# Jacobian discrepancies and rational singularities

Tommaso de Fernex; Roi Docampo

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 1, page 165-199
- ISSN: 1435-9855

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topde Fernex, Tommaso, and Docampo, Roi. "Jacobian discrepancies and rational singularities." Journal of the European Mathematical Society 016.1 (2014): 165-199. <http://eudml.org/doc/277541>.

@article{deFernex2014,

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 $\textit \{Jacobian discrepancy\}$, 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 $\mathbb \{Q\}$-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.},

author = {de Fernex, Tommaso, Docampo, Roi},

journal = {Journal of the European Mathematical Society},

keywords = {discrepancy; Jacobian; adjunction; Nash blow-up; jet scheme; multiplier ideal; rational singularity; Du Bois singularity; discrepancy; Jacobian; adjunction; Nash blow-up; jet scheme; multiplier ideal; rational singularity; Du Bois singularity},

language = {eng},

number = {1},

pages = {165-199},

publisher = {European Mathematical Society Publishing House},

title = {Jacobian discrepancies and rational singularities},

url = {http://eudml.org/doc/277541},

volume = {016},

year = {2014},

}

TY - JOUR

AU - de Fernex, Tommaso

AU - Docampo, Roi

TI - Jacobian discrepancies and rational singularities

JO - Journal of the European Mathematical Society

PY - 2014

PB - European Mathematical Society Publishing House

VL - 016

IS - 1

SP - 165

EP - 199

AB - 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 $\textit {Jacobian discrepancy}$, 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 $\mathbb {Q}$-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.

LA - eng

KW - discrepancy; Jacobian; adjunction; Nash blow-up; jet scheme; multiplier ideal; rational singularity; Du Bois singularity; discrepancy; Jacobian; adjunction; Nash blow-up; jet scheme; multiplier ideal; rational singularity; Du Bois singularity

UR - http://eudml.org/doc/277541

ER -

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