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

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 ${\mathbb{P}}_k^r\times {\mathbb{P}}_k^s$, when $k$ is perfect. By reduction to positive characteristic, it follows that USTP holds effectively for the affine cone over ${\mathbb{P}}_{\mathbb{C}}^r\times {\mathbb{P}}_{\mathbb{C}}^s$ and more generally for complex varieties of diagonal $F$-regular type.

22 pages. Comments welcome

References in corpus (4)

• Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers
• Uniform Symbolic Topologies via Multinomial Expansions
• Uniform Symbolic Topologies in Normal Toric Rings
• Globalizing F-invariants

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• Symbolic and Ordinary Powers of Ideals in Hibi Rings