Divisorial Valuations via Arcs

Fernex, Tommaso de; Ein, Lawrence; Ishii, Shihoko

doi:10.2977/prims/1210167333

Cited by 41 publications

(59 citation statements)

References 13 publications

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“…For a full treatment, including proofs, we refer the reader to [ELM04,Voj07,Ish08,dFEI08,Mor09,EM09].…”

Section: Generalities On Arc Spacesmentioning

confidence: 99%

The arc space of the Grassmannian

Docampo¹,

Nigro²

2017

Advances in Mathematics

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Abstract. We study the arc space of the Grassmannian from the point of view of the singularities of Schubert varieties. Our main tool is a decomposition of the arc space of the Grassmannian that resembles the Schubert cell decomposition of the Grassmannian itself. Just as the combinatorics of Schubert cells is controlled by partitions, the combinatorics in the arc space is controlled by plane partitions (sometimes also called 3d partitions). A combination of a geometric analysis of the pieces in the decomposition and a combinatorial analysis of plane partitions leads to invariants of the singularities. As an application we reduce the computation of log canonical thresholds of pairs involving Schubert varieties to an easy linear programming problem. We also study the Nash problem for Schubert varieties, showing that the Nash map is always bijective in this case.

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“…For a full treatment, including proofs, we refer the reader to [ELM04,Voj07,Ish08,dFEI08,Mor09,EM09].…”

Section: Generalities On Arc Spacesmentioning

confidence: 99%

The arc space of the Grassmannian

Docampo¹,

Nigro²

2017

Advances in Mathematics

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“…Following the terminology introduced in [dFEI08], we refer to the restriction O X (1) := O P( ω X ) (1)| X the Mather canonical line bundle of X, and to any…”

Section: Nash Blow-upmentioning

confidence: 99%

“…The above terminology is motivated by the relationship with Mather-Chern classes, see Remark 1.5 of [dFEI08] for a discussion. Note that in [dFEI08] the symbol K X was used to denote the Mather canonical line bundle.…”

Section: Nash Blow-upmentioning

confidence: 99%

“…and K Y /X is the unique f -exceptional divisor linearly equivalent to K Y − f * K X for a choice of a canonical divisor K Y on Y and of a Mather canonical divisor K X (see Proposition 1.7 of [dFEI08]). Furthermore, if f : Y → X factors through the blow-up of j X , and we write…”

Section: Discrepanciesmentioning

confidence: 99%

“…The latter are the main ingredient of the change-of-variables formula in motivic integration [DL99] and are determined by the Nash blow-up of the variety. 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%

See 2 more Smart Citations

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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Singularities, the Space of Arcs and Applications to Birational Geometry

Ishii

2023

Handbook of Geometry and Topology of Singularities IV

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Divisorial Valuations via Arcs

Cited by 41 publications

References 13 publications

The arc space of the Grassmannian

The arc space of the Grassmannian

Jacobian discrepancies and rational singularities

Singularities, the Space of Arcs and Applications to Birational Geometry

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