Duke Mathematical Journal

Hecke correspondence, stable maps, and the Kirwan desingularization

Young-Hoon Kiem

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We prove that the moduli space M̲0,0(N,2) of stable maps of degree 2 to the moduli space N of rank 2 stable bundles of fixed odd determinant OX(-x) over a smooth projective curve X of genus g3 has two irreducible components that intersect transversely. One of them is Kirwan's partial desingularization M~X of the moduli space MX of rank 2 semistable bundles with determinant isomorphic to OX(y-x) for some yX. The other component is the partial desingularization of the geometric invariant theory (GIT) quotient PHom(sl(2),W)//PGL(2) for a vector bundle W=R1π*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 M̲0,0(N,2) are related by explicit contractions

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

First available in Project Euclid: 29 January 2007

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Zentralblatt MATH identifier

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 14H60: Vector bundles on curves and their moduli [See also 14D20, 14F05]
Secondary: 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]


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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