Jet schemes and singularities

Ein, Lawrence; Mustaţă, Mircea

doi:10.1090/pspum/080.2/2483946

Cited by 99 publications

(132 citation statements)

References 7 publications

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“…The latter, independently proven also by Ishii in [Ish11], extends to the setting considered here the main theorems of [EMY03,EM04,Kaw08,EM09], and has a number of consequences that were previously obtained in [Mus01,EM04,dFEM10] for normal, locally complete intersection varieties. These include the semi-continuity of minimal log J-discrepancies (see Corollaries 4.15 and 4.14, see also [Ish11]) and the fact that the set of all log J-canonical thresholds in any fixed dimension satisfies the ascending chain condition (see Corollary 4.13).…”

Section: Introductionsupporting

confidence: 67%

“…Proposition (2.8) of [AK70]) and the canonical map Ω n X → ω X (cf. Proposition 9.1 of [EM09]) factors through an inclusion ω X ֒→ ω X .…”

Section: Nash Blow-upmentioning

confidence: 99%

“…Geometric properties of Mather discrepancies were investigated in [dFEI08], and the recent work of Ishii [Ish11] is devoted to a study of singularities from the point of view of Mather discrepancies. When X is Q-Gorenstein (that is, X is normal and its canonical class is Q-Cartier), the relationship between Jacobian discrepancies, Mather discrepancies and usual discrepancies is implicit in the works [Kaw08,EM09,Eis10].…”

Section: Introductionmentioning

confidence: 99%

“…We are grateful to the referee for the careful reading and many comments and corrections. The article [EM09] has been of invaluable help in the preparation of this paper. The paper was completed while the first author was visiting theÉcole Normale Supérieure, and he wishes to thank the institution and Olivier Debarre for providing support and a stimulating environment.…”

Section: Introductionmentioning

confidence: 99%

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Jacobian discrepancies and rational singularities

Fernex¹,

Docampo²

2013

J. Eur. Math. Soc.

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Abstract. Inspired by several works on jet schemes and motivic integration, we consider an extension to singular varieties of the classical definition of discrepancy for morphisms of smooth varieties. The resulting invariant, which we call Jacobian discrepancy, is closely related to the jet schemes and the Nash blow-up of the variety. This notion leads to a framework in which adjunction and inversion of adjunction hold in full generality, and several consequences are drawn from these properties. The main result of the paper is a formula measuring the gap between the dualizing sheaf and the Grauert-Riemenschneider canonical sheaf of a normal variety. As an application, we give characterizations for rational and Du Bois singularities on normal Cohen-Macaulay varieties in terms of Jacobian discrepancies. In the case when the canonical class of the variety is Q-Cartier, our result provides the necessary corrections for the converses to hold in theorems of Elkik, of Kovács, Schwede and Smith, and of Kollár and Kovács on rational and Du Bois singularities.

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Section: Introductionsupporting

confidence: 67%

“…Proposition (2.8) of [AK70]) and the canonical map Ω n X → ω X (cf. Proposition 9.1 of [EM09]) factors through an inclusion ω X ֒→ ω X .…”

Section: Nash Blow-upmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Jacobian discrepancies and rational singularities

Fernex¹,

Docampo²

2013

J. Eur. Math. Soc.

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“…Mais au-delà du problème de Nash, c'est important d'étudier les espaces de m-jets et l'espace de arcs, parce qu'ils fournissent des informations sur la géométrie de la variété sous-jacente, par exemple, voir les articles suivants : [Mus01], [EM09], [DL99] y [DL02]. Les espaces d'arcs et les espaces de m-jets ont été utilisés aussi pour solutionner certains problèmes qui ont été considérés comme des problèmes difficiles à résoudre, par exemple, la Conjecture de Batyrev sur les variétés de Calabi-Yau (voir [Kon95]).…”

