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Partial resolutions of Hilbert type, Dynkin diagrams, and generalized Kummer varieties

D. Kaledin·M. Verbitsky
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Abstract

We study the partial resolutions of singularities related to Hilbert schemes of points on an affine space. Consider a quotient of a vector space $V$ by an action of a finite group $G$ of linear transforms. Under some additional assumptions, we prove that the partial desingularization of Hilbert type is smooth only if the action of $G$ is generated by complex reflections. This is used to study the subvarieties of a Hilbert scheme of a complex torus. We show that any subvariety of a generic deformation of a Hilbert scheme of a torus is birational to a quotient of another torus by an action of a Weyl group of some semisimple Lie algebra. In Appendix, we produce counterexamples to a false theorem stated in our preprint math.AG/9801038.

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D. Kaledin, M. Verbitsky. 1998-12-14. Partial resolutions of Hilbert type, Dynkin diagrams, and generalized Kummer varieties. https://arxiv.org/abs/math/9812078

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