Projective models of the supersingular K3 surface with Artin invariant 1 in characteristic 5

Details Projective models of the supersingular K3 surface with Artin invariant 1 in characteristic 5 Projective models of the supersingular K3 surface with Artin invariant 1 in characteristic 5

2012-01-22

arxiv.org/abs/1201.4533v1

Mathematics

Algebraic Geometry

38 pages, no figures

Scientific paper

Let X be a supersingular K3 surface in characteristic 5 with Artin invariant 1, which is unique up to isomorphisms. Then X has a polarization of degree 2 that realizes X as the double cover of the projective plane branching along the Fermat sextic curve. We present a list of polarizations of X with degree 2 whose intersection number with this Fermat sextic polarization is less than or equal to 5, and give the defining equations of the corresponding projective models. We also present a method to describe birational morphisms between these projective models explicitly. As a by-product, a non-projective automorphism of the Fermat sextic double plane is obtained.

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