We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Jacobian discrepancies and rational singularities.
- Authors
de Fernex, Tommaso; Docampo, Roi
- 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 Kov'acs, Schwede and Smith, and of Koll'ar and Kov'acs on rational and Du Bois singularities.
- Subjects
MATHEMATICAL singularities; JACOBIAN determinants; MORPHISMS (Mathematics); SMOOTHNESS of functions; COHEN-Macaulay modules
- Publication
Journal of the European Mathematical Society (EMS Publishing), 2014, Vol 16, Issue 1, p165
- ISSN
1435-9855
- Publication type
Article
- DOI
10.4171/JEMS/430