Arkiv för Matematik

  • Ark. Mat.
  • Volume 47, Number 1 (2009), 143-181.

The Hilbert scheme of points for supersingular abelian surfaces

Stefan Schröer

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Abstract

We study 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 on surfaces. Along the line, we find examples of canonical singularities that are not rational singularities.

Article information

Source
Ark. Mat., Volume 47, Number 1 (2009), 143-181.

Dates
Received: 19 February 2007
Revised: 8 August 2007
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907062

Digital Object Identifier
doi:10.1007/s11512-007-0065-6

Mathematical Reviews number (MathSciNet)
MR2480919

Zentralblatt MATH identifier
1190.14032

Rights
2008 © Institut Mittag-Leffler

Citation

Schröer, Stefan. The Hilbert scheme of points for supersingular abelian surfaces. Ark. Mat. 47 (2009), no. 1, 143--181. doi:10.1007/s11512-007-0065-6. https://projecteuclid.org/euclid.afm/1485907062


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