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 eprints
 Pub Date:
 December 1998
 DOI:
 10.48550/arXiv.math/9812078
 arXiv:
 arXiv:math/9812078
 Bibcode:
 1998math.....12078K
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Complex Variables;
 Mathematics  Differential Geometry;
 Mathematics  Representation Theory
 EPrint:
 33 pages