A geometric characterization of normal two-dimensional singularities of multiplicity two with \(p_ a\leq 1\)
type: Item
instance of: scholarly article
title: A geometric characterization of normal two-dimensional singularities of multiplicity two with \(p_ a\leq 1\)
zbMATH Open document ID: 0545.14030
author: Masataka Tomari
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1984-01-01T00:00:00Z
review text: The goal of this paper is the following theorem: A normal two-dimensional singularity (V,p) of multiplicity two over \({\mathbb{C}}\) satisfies the condition \(p_ a(V,p)\leq 1\) if and only if each normalization in Zariski's canonical resolution of (V,p) is trivial or obtained by blowing-up along (reduced) \({\mathbb{P}}^ 1\) in the singular locus. This is an analogy of the well-known characterization of the rational double point by the absolute isolatedness. Our result is proved as a corollary of four lemmas; an inequality about \(p_ a\), a formula for computation of the geometric genus due to \textit{E. Horikawa} [Invent. Math. 31, 43-85 (1975; Zbl 0317.14018)], a criterion for the condition \(p_ a\leq 1\) due to \textit{S. S.-T. Yau} [Trans. Am. Math. Soc. 257, 269-329 (1980; Zbl 0465.32008)], and an elementary computation on one-dimensional singularity with multiplicity \(\leq 5\). The effective use of the arithmetic genus \(p_ a\) of the singularity is new at all.
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14E15
zbMATH DE Number: 3867505
zbMATH Keywords: arithmetic genus of singularity
zbMATH Keywords: normal two-dimensional singularity
zbMATH Keywords: resolution
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195181824
OpenAlex ID: W2067425978
cites work: Local Rings of High Embedding Dimension
cites work: Some Numerical Criteria for Contractability of Curves on Algebraic Surfaces
cites work: On Isolated Rational Singularities of Surfaces
cites work: Über die Auflösung gewisser Singularitäten von holomorphen Abbildungen
cites work: The fundamental divisor of normal double points of surfaces
cites work: On the Factoriality of Local Rings of Small Embedding Codimension
cites work: On deformations of quintic surfaces
cites work: The Structure of an Isolated Multiple Point of a Surface. I
cites work: Normal Two-Dimensional Singularities. (AM-71)
cites work: On Rational Singuarities
cites work: On Minimally Elliptic Singularities
cites work: On normal two-dimensional double point singularities
cites work: Q4063570
cites work: Algebricite de certains points singuliers algebroides
cites work: Analytic Sheaf Cohomology Groups of Dimension n of n-Dimensional Complex Spaces
cites work: Q5568962
cites work: Elliptic Singularities of Surfaces
cites work: On plurigenera of normal isolated singularities. I
cites work: On Maximally Elliptic Singularities
cites work: Hypersurface Weighted Dual Graphs of Normal Singularities of Surfaces
cites work: The reduction of the singularities of an algebraic surface
DOI: 10.2977/PRIMS/1195181824
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description: scientific article; zbMATH DE number 3867505