Cohomological characterizations of smoothness and the case of rational surface singularities

Abstract

We establish characterizations of smoothness for a complex affine algebraic variety at a given Cohen-Macaulay point. Our main result treats the case of surfaces with (at most) rational singularities, by a technique that requires the vanishing of suitable Ext modules rather than of sheaf cohomology groups. We also prove results in arbitrary dimension which, in addition to smoothness, detect curves and surfaces as a global feature.