Nondiscreteness of $F$-thresholds

TL;DR

This paper constructs examples of two-dimensional normal graded domains where the set of F-thresholds is not discrete, and explores the relationship between F-thresholds in characteristic zero and positive characteristic.

Contribution

It provides the first examples of non-discrete F-threshold sets in two-dimensional normal domains and analyzes their behavior under reduction mod p.

Findings

Examples of non-discrete F-threshold sets in 2D normal domains.

F-thresholds in characteristic p relate to denominators involving p.

Discrepancy between characteristic zero and positive characteristic F-thresholds.

Abstract

We give examples of two dimensional normal ${\mathbb Q}$-Gorenstein graded domains, where the set of $F$-thresholds of the maximal ideal is not discrete, thus answering a question by Musta\c{t}\u{a}-Takagi-Watanabe. We also prove that, for a two dimensional standard graded domain $(R, {\bf m})$ over a field of characteristic $0$, with graded ideal $I$, if $({\bf m}_p, I_p)$ is a reduction mod $p$ of $({\bf m}, I)$ then $c^{I_p}({\bf m}_p) \neq c^I_{\infty}({\bf m})$ implies $c^{I_p}({\bf m}_p)$ has $p$ in the denominator.