Asian Journal of Mathematics

Volume 19 (2015)

Number 3

Genus $1$ fibrations on the supersingular $\mathrm{K}3$ surface in characteristic $2$ with Artin invariant $1$

Pages: 555 – 582

DOI: https://dx.doi.org/10.4310/AJM.2015.v19.n3.a7

Authors

Noam D. Elkies (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Matthias Schütt (Institut für Algebraische Geometrie, Leibniz Universität, Hannover, Germany)

Abstract

The supersingular $\mathrm{K}3$ surface $X$ in characteristic $2$ with Artin invariant $1$ admits several genus $1$ fibrations (elliptic and quasi-elliptic). We use a bijection between fibrations and definite even lattices of rank $20$ and discriminant $4$ to classify the fibrations, and we exhibit isomorphisms between the resulting models of $X$. We also study a configuration of $(-2)$-curves on $X$ related to the incidence graph of points and lines of $\mathbb{P}^2(\mathbb{F}_4)$.

Keywords

$\mathrm{K}3$ surface, supersingular, elliptic fibration, quasi-elliptic

2010 Mathematics Subject Classification

Primary 14J27, 14J28. Secondary 06B05, 11G25, 14N20, 51A20.

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Published 19 June 2015