The Artin invariant of supersingular weighted Delsarte \(K3\) surfaces (Q2565390)

Let \(k\) be an algebraically closed field of positive characteristic \(p\). Let \(X_k\) be a \(K3\) surface defined over \(k\). Denote by \(\text{NS} (X_k)\) the Néron-Severi group of \(X_k\). It is known that \(\text{NS} (X_k)\) is a finitely generated abelian group with \(\mathbb{Z}\)-rank at most 22: put \(\rho(X_k)= \text{NS}(X_k)\). As \textit{T. Shioda} [J. Reine Angew. Math. 381, 205-210 (1987; Zbl 0618.14014)], we call \(X_k\) a supersingular \(K3\) surface if \(\rho(X_k)=22\). Write disc\(\text{NS} (X_k)\) for the determinant of the intersection matrix of \(\text{NS}(X_k)\). If \(X_k\) is supersingular, then \(\text{disc NS}(X_k)=-p^{2\sigma_0(X_k)}\) for some integer \(\sigma_0= \sigma_0(X_k)\) satisfying \(1\leq\sigma_0\leq 10\) [\textit{M. Artin}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 543-567 (1974; Zbl 0322.14014)]. The integer \(\sigma_0\) may be called the Artin invariant of \(X_k\). In: Algebraic geometry, Proc. Summer Meet., Copenh. 1978, Lect. Notes Math. 732, 564-591 (1979; Zbl 0414.14019), \textit{T. Shioda} showed that \(\sigma_0\) takes all the 10 possible values: furthermore, in his first cited paper, he gave concrete examples of \(K3\) surfaces for all values of \(\sigma_0\) except for \(\sigma_0=7\) and 10. In this paper, we apply Shioda's method (which is based on Ekedahl's algorithm of computing \(\sigma_0)\) to weighted Delsarte surfaces and construct supersingular \(K3\) surfaces with Artin invariant 10.