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.