Classification of Enriques log surfaces with \(d = 1\)
type: Item
instance of: scholarly article
title: Classification of Enriques log surfaces with \(d = 1\)
zbMATH Open document ID: 1061.14032
author: S. A. Kudryavtsev
published in: Mathematical Notes
publication date: 2004-12-07T00:00:00Z
review text: The author continues his classification of log Enriques surfaces \(X\), i.e., \(X\) is projective and log terminal (= quotient singularity because \(\dim X = 2\) now) with \(K_X = 0\) (numerically). Following Shokurov, the author defines \(\delta(X)\) to be the number of exceptional divisors with discrepancy \(\leq -6/7\). \textit{V.V. Shokurov} [J. Math. Sci., New York 102, 3876--3932 (2000; Zbl 1177.14078)] proved that \(0 \leq \delta(X) \leq 2\). In an earlier paper [Math. Notes 72, 660--666 (2002; Zbl 1047.14023)], the author has classified those \(X\) with \(\delta(X) = 2\). Here the author finishes the case \(\delta(X) = 1\). It turns out that the canonical index \(I = I(X)\) (the minimal positive integer such that \(IK_X \sim 0\)) is one of \(7, 11, 13, 17\). Log Enriques surface were first defined in the reviewer's papers [J. Math. Kyoto Univ. 31, 419--466 (1991; Zbl 0759.14029); ibid., 33, 357--397 (1993; Zbl 0804.14017)] where prime canonical index cases are classified. They were further studied by \textit{R. Blache} [J. Algebr. Geom. 4, 137--179 (1995; Zbl 0836.14018)] where an inequality from the reviewer's paper was further sharpened to \(I(X) \leq 21\). No examples with \(I(X) = 20, 21\) have been constructed yet. Examples with \(I(X) = prime\) are constructed in reviewer's and also Blache's paper.
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14E30
zbMATH DE Number: 2121445
reviewed by: De-Qi Zhang
MaRDI profile type: Publication
full work available at URL: b:matn.0000036744.16067.fa
OpenAlex ID: W2024047023
DOI: 10.1023/B:MATN.0000036744.16067.FA
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description: scientific article; zbMATH DE number 2121445