Rational Singularities and q-Birational Morphisms

In this paper, we generalize the notion of rational singularities for any reflexive sheaf of rank Formula: see text, link our notion of rational singularities with the notion of rational singularities in, S. Kovacs, Irrational centers, Pure Appl. Math. Q. 7(4) (2011) 1495–1515, and prove generalizations of standard facts about rational singularities. Moreover, by using a definition of non-rational locus, we introduce the notion of Formula: see text as a dual notion of well-known Serre’s notion of Formula: see text, and prove a theorem about Formula: see text-birational morphisms.