The Hilbert scheme of points for supersingular abelian surfaces

2006/06/12 by Stefan Schroeer, Schroeer, Stefan
Mathematics · Physics and Astronomy · #14B05 #14C05 #14D06 #14K15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Nonlinear Waves and Solitons #math.AG #msc:14B05 #msc:14C05 #msc:14D06 #msc:14K15

paper · pdf · doi:10.48550/arxiv.math/0606267

31 pages, 5 figures

arxiv created 2006/06/12 · openalex publication_date 2006/06/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We analyse the geometry of Hilbert schemes of points on abelian surfaces and Beauville's generalized Kummer varieties in positive characteristics. The main result is that, in characteristic two, the addition map from the Hilbert scheme of two points to the abelian surface is a quasifibration, such that all fibers are nonsmooth. In particular, the corresponding generalized Kummer surface is nonsmooth, and minimally elliptic singularities occur in the supersingular case. We unravel the structure of the singularities in dependence of p-rank and a-number of the abelian surface. To do so, we establish a McKay Correspondence for Artin's wild involutions.

Citations

• The deformation theory of local complete intersections
• Cohen-Macaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci

Related

• Hilbert schemes of points on some classes of surface singularities
• Hilbert schemes of points on surfaces
• G-invariant Hilbert Schemes on Abelian Surfaces and Enumerative Geometry of the Orbifold Kummer Surface
• Supersingular reduction of Kummer surfaces in residue characteristic 2
• Euler characteristics of Hilbert schemes of points on simple surface\n singularities