On the behavior of $F$-thresholds with respect to the fibers of blow-ups and $F$-rationality

TL;DR

This paper investigates the relationship between $F$-thresholds of local rings and their associated graded rings, demonstrating that $F$-rationality of the graded ring implies $F$-rationality of the local ring.

Contribution

It establishes a general inequality relating $F$-thresholds of local and graded rings and shows the descent of $F$-rationality from graded rings to local rings.

Findings

Proves a general inequality for $F$-thresholds

Shows $F$-rationality descends from graded to local rings

Provides new insights into $F$-singularities

Abstract

We establish a general inequality comparing the $F$-thresholds of a local ring and its associated graded ring. As an application, we deduce that the $F$-rationality of the graded ring descends to the local ring.