Supersingular \(K3\) surfaces in odd characteristic and sextic double planes
Let \(X\) be a \(K3\) surface defined over an algebraically closed field of positive characteristic \(p\neq 2\). According to Shioda, \(X\) is said to be supersingular if its Picard lattice has rank 22. The main result of this paper is the following: Theorem. If \(X\) is a supersingular \(K3\) surface, then there exists a separable degree 2 morphism \(f: X\to{\mathbb P}^2\). (In other words, \(X\) is a so-called sextic double plane.) The idea of the proof is to use lattice theoretic arguments to show the existence of a nef class \(L\in \text{Pic}(X)\) with \(L^2=2\). Then the morphism \(f\) is just the map given by the linear system \(| L| \). The author also describes the branch locus of \(f\), which is a plane sextic with at most simple singularities. Hence, if \(X\to Y\to{\mathbb P}^2\) is the Stein factorization of \(f\), then the surface \(Y\) is birational to \(X\) and it has at most canonical singularities. The characteristic 2 case had already been treated by the author in an earlier paper [\textit{I. Shimada}, Math. Comput. 73, No. 248, 1989--2017 (2004; Zbl 1052.14041)], in which he had shown the existence of an inseparable degree 2 map \(f\colon X\to {\mathbb P}^2\).
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