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

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

1998-12-14

arxiv.org/abs/math/9812078v1

Mathematics

Algebraic Geometry

33 pages

Scientific paper

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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