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Abstract: The main purpose of this paper is to establish some useful partial resolutions of singularities for pairs from the minimal model theoretic viewpoint. We first establish the existence of log canonical modifications of normal pairs under some suitable assumptions. It recovers Kawakita's inversion of adjunction on log canonicity in full generality. We also discuss the existence of semi-log canonical modifications for demi-normal pairs and construct dlt blow-ups with several extra good properties. As applications, we study lengths of extremal rational curves and so on.

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Topics: Inversion (discrete mathematics) (58%), Adjunction (51%)

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5 results found

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Abstract: We prove the existence of flips for $\mathbb Q$-factorial NQC generalized lc pairs, and the cone and contraction theorems for NQC generalized lc pairs. This answers a question of C. Birkar which was conjectured by J. Han and Z. Li. As an immediate application, we show that we can run the minimal model program for $\mathbb Q$-factorial NQC generalized lc pairs. In particular, we complete the minimal model program for $\mathbb Q$-factorial NQC generalized lc pairs in dimension $\leq 3$ and pseudo-effective $\mathbb Q$-factorial NQC generalized lc pairs in dimension $4$.

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2 Citations

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Abstract: We establish adjunction and inversion of adjunction for log canonical centers of arbitrary codimension in full generality.

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Topics: Adjunction (60%), Inversion (discrete mathematics) (57%)

2 Citations

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Abstract: We first announce our recent result on adjunction and inversion of adjunction. Then we clarify the relationship between our inversion of adjunction and Hacon's inversion of adjunction for log canonical centers of arbitrary codimension.

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Topics: Adjunction (59%), Inversion (discrete mathematics) (57%)

1 Citations

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Abstract: We prove the existence of a crepant sdlt model for slc pairs whose irreducible components are normal in codimension one.

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Topics: Codimension (52%)

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Abstract: The notion of quasi-log schemes was first introduced by Florin Ambro in his epoch-making paper: Quasi-log varieties. In this paper, we establish the basepoint-free theorem of Reid--Fukuda type for quasi-log schemes in full generality. Roughly speaking, it means that all the results for quasi-log schemes claimed in Ambro's paper hold true. The proof is Kawamata's X-method with the aid of the theory of basic slc-trivial fibrations. For the reader's convenience, we make many comments on the theory of quasi-log schemes in order to make it more accessible.

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14 results found

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Caucher Birkar^{1}, Paolo Cascini^{2}, Christopher D. Hacon^{3}, James McKernan^{2}•Institutions (3)

Abstract: Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.

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Topics: Minimal models (57%), Birational geometry (53%), Dimension (vector space) (52%) ... read more

1,393 Citations

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Abstract: The central theme of this article is the study of deformations of surface singularities using recent advances in three dimensional geometry. The basic idea is the following. Let X 0 be a surface singularity and consider a one parameter deformation {Xo: t e A } . Then the total space X = U X t is a three dimensional object. One can attempt to use the geometry of X to get information about the surface X~. In general X is very singular and so one can try to study it via a suitable resolution of singularities f : X ' -~ X. The existence of a resolution was established by Zariski; the problem is that there are too many of them, none particularly simple. Mori and Reid discovered that the best one can hope for is a partial resolution f : X ' ~ X where X' possesses certain mild singularities but otherwise is a good analog of the minimal resolution of surface singularities. The search for such a resolution is known as Mori 's program (see e.g. [-Ko3, KMM]). After substantial contributions by several mathematicians (Benveniste, Kawamata, Kollfir, Mori, Reid, Shokurov, Vichweg) this was recently completed by Mori [Mo 3]. A special case, which is nonetheless sufficient for the applications presented here, was settled by several persons. A proof was first announced by Tsunoda [TsM], later followed by Shokurov [Sh], Mori [Mo2] and Kawamata [Kaw2]. A precise formulation of the result we need will be provided at the end of the introduction. In certain situations X0 will impose very strong restrictions on X ' and one can use this to obtain information about X and X~ for t 40 . The first application is in chapter two. Teissier [Tel posed the following problem. Let {X~ : s~S} be a flat family of surfaces parameterized by the connected space S. Let X s be the minimal resolution of X~. In general {Xs: s e S } is not a flat family of surfaces, and it is of interest to find necessary and sufficient conditions for this to hold.

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Topics: Isolated singularity (58%), Resolution of singularities (52%), Surface (mathematics) (51%)

553 Citations

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Abstract: In this paper, we prove the cone theorem and the contraction theorem for pairs (X;B), where X is a normal variety and B is an effective R-divisor on X such that KX +B is R-Cartier.

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Topics: Minimal model program (55%), Contraction (operator theory) (51%)

207 Citations

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Abstract: A recurring difficulty in the Minimal Model Program (MMP) is that while log terminal singularities are quite well behaved (for instance, they are rational), log canonical singularities are much more complicated; they need not even be CohenMacaulay. The aim of this paper is to prove that, as conjectured in [Kol92, 1.13], log canonical singularities are Du Bois. The concept of Du Bois singularities, abbreviated as DB, was introduced by Steenbrink in [Ste83] as a weakening of rationality. It is not clear how to define Du Bois singularities in positive characteristic, so we work over a field of characteristic 0 throughout the paper. The precise definition is rather involved, see (1.10), but our main applications rely only on the following consequence:

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Topics: Minimal model program (53%)

195 Citations

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Abstract: We prove inversion of adjunction on log canonicity.

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Topics: Inversion (discrete mathematics) (60%), Ideal sheaf (59%), Invertible sheaf (57%) ... read more

147 Citations