Journal of Algebraic Geometry

Author(s): Stefan Schröer
Journal: J. Algebraic Geom. 16 (2007), 305-346.
Posted: December 4, 2006
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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

and

instead of type

. 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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