Posted Content•
Crepant semi-divisorial log terminal model
TL;DR: In this paper, the existence of a crepant sdlt model for slc pairs whose irreducible components are normal in codimension one is proved. But it is not known whether the model can be applied to the case of non-normal components.
Abstract: We prove the existence of a crepant sdlt model for slc pairs whose irreducible components are normal in codimension one.
References
More filters
TL;DR: In this paper, the authors studied the study of surface singularities using recent advances in 3D geometry and proved the existence of a minimal resolution of singularities for a given set of surfaces.
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.
619 citations
TL;DR: In this article, it was shown that any LMMP/Z on K ≥ 0 with scaling of an ample/Z divisor terminates with a good log minimal model or a Mori fibre space.
Abstract: Let (X/Z,B+A) be a Q-factorial dlt pair where B,A≥0 are Q-divisors and K
X
+B+A∼
Q
0/Z. We prove that any LMMP/Z on K
X
+B with scaling of an ample/Z divisor terminates with a good log minimal model or a Mori fibre space. We show that a more general statement follows from the ACC for lc thresholds. An immediate corollary of these results is that log flips exist for log canonical pairs.
99 citations
TL;DR: In this paper, the abundance theorem for slc three-folds has been proved for B-pluricanonical representations, and it is shown that the abundance of three-fold representations can be reduced to 2.0.
Abstract: 0. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 1. Definitions and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 2. Reduced boundaries of dlt n-folds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 3. Finiteness of B-pluricanonical representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 4. The abundance theorem for slc threefolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
98 citations
TL;DR: In this article, the existence of log canonical modifications for a log pair was proved and it was shown that K-semistable polarized varieties can only have semi-log-canonical singularities.
Abstract: We prove the existence of log canonical modifications for a log pair. As an application, together with Koll\"ar's gluing theory, we remove the assumption in the first named author's work [Odaka11], which shows that K-semistable polarized varieties can only have semi-log-canonical singularities.
39 citations
TL;DR: In this paper, the authors constructed a composition of blowings-up such that the transformed pair is everywhere semi-simple normal crossings, and f is an isomorphism over the semi-simple normal crossings locus of (X, D).
Abstract: Let X denote a reduced algebraic variety and D a Weil divisor on X. The pair (X, D) is said to be semi-simple normal crossings (semi-snc) at \({a \in X}\) if X is simple normal crossings at a (i.e., a simple normal crossings hypersurface, with respect to a local embedding in a smooth ambient variety), and D is induced by the restriction to X of a hypersurface that is simple normal crossings with respect to X. We construct a composition of blowings-up \({f:\tilde{X}\rightarrow {X}}\) such that the transformed pair \({(\tilde{X}, \tilde{D})}\) is everywhere semi-simple normal crossings, and f is an isomorphism over the semi-simple normal crossings locus of (X, D). The result answers a question of Kollar.
24 citations