Kummer surfaces for the self-product of the cuspidal rational curve
Author:
Stefan Schröer
Journal:
J. Algebraic Geom. 16 (2007), 305-346
DOI:
https://doi.org/10.1090/S1056-3911-06-00438-3
Published electronically:
December 4, 2006
MathSciNet review:
2274516
Full-text PDF
Abstract |
References |
Additional Information
Abstract: The classical Kummer construction attaches a K3 surface to an abelian surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by the self-product of the rational cuspidal curve, and the sign involution by suitable infinitesimal group scheme actions, we give the correct Kummer-type construction for this situation. We encounter rational double points of type $D_4$ and $D_8$ instead of type $A_1$. It turns out that the resulting surfaces are supersingular K3 surfaces with Artin invariant one and two. They lie in a 1-dimensional family obtained by simultaneous resolution, which exists after purely inseparable base change.
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Additional Information
Stefan Schröer
Affiliation:
Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany
MR Author ID:
630946
Email:
schroeer@math.uni-duesseldorf.de
Received by editor(s):
May 19, 2005
Received by editor(s) in revised form:
August 30, 2005, October 19, 2005, and November 11, 2005
Published electronically:
December 4, 2006
