Classification of Enriques log surfaces with \(d = 1\)
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.
scientific article; zbMATH DE number 2121445
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