Flips in dimension three via crepant descent method. (English) Zbl 1044.14004

Hayakawa, Takayuki

Proc. Japan Acad., Ser. A 79, No. 2, 46-51 (2003).

The flip theorem proved by 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 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 [J. Kollár and 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.

Reviewer: Massimiliano Mella (Ferrara)

MSC:

14E30	Minimal model program (Mori theory, extremal rays)
14J30	\(3\)-folds
14E05	Rational and birational maps
14E15	Global theory and resolution of singularities (algebro-geometric aspects)

Keywords:

flip; crepant descent; canonical pair; birational classification; 3-folds; log flops; crepant divisors

Citations:

Zbl 0649.14023; Zbl 0651.14005; Zbl 0773.14004

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