Multigraded rings, diagonal subalgebras, and rational singularities

TL;DR

This paper investigates the properties of F-rationality and F-regularity in multigraded rings and their diagonal subalgebras, revealing new examples of rings with specific singularity characteristics and divisor class group properties.

Contribution

It introduces new results on the F-singularity properties of diagonal subalgebras of bigraded rings, including examples with rational singularities that are not F-regular.

Findings

Existence of bigraded hypersurfaces with invariant rings having rational singularities but not F-regular.

Construction of rings with finitely generated but non-discrete divisor class groups.

Analysis of F-rationality and F-regularity in the context of multigraded and diagonal subalgebras.

Abstract

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. As a consequence of some of the results obtained here, it is shown that there exist standard bigraded hypersurfaces whose rings of invariants under torus actions have rational singularities, but are not of F-regular type. Another application is the construction of families of rings with divisor class groups that are finitely generated, but not discrete in the sense of Danilov.