Kummer surfaces associated with group schemes
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© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Shigeyuki Kond¯o · Stefan Schröer
Kummer surfaces associated with group schemes Received: 11 May 2020 / Accepted: 30 October 2020 Abstract. We introduce Kummer surfaces X = Km(C ×C) with the group scheme G = μ2 acting on the self-product of the rational cuspidal curve in characteristic two. The resulting quotients are normal surfaces having a configuration of sixteen rational double points of type A1 , together with a rational double point of type D4 . We show that our Kummer surfaces are precisely the supersingular K3 surfaces with Artin invariant σ ≤ 3, and characterize them by the existence of a certain configuration of thirty curves. After contracting suitable curves, they also appear as normal K3-like coverings for simply-connected Enriques surfaces.
Contents Introduction . . . . . . . . . . . . . . . . . . . . . 1. Some restricted Lie algebras . . . . . . . . . . . 2. Diagonal actions and rational points . . . . . . . 3. Kummer surfaces associated with group schemes 4. Characterization with configurations of curves . 5. Characterization with Artin invariants . . . . . . 6. Enriques surfaces and K3-like coverings . . . . References . . . . . . . . . . . . . . . . . . . . . .
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Introduction For each abelian surface A in characteristic p = 2 the quotient Z = A/{±1} by the sign involution is a normal surface with sixteen rational double points of type A1 , and the minimal resolution of singularities X = Km(A) is a K3 surface called Kummer surface. Over the complex numbers a K3 surface is a Kummer surface if and only if it contains sixteen disjoint (−2)-curves [15]. In this paper we call a non-singular rational curve on a K3 surface a (−2)-curve for simplicity. If A = J is S. Kond¯o: Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan. e-mail: [email protected] S. Schröer (B): Mathematisches Institut, Heinrich-Heine-Universität, 40204 Düsseldorf, Germany. e-mail: [email protected] Mathematics Subject Classification: 14J28 · 14L15 · 14J27
https://doi.org/10.1007/s00229-020-01257-4
S. Kond¯o, S. Schröer
the jacobian variety of a curve of genus two, the Kummer surface X contains thirtytwo distinguished (−2)-curves forming the so-called (166 )-configuration (e.g. [7], Chapter 6, page 787, Figure 21), and the existence of these thirty-two (−2)-curves characterizes the Kummer surface associated with a curve of genus 2 [14]. Shioda [24] showed that a Kummer surface X = Km(A) in odd characteristics is supersingular if and only if the abelian surface A is supersingular. For p = 2, however, Shioda [23] and Katsura [9] observed that the singularities on Z = A/{±1} are more complicated, and that X is a K3 surface if and only if A i
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