularization

Young-Hoon Kiem

Abstract

We prove that the moduli space $\overline{\mathbf{M}}_{0,0}(\mathcal{N},2)$ of stable maps of degree $2$ to the moduli space $\mathcal{N}$ of rank $2$ stable bundles of fixed odd determinant $\mathcal{O}_X(-x)$ 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 $\widetilde{\mathcal{M}}_X$ of the moduli space $\mathcal{M}_X$ of rank $2$ semistable bundles with determinant isomorphic to $\mathcal{O}_X(y-x)$ for some $y\in X$. The other component is the partial desingularization of the geometric invariant theory (GIT) quotient $\mathbb{P}\mathrm{Hom} ({\rm sl}(2)^\vee, \mathcal{W})//{\rm PGL}(2)$ for a vector bundle $\mathcal{W}=R^1{\pi}_*\mathcal{L}^{-2}(-x)$ of rank $g$ over the Jacobian of $X$. We also show that the Hilbert scheme $\mathbf{H}$, the Chow scheme $\mathbf{C}$ of conics in $\mathcal{N}$, and $\overline{\mathbf{M}}_{0,0}(\mathcal{N},2)$ are related by explicit contractions

Article information

Source
Duke Math. J., Volume 136, Number 3 (2007), 585-618.

Dates
First available in Project Euclid: 29 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1170084899

Digital Object Identifier
doi:10.1215/S0012-7094-07-13636-6

Mathematical Reviews number (MathSciNet)
MR2309175

Zentralblatt MATH identifier
1119.14033

Citation

Kiem, Young-Hoon. Hecke correspondence, stable maps, and the Kirwan desingularization. Duke Math. J. 136 (2007), no. 3, 585--618. doi:10.1215/S0012-7094-07-13636-6. https://projecteuclid.org/euclid.dmj/1170084899

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