Singularities of pairs

Kollár, János

doi:10.1090/pspum/062.1/1492525

Cited by 316 publications

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“…Hence, ðY; S 0 Þ is also canonical and, hence, plt. By [Ko-Mo98, Theorem 5.51] and [Ko97], S 0 is normal and canonical. & LEMMA 2.5.…”

Section: Some Easy Lemmasmentioning

confidence: 99%

Terminal 3-fold Divisorial Contractions of a Surface to a Curve I

Tziolas¹

2003

Compositio Mathematica

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Abstract. Let G & X be a smooth curve on a 3-fold which has only index 1 terminal singularities along G. In this paper we investigate the existence of extremal terminal divisorial contractions E & Y À! G & X, contracting an irreducible surface E to G. We consider cases with respect to the singularities of the general hypersurface section S of X through G. We completely classify the cases when S is A i , i 4 3, and D 2n for any n. Mathematics Subject Classifications (2000). Primary: 14E30, 14E35.Key words. algebraic geometry. IntroductionOne of the main objectives of birational geometry is to identify in each birational class of varieties some distinguished members which are 'simple' and are called minimal models, and then study the structure of birational maps between them. In dimension two, satisfactory results were known for over one hundred years. In higher dimensions, the minimal model program (MMP) was developed to search for minimal models. After contributions of Reid, Mori, Kawamata, Kolla´r, Shokurov and others, the program was completed in dimension three by Mori in 1988. A projective variety X is called a minimal model iff it is Q-factorial, terminal and K X is nef. According to Mori's theorem, for any Q-factorial, terminal projective 3-fold X, there is a sequence of birational maps X À! X 0 , such that X 0 is either a minimal model or has the structure of a Mori fiber space. The birational maps that appear are divisorial contractions and flips. Any birational map between minimal models is an isomorphism in codimension one and a composition of flops . Terminal flops were classified by the work of Kolla´r [Ko91].The structure of birational maps between Fano fiber spaces is complicated. The Sarkisov program was developed by Sarkisov, Reid and Corti to factorize birational maps between these spaces as a composition of 'elementary links ' [Cor95]. These links consist of flops, flips and divisorial contractions. Therefore to understand the structure of birational maps between Fano fiber spaces, it is important to understand divisorial contractions and flips. Flips were classified by Kolla´r and Mori . The structure of divisorial contractions is still an open problem.

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“…Hence, ðY; S 0 Þ is also canonical and, hence, plt. By [Ko-Mo98, Theorem 5.51] and [Ko97], S 0 is normal and canonical. & LEMMA 2.5.…”

Section: Some Easy Lemmasmentioning

confidence: 99%

Terminal 3-fold Divisorial Contractions of a Surface to a Curve I

Tziolas¹

2003

Compositio Mathematica

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“…In this situation, we say that Theorem 1.2 is a theorem on Analytic Inversion of Adjunction. For more on Inversion of Adjunction, we refer the reader to the book [4] of R. Lazarsfeld or the notes [3] of J. Kollár.…”

Section: And Note That This Function Takes Values Inmentioning

confidence: 99%