Flips in dimension three via crepant descent method
type: Item
instance of: scholarly article
title: Flips in dimension three via crepant descent method
zbMATH Open document ID: 1044.14004
author: Takayuki Hayakawa
published in: Proceedings of the Japan Academy. Series A
publication date: 2003-01-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1116443608
review text: The flip theorem proved by \textit{S. Mori} [J. Am. Math. Soc. 1, No. 1, 117--253 (1988; Zbl 0649.14023)], is one the main breakthrough in the birational classification of projective varieties. Mori's proof is based on a deep analysis of the possible singularities of an anticanonical section of a flipping contraction and relies on \textit{Y. Kawamata}'s results [Ann. Math. (2) 127, No. 1, 93--163 (1988; Zbl 0651.14005)] to conclude from this the existence of flips for 3-folds. In the note under review the author gives an alternative conclusion of the proof. The existence of a general elephant [\textit{J. Kollár} and \textit{S. Mori}, J. Am. Math. Soc. 5, No. 3, 533--703 (1992; Zbl 0773.14004)], reduces the existence of flips to the existence of log flops. The latter are proved to exist by descending induction on the number of crepant divisors via the study of terminal extractions.
reviewed by: Massimiliano Mella
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14E15
zbMATH DE Number: 2024719
zbMATH Keywords: flip
zbMATH Keywords: crepant descent
zbMATH Keywords: canonical pair
zbMATH Keywords: birational classification
zbMATH Keywords: 3-folds
zbMATH Keywords: log flops
zbMATH Keywords: crepant divisors
MaRDI profile type: Publication
OpenAlex ID: W1980976014
cites work: Q4296337
cites work: 3-fold flips after Shokurov
cites work: Elementary contractions of Gorenstein threefolds
cites work: Blowing ups of 3-dimensional terminal singularities
cites work: Blowing ups of 3-dimensional terminal singularities. II
cites work: Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces
cites work: TERMINATION OF LOG FLIPS FOR ALGEBRAIC 3-FOLDS
cites work: Q3825176
cites work: Flops
cites work: Q3136464
cites work: Classification of Three-Dimensional Flips
cites work: Q3122845
cites work: Threefolds whose canonical bundles are not numerically effective
cites work: Flip Theorem and the Existence of Minimal Models for 3-Folds
cites work: Q3222980
cites work: AN ADDENDUM TO THE PAPER “3-FOLD LOG FLIPS”
DOI: 10.3792/PJAA.79.46
description: scientific article; zbMATH DE number 2024719