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
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.

References [Enhancements On Off] (What's this?)

Additional Information

Stefan Schröer
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany

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

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2017 University Press, Inc.
AMS Website