Supplement to classification of threefold divisorial contractions
type: Item
instance of: scholarly article
title: Supplement to classification of threefold divisorial contractions
zbMATH Open document ID: 1256.14014
DOI: 10.1215/00277630-1548493
author: Masayuki Kawakita
published in: Nagoya Mathematical Journal
publication date: 2012-06-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/1103.1182
full work available at URL: https://projecteuclid.org/euclid.nmj/1337690051
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14J30
zbMATH DE Number: 6050955
zbMATH Keywords: minimal model program
zbMATH Keywords: Mori's program
zbMATH Keywords: divisorial contraction
zbMATH Keywords: threefold
zbMATH Keywords: terminal
MaRDI profile type: Publication
OpenAlex ID: W2080427543
arXiv ID: 1103.1182
cites work: Blowing ups of 3-dimensional terminal singularities
cites work: Blowing ups of 3-dimensional terminal singularities. II
cites work: Divisorial contractions to 3-dimensional terminal singularities with discrepancy one
cites work: Divisorial contractions in dimension three which contract divisors to smooth points
cites work: Q4550590
cites work: General elephants of three-fold divisorial contractions
cites work: Three-fold divisorial contractions to singularities of higher indices
cites work: Q4352472
cites work: On 3-dimensional terminal singularities
review text: Let \(f: Y\rightarrow X\) be a \(K_Y\)-negative extremal contraction of a normal \(\mathbb{Q}\)-Gorenstein terminal threefold of relative Picard number \(\rho(Y/X)=1\). The map \(f\) is projective with connected fibers and \(-K_Y\) is \(f\)-ample. Assume that it contracts a prime divisor \(E\subseteq Y\) to a point \(P\in X\), then \(E=\)Exc\((f)\) and \(K_Y=f^*K_X+a(E,X)E\) for some rational number \(a(E,X)>0.\) A map like is indispensable in the minimal model program of terminal threefolds. From the general theory of singularities in the minimal model program, we know that \(P\in X\) is normal, \(\mathbb{Q}\)-Gorenstein, and terminal. Three dimensional terminal singularities were classified by \textit{S. Mori} [Nagoya Math. J. 98, 43--66 (1985; Zbl 0589.14005)].NEWLINENEWLINEIn the paper under review, the author studies the case where \((X,P)\) is of type \(e1\) with \(P=cD/2\) and \(a(E,X)=2\) via the technique developed in [\textit{M. Kawakita}, Duke Math. J. 130, No. 1, 57--126 (2005; Zbl 1091.14008)]. Combined with \textit{M. Kawakita}, [Duke Math. J. 130, No. 1, 57--126 (2005; Zbl 1091.14008)] and \textit{T. Hayakawa}, [Publ. Res. Inst. Math. Sci. 35, No. 3, 515--570 (1999; Zbl 0969.14008); Publ. Res. Inst. Math. Sci. 36, No. 3, 423--456 (2000; Zbl 1017.14006)], the author completes the classification of the \(K_Y\)-extremal divisorial contraction \(f: Y\rightarrow X\) of \(\rho(Y/X)=1\) that contracts the exceptional divisor \(E\) to a non-Gorenstein point \(P\). As a corollary, every such a contraction is a weighted blowup.NEWLINENEWLINEOn the other hand, if \(P\in X\) a Gorenstein point, in [\textit{A. Corti} (ed.) and \textit{M. Reid} (ed.), Explicit birational geometry of 3-folds. London Mathematical Society Lecture Note Series. 281. Cambridge: Cambridge University Press. (2000; Zbl 0942.00009)] it was shown that \(f\) is a weighted blow up if \(P\in X\) is a rational double point. Later on, the author showed that a contraction is a weighted blow up if \…
reviewed by: Ching-Jui Lai
description: scientific article; zbMATH DE number 6050955