Section: Introductionunclassified

Familles d'espaces de m -jets et d'espaces d'arcs

Leyton-Álvarez

2016

Journal of Pure and Applied Algebra

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MAXIMILIANO LEYTON-ÁLVAREZRésumé. Soit V une variété algébrique définie sur un corps K algébriquement clos et de caractéristique nulle. Comme les espaces de m-jets Vm et l'espace d'arcs V∞ fournissent des informations sur la géométrie de la variété V , il est donc naturel de se poser les questions suivantes : Quand est-ce qu'une déformation de V induit une déformation des espaces Vm, 1 ≤ m ≤ ∞ ? Si l'on considère une déformation de V qui admet une résolution simultanée à plat, comment variera l'image de l'application de Nash NV ? Dans la section 3, on donne quelques réponses partielles à ces questions.Dans la section 4, on montre deux familles d'hypersufaces de A 4 K où l'application de Nash est bijective. De plus, dans chaque cas, on donne explicitement une désingularization où toutes les composantes irréductibles de la fibre exceptionnelle sont des diviseurs essentiels. Il faut remarquer que dans la littérature on ne trouve pas beaucoup d'exemples de ce type.ABSTRACT. Families of m-jet spaces and arc spaces.Let V be an algebraic variety defined over an algebraically closed field K of characteristic zero. The m-jet spaces Vm and the space of arcs V∞ provide the information on the geometry of the variety V , therefore it is natural to ask the following questions: When will a deformation of V induce a deformation of the spaces Vm, 1 ≤ m ≤ ∞? If one considers a deformation of V which admits a flat simultaneous resolution, how will the image of the Nash application NV vary? In section 3 some partial answers to these questions can be found.In section 4 two families of hypersurfaces of A 4 K are shown in which the Nash application is bijective. What's more, in each case a desingularization in which all the irreducible components of the exceptional fiber are essential divisors is explicitly given. It is important to note that very few examples of this type are found in the literature. IntroductionSoit V une variété algébrique sur un corps K algébriquement clos ou une variété analytique complexe (K := C). Dans la fin des années 60, John Nash a introduit l'espace d'arcs V ∞ dans le but d'obtenir des informations sur la géométrie locale du lieu singulier Sing V de V (voir [Nas95]). L'espace d'arcs V ∞ peut être interprété comme l'ensemble de tous les arcs Spec K[[t]] → V , muni d'une structure "naturelle" de schéma sur K. Une composante de Nash associée à V est une famille d'arcs sur V passant par le lieu singulier de V . Un diviseur essentiel E sur V est grosso modo un diviseur exceptionnel dans une désingularisation de V qui apparaît comme une composante irréductible de la fibre exceptionnelle de toute désin-gularisation possible de V . Nash a défini une application de l'ensemble des composantes de Nash associées à V dans l'ensemble des diviseurs essentiels sur V et il a démontré qu'elle est injective ; cette application est connue sous le nom d'application de Nash, notée N V . Nash pose donc la question suivante : Est-ce que la application N V est bijective ? (pour plus de détails voir la section 2.2). Étant donné exp...

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Singular Support of a Vertex Algebra and the Arc Space of Its Associated Scheme

Arakawa

Linshaw

2019

Representations and Nilpotent Orbits of Lie Algebraic Systems

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Attached to a vertex algebra V are two geometric objects. The associated scheme of V is the spectrum of Zhu's Poisson algebra R V . The singular support of V is the spectrum of the associated graded algebra gr(V) with respect to Li's canonical decreasing filtration. There is a closed embedding from the singular support to the arc space of the associated scheme, which is an isomorphism in many interesting cases. In this note we give an example of a non-quasi-lisse vertex algebra whose associated scheme is reduced, for which the isomorphism is not true as schemes but true as varieties.

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Jet schemes and singularities

Cited by 99 publications

References 7 publications

Jacobian discrepancies and rational singularities

Jacobian discrepancies and rational singularities

Familles d'espaces de m -jets et d'espaces d'arcs

Singular Support of a Vertex Algebra and the Arc Space of Its Associated Scheme

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