F-Splittings of seminormal monoid algebras

TL;DR

This paper investigates the Frobenius-based invariants of seminormal affine toric varieties in characteristic p, providing combinatorial tools to compute F-splitting ratios and stable ideals, including the test ideal.

Contribution

It introduces a combinatorial approach to describe potential Frobenius splittings of seminormal monoid algebras, enabling explicit calculations of invariants.

Findings

Derived a formula for F-splitting ratio

Computed ideals stable under the Cartier algebra

Provided methods to determine the test ideal

Abstract

We compute a number of invariants of singularities defined via the Frobenius morphism for seminormal affine toric varieties over fields of characteristic p > 0. Our main technical tool is a combinatorial description of the potential splittings of iterates of Frobenius for seminormal monoid algebras. This allows us to give an easy formula for the F-splitting ratio of such rings as well as to compute the ideals stable under the Cartier algebra, including the test ideal.