\(E1\)-blowing ups of terminal singularities
type: Item
instance of: scholarly article
title: \(E1\)-blowing ups of terminal singularities
zbMATH Open document ID: 1038.14005
DOI: 10.1023/A:1022879914227
published in: Journal of Mathematical Sciences (New York)
publication date: 2003-09-16T00:00:00Z
review text: Let \(o\in Z\) be a 3-dimensional terminal singularity. A plt-blowup is a projective birational morphism \(f:X \to Z\) with an irreducible exceptional divisor \(S\) such that \((X,S)\) is purely log terminal. The exceptional divisor \(S\) is a normal surface, and there is an adjunction formula \((K_X+S)| _S = K_X+\text{Diff}_S\), where \(\text{Diff}_S\) is a naturally defined \(\mathbb{Q}\)-Weil divisor on \(S\). The pair \((S,\text{Diff}_S)\) is klt and \(-(K_S+\text{Diff}_S)\) is ample over \(f(S)\). This leads to the question of classifying the following situations: \(f(S)\) is a point, \(S\) is projective and \((S,\text{Diff}_S)\) is a log del Pezzo surface. \(f(S)\) is a curve and \(S/f(S)\) is a log conic bundle. In previous work, \textit{Yu. G. Prokhorov} and \textit{V. V. Shokurov} [Izv. Math. 65, 1169--1196 (2001); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 65, No. 6, 99--128 (2001; Zbl 1068.14018)] have raised the question on the classification of singularities and Mori contractions in terms of complements. Shokurov's proposal to subdivide all regular 2-dimensional complements into the seven types \(A1\), \(E1\), \(D2\), \(E2\), \(E3\), \(E4\) and \(E6\) is parallel to the Kodaira-classification of singular fibres on an elliptic fibration. Problem: Find the relation between the classification of terminal singularities and the classification of 2-dimensional regular complements. In the present paper, the author studies the case \(E1\), which is an analog of the Kodaira-type-\(I_0\) fibres. Main results: Let \(o\in Z\) be as above and \(f:(X,S)\to Z\) be a plt-blowup. Assume that \((S,\text{Diff}_S)\) has a complement of type \(E1\) and no complements of types \(A1\), \(D2\). (i) \(S\) is a projective del Pezzo surface with DuVal singularities, i.e. \(f(S)=o\). (ii) \(\text{Diff}_S= (1-1/m)C\), where \(2\leq m \leq 6\) and \(C\) is a smooth elliptic curve contained in the smooth locus of \(S\). (iii) \(X\) has only terminal singular points outside \(C…
reviewed by: Marko Roczen
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14J27
zbMATH DE Number: 1982004
zbMATH Keywords: plt-blowup
zbMATH Keywords: terminal singularity
zbMATH Keywords: two-dimensional regular complement
author: Yuri Prokhorov
MaRDI profile type: Publication
full work available at URL: a:1022879914227
OpenAlex ID: W113170874
description: scientific article; zbMATH DE number 1982004