Logarithmic Enriques surfaces. II
[For part I see J. Math. Kyoto Univ. 31, No. 2, 419-466 (1991; Zbl 0759.14029).] A normal projective surface \(\overline V\) defined over \(\mathbb{C}\) is called a log Enriques surface if \(\overline V\) has only quotient singularities, some multiple \(NK_{\overline V}\) of the canonical divisor \(K_{\overline V}\) is linearly equivalent to zero for \(N \geq 1\) and \(H^ 1(\overline V, {\mathcal O}_{\overline V}) = 0\). The smallest positive integer \(I\) with \(IK_{\overline V}\sim 0\) is called the index of \(\overline V\), which necessarily satisfies \(1 \leq I \leq 66\) and \(I \leq 19\) if \(I\) is prime. Associated with the relation \(IK_{\overline V} \sim 0\), there exists the canonical covering \(\pi : \overline U \to \overline V\) which is a \(\mathbb{Z}/I\mathbb{Z}\)-covering and étale outside \(\text{Sing} (\overline V)\). Moreover \(\overline U\) is either an abelian surface or a \(K3\)-surface admitting at most rational double points. As a sequel to the author's previous paper [cf. part I, loc. cit.], where the cases \(I \leq 2\) or \(\overline U\) is smooth are treated, the author considers the case where \(I \geq 3\), \(I\) is a prime and \(\overline U\) is a \(K3\)-surface. He describes completely the possible distributions of singular points on \(\overline V\) and \(\overline U\) after modifying \(\overline V\) and \(\overline U\) by the canonical birational morphisms. Particularly interesting is the explicit construction of such surfaces with indices 3, 5, 7, 11 and 13.
scientific article; zbMATH DE number 558897