Jet schemes, log discrepancies and inversion of adjunction

Ein, Lawrence; Mustaţă, Mircea; Yasuda, Takehiko

doi:10.1007/s00222-003-0298-3

Cited by 74 publications

(97 citation statements)

References 11 publications

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“…These results are due to Mustaţǎ [36] and his collaborators [15,17,45]. The precise statement is as follows:…”

Section: Birational Invariants Via Motivic Integrationmentioning

confidence: 93%

“…Recent work of Mustaţǎ [35] supports these predictions by showing that the arc spaces contain information about singularities, for example rational singularities of X can be detected by the irreducibility of the jet schemes for complete intersections. In subsequent investigations he and his collaborators show that certain invariants of birational geometry, such as the log canonical threshold of a pair, for example, can be read off from the dimensions of certain components of the jet schemes, see [36,15,17] and Section 5 where we will discuss some of these results in detail. Let X be a (smooth) scheme of finite type over k of dimension n. An m-jet of X is an order m infinitesimal curve in X , i.e.…”

Section: 2mentioning

confidence: 99%

“…This is achieved as one would expect by declaring a set measurable if it is approximated in a suitable sense by stable sets. This is essentially carried out in [33], though there are some inaccuracies; but everything should be fine if one works over C and makes some 17 The constructible subsets of a scheme Y are the smallest algebra of sets containing the closed sets in Zariski topology. 18 Reid [40], Batyrev [2] and Looijenga [33] use this definition which gives the volume µ X (J m (X )) ∈ M k of X virtual dimension n. Denef, Loeser [13] and Craw [7] use an additional factor L −n to give µ X (J m (X )) virtual dimension 0.…”

Section: Definition 24 a Subsetmentioning

confidence: 99%

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A short course on geometric motivic integration

Blickle¹

2011

Motivic Integration and Its Interactions With Model Theory and Non-Archimedean Geometry

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ABSTRACT. These notes grew out of the authors effort to understand the theory of motivic integration. They give a short but thorough introduction to the flavor of motivic integration which nowadays goes by the name of geometric motivic integration. Motivic inegration was introduced by Kontsevich and the foundations were worked out by Denef, Loeser, Batyrev and Looijenga. We focus on the smooth complex case and present the theory as self contained as possible. As an illustration we give some applications to birational geometry which originated in the work of Mustaţǎ. CONTENTS

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“…These results are due to Mustaţǎ [36] and his collaborators [15,17,45]. The precise statement is as follows:…”

Section: Birational Invariants Via Motivic Integrationmentioning

confidence: 93%

Section: 2mentioning

confidence: 99%

Section: Definition 24 a Subsetmentioning

confidence: 99%

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A short course on geometric motivic integration

Blickle¹

2011

Motivic Integration and Its Interactions With Model Theory and Non-Archimedean Geometry

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“…By [EMY03,Theorem 0.3], LSC holds on smooth varieties. This immediately yields the following result.…”

Section: Conjectures About Minimal Log Discrepanciesmentioning

confidence: 99%

“…Shokurov proved that these two conjectures imply the termination of flips [Sho04]. For smooth varieties, LSC holds by the fascinating paper [EMY03] and if all varieties in a sequence of flips are smooth, ACC holds for trivial reasons. However, even if we start with a smooth variety X, the MMP easily carries us out of the class of smooth varieties.…”

Section: Introductionmentioning

confidence: 99%

Deformations of singular symplectic varieties and termination of the log minimal model program

Lehn¹,

Pacienza²

2016

Alg. Geom.

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We generalize Huybrechts' theorem on deformation equivalence of birational irreducible symplectic manifolds to the singular setting. More precisely, under suitable natural hypotheses, we show that two birational symplectic varieties are locally trivial deformations of each other. As an application we show the termination of any log minimal model program for a pair (X, ∆) of a projective irreducible symplectic manifold X and an effective R-divisor ∆. To prove this result we follow Shokurov's strategy and show that LSC and ACC for minimal log discrepancies hold for all the models appearing along any log MMP of the initial pair.

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