Classification of Enriques surfaces covered by the supersingular \(K3\) surface with Artin invariant 1 in characteristic 2
A \(K3\) surface defined over an algebraically closed field of characteristic~\(p\) is called supersingular if its Néron-Severi group has rank~\(22\). Then, \(p > 0\) and the determinant of the Néron-Severi lattice is equal to \(-p^{2\sigma}\), for some \(1\leq \sigma \leq 10\) called Artin invariant. A supersingular \(K3\) surface with of Artin invariant \(\sigma = 1\) is unique up to isomorphism for every \(p\). The author classifies all Enriques surfaces \(X\) defined over an algebraically closed field of characteristic~\(p = 2\) such that the minimal resolution \(Y\) of the canonical double cover \(\bar Y \rightarrow X\) is isomorphic to the supersingular \(K3\) surface of Artin invariant~\(1\). It turns out that there exist exactly three types of such Enriques surfaces, called \(\mathrm{MI}\), \(\mathrm{VII}\) and \(\mathrm{MII}\), for which explicit models are provided. The first type had already been studied by \textit{T. Katsura} and the author [Pure Appl. Math. Q. 11, No. 4, 683--709 (2015; Zbl 1367.14015)]. The proof is based on a detailed study of the nodal curves and elliptic fibrations of \(X\), and of the singularities of \(\bar Y\) (necessarily rational double points). A key ingredient is the classification of the genus 1 fibrations on \(Y\) by \textit{N. D. Elkies} and \textit{M. Schütt} [Asian J. Math. 19, No. 3, 555--582 (2015; Zbl 1319.14043)]. The problem of enumerating all Enriques surfaces covered by a given \(K3\) surface has also been studied in characteristic \(p = 0\) by \textit{H. Ohashi} [Publ. Res. Inst. Math. Sci. 43, No. 1, 181--200 (2007; Zbl 1133.14038)] and by \textit{I. Shimada} and the reviewer [``Enriques involutions on singular \(K3\) surfaces of small discriminants'', Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) XXI, 1667--1701 (2020)], and in characteristic \(p > 2\) by \textit{K. Behrens} [``On the number of Enriques quotients for supersingular \(K3\) surfaces'', Preprint, \url{arXiv:2003.02132}].
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