The Uniform Symbolic Topology Property for Diagonally $F$-regular Algebras

TL;DR

This paper introduces diagonally $F$-regular algebras in positive characteristic, establishing the effective validity of the Uniform Symbolic Topology Property (USTP) for a broad class of smooth and certain singular algebras, with implications for complex varieties.

Contribution

It defines diagonally $F$-regular algebras and proves they satisfy USTP, extending the property to include many singular and smooth algebras, with applications to complex algebraic varieties.

Findings

USTP holds effectively for diagonally $F$-regular algebras.

All essentially smooth $k$-algebras are diagonally $F$-regular.

Certain singular algebras, like affine cones over products of projective spaces, satisfy USTP.

Abstract

Let $k$ be a field of positive characteristic. Building on the work of the second named author, we define a new class of $k$-algebras, called diagonally $F$-regular algebras, for which the so-called Uniform Symbolic Topology Property (USTP) holds effectively. We show that this class contains all essentially smooth $k$-algebras. We also show that this class contains certain singular algebras, such as the affine cone over $\mathbb{P}^r_{k} \times \mathbb{P}^s_{k}$, when $k$ is perfect. By reduction to positive characteristic, it follows that USTP holds effectively for the affine cone over $\mathbb{P}^r_{\mathbb{C}} \times \mathbb{P}^s_{\mathbb{C}}$ and more generally for complex varieties of diagonal $F$-regular type.