Multigraded Rings, Diagonal Subalgebras, and Rational Singularities

We study the properties of F-rationality and F-regularity in multigraded rings and their diagonal subalgebras. The main focus is on diagonal subalgebras of bigraded rings: these constitute an interesting class of rings since they arise naturally as homogeneous coordinate rings of blow-ups of projective varieties. LetX be a projective variety over a fieldK, with homogeneous coordinate ringA. Let a ⊂ A be a homogeneous ideal, and V ⊂ X the closed subvariety defined by a. For g an integer, we use ag to denote the K-vector space consisting of homogeneous elements of a of degree g. If g ≫ 0, then ag defines a very ample complete linear system on the blow-up of X along V , and hence K[ag] is a homogeneous coordinate ring for this blow-up. Since the ideals a define the same subvariety V , the rings K[(a)g] are homogeneous coordinate ring for the blow-up provided g ≫ h > 0. Suppose that A is a standard N-graded K-algebra, and consider the N-grading on the Rees algebra A[at], where deg rt = (i, j) for r ∈ Ai. The connection with diagonal subalgebras stems from the fact that if a is generated by elements of degree less than or equal to g, then