Normal algebraic surfaces with trivial tricanonical divisors
type: Item
instance of: scholarly article
title: Normal algebraic surfaces with trivial tricanonical divisors
zbMATH Open document ID: 0897.14009
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1998-10-15T00:00:00Z
review text: A normal projective algebraic surface \(S\) with at worst quotient singularities is called a log Enriques surface (LES) if \(H^1 ({\mathcal O}_S) =0\) and some multiple of \(K_S\) is trivial. The smallest such multiple \(I\) is defined as the index of \(S\). Consider the canonical Galois cover \(\pi:Y=\text{proj} \bigoplus^{I-1}_{i=0} {\mathcal O}_S (iK_S)\to S\) with Galois group \(\langle \sigma \rangle\) of order \(I\). \(Y\) is either an abelian or K3 surface with at worst Du Val singularities. \(S\) is said of actual type \(A_n\) \((D_n)\) if \(Y\) has exactly one Du Val points of the corresponding type. The article under review studies rational LES whose canonical cover has a singular point of type \(A_{17}\) or \(D_{17}\) and of index 3. Using the unique algebraic K3 surface of Picard number 20 and discriminant 3 constructed by Shioda and Inose [see ``On the most algebraic K3 surfaces and the most extremal log Enriques surfaces'' by \textit{K. Oguiso} and \textit{D.-Q. Zhang}, in Am. J. Math. 118, No. 6, 1277-1297 (1996; Zbl 0889.14016)], the author constructs three index 3 rational LES of actual type \(A_{17}\) by contracting the three pairs of rational curves located at the ends of the \(D_{19}\) graph in the K3 surface followed by taking quotient under an order 3 automorphism. It is proved that any rational LES of index 3 of type \(A_{17}\) (i.e.; whose canonical cover has an \(A_{17}\) singularity) must be isomorphic to one of the three examples by showing that there are two smooth rational curves on the minimal resolution \(X\) of the canonical cover \(Y\) jointly with the graph corresponding to the \(A_{17}\) singularity form a \(D_{19}\) graph and \(X\) must be the unique Shioda-Inose surface. By results from early works by the author [``Normal algebraic surfaces with trivial two or four times of the canonical divisors''] and the author and \textit{K. Oguiso} [``K3 surfaces with order 5 automorphisms''] the index of a rational LES of type \(A_{17}\) ha…
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J26
zbMATH DE Number: 1117532
zbMATH Keywords: trivial tricanonical divisors
zbMATH Keywords: log Enriques surface
author: De-Qi Zhang
reviewed by: Tie Luo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195145323
OpenAlex ID: W2072867587
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DOI: 10.2977/PRIMS/1195145323
description: scientific article; zbMATH DE number 1117532