Logarithmic Enriques surfaces. II
type: Item
instance of: scholarly article
title: Logarithmic Enriques surfaces. II
zbMATH Open document ID: 0804.14017
published in: Journal of Mathematics of Kyoto University
publication date: 1995-01-16T00:00:00Z
review text: [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.
reviewed by: M. Miyanishi
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J17
zbMATH DE Number: 558897
zbMATH Keywords: index of normal surface
zbMATH Keywords: log Enriques surface
zbMATH Keywords: normal quotient singularities
zbMATH Keywords: covering
zbMATH Keywords: distribution of singular points
author: De-Qi Zhang
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519265
OpenAlex ID: W4250722627
DOI: 10.1215/KJM/1250519265
Recommended article: Logarithmic Enriques surfaces
Recommended article: NORMAL ALGEBRAIC SURFACES WITH TRIVIAL TWO OR FOUR TIMES OF THE CANONICAL DIVISOR
Recommended article: Q4357383
Recommended article: Classification of Enriques log surfaces with \(d = 1\)
Recommended article: Normal algebraic surfaces with trivial bicanonical divisor
Recommended article: Normal algebraic surfaces with trivial tricanonical divisors
Recommended article: On extremal log Enriques surfaces. II
Recommended article: THE CLASSIFICATION OF LOG ENRIQUES SURFACES OF RANK 18
Recommended article: On the most algebraic K3 surfaces and the most extremal log Enriques surfaces
Recommended article: Q4035325
description: scientific article; zbMATH DE number 558897