Rational singularities

Abstract

A resolution-free definition of rational singularities is introduced, and it is proved that for a variety admitting a resolution of singularities, so in particular in characteristic zero, this is equivalent to the usual definition. It is also demonstrated that pseudo-rational singularities are rational. The main theorem, a Kempf-type criterion for rational singularities, is new even in characteristic zero contrasting an earlier result of Cutkosky that a similar result with slightly different assumptions could not hold. Several applications are included. In particular, an Elkik-type theorem, proving that Cohen-Macaulay dlt singularities are rational in arbitrary characteristic.

There's likely an error in the proof of the main theorem. Many intermediate results are correct and will be included elsewhere. The main result may be correct under stronger assumptions

Cited by in corpus (5)

• Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic
• Homogeneous contact manifolds and resolutions of Calabi-Yau cones
• Central elements in affine mod $p$ Hecke algebras via perverse ${\mathbb{F}}_p$-sheaves
• The torsion in the cohomology of wild elliptic fibers
• On the local étale fundamental group of KLT threefold singularities