Almost minimal embeddings of quotient singular points into rational surfaces
type: Item
instance of: scholarly article
title: Almost minimal embeddings of quotient singular points into rational surfaces
zbMATH Open document ID: 0942.14009
author: Hideo Kojima
published in: Journal of Mathematics of Kyoto University
publication date: 1999-09-02T00:00:00Z
review text: Let \(V\) be a normal algebraic surface over an algebraically closed field of characteristic zero. Suppose that the singular locus of \(V\) consists of the only point \(P \in V\) which is a quotient singularity. Let \(f \colon X \rightarrow V\) be a minimal resolution and let \(D\) be the reduced exceptional divisor with respect to \(f.\) The singular point \(P\) is called almost minimal when the pair \((X,D)\) is [see \textit{M. Miyanishi} and \textit{S. Tsunoda}, Jap. J. Math., New Ser. 10, 195-242 (1984; Zbl 0596.14023)]. Making use of Mori theory [\textit{S. Mori}, Ann. Math., II. Ser. 116, 133-176 (1982; Zbl 0557.14021)], the author studies such singularities with logarithmic Kodaira dimension \(\overline{\kappa}(X-D) = -\infty\) and obtains a classification theorem when the support of \(D\) is contained in a fiber of a certain \({\mathbb P}^1\)-fibration. Then the theory of rational surfaces with almost minimal singularities is considered. In fact, the author gives a classification of almost minimal pairs \((X,D)\) in the cases where \(\overline{\kappa}(X-D) = 1\) or \(D\) is irreducible. In conclusion some intersecting examples with \(\overline{\kappa}(X-D) = 0\) are described in details.
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14E25
Mathematics Subject Classification ID: 14J17
zbMATH DE Number: 1295043
zbMATH Keywords: normal surface
zbMATH Keywords: rational surface
zbMATH Keywords: almost minimal singularity
zbMATH Keywords: quotient singularity
zbMATH Keywords: minimal resolution
zbMATH Keywords: exceptional divisor
zbMATH Keywords: canonical divisor
zbMATH Keywords: extremal curve
zbMATH Keywords: logarithmic Kodaira dimension
zbMATH Keywords: Hirzebruch surface
zbMATH Keywords: Picard number
zbMATH Keywords: dual graph
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518161
OpenAlex ID: W1891901269
DOI: 10.1215/KJM/1250518161
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description: scientific article; zbMATH DE number 1295043