Partial resolutions of Hilbert type, Dynkin diagrams, and generalized Kummer varieties
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.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- December 1998
- DOI:
- 10.48550/arXiv.math/9812078
- arXiv:
- arXiv:math/9812078
- Bibcode:
- 1998math.....12078K
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Complex Variables;
- Mathematics - Differential Geometry;
- Mathematics - Representation Theory
- E-Print:
- 33 pages