Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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
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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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Stefan Schröer
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany
Email: schroeer@math.uni-duesseldorf.de

DOI: https://doi.org/10.1090/S1056-3911-06-00438-3
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

American Mathematical Society