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

Ghazaleh FakhriVaighan

TL;DR

The paper proves a general inequality comparing $F$-thresholds of a local ring and its associated graded ring: for an $ rak m$-primary ideal $ rak b$ and $I= ext{gr}_{ rak m}( rak b)$, $A= ext{gr}_{ rak m}(R)$, one has $c^{ rak b}( rak m) \u2264 c^{I}( rak n)$. This degeneration-to-initial-form framework transfers Frobenius numerical data from the graded fiber back to the local ring, yielding corollaries such as $c^{ rak m}( rak m) ^{ rak n}( rak n)$. A key application is that if $ ext{gr}_{ rak m}(R)$ is $F$-rational, then $R$ is $F$-rational, providing descent of $F$-rationality from the graded fiber to the local ring. The results connect Frobenius asymptotics, tight closure, and reductions via superficial elements and Sally’s machine, and complement known flat-descent principles for $F$-rationality. These findings enhance our understanding of how singularity properties behave under blow-ups and degenerations.

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.

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

TL;DR

The paper proves a general inequality comparing -thresholds of a local ring and its associated graded ring: for an -primary ideal and , , one has . This degeneration-to-initial-form framework transfers Frobenius numerical data from the graded fiber back to the local ring, yielding corollaries such as . A key application is that if is -rational, then is -rational, providing descent of -rationality from the graded fiber to the local ring. The results connect Frobenius asymptotics, tight closure, and reductions via superficial elements and Sally’s machine, and complement known flat-descent principles for -rationality. These findings enhance our understanding of how singularity properties behave under blow-ups and degenerations.

Abstract

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

Paper Structure

This paper contains 4 sections, 18 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Behavior of $F$-thresholds with Respect to Fiber of Blow-ups
  4. Behavior of $F$-rationality with Respect to Fiber of Blow-ups

Key Result

Theorem A

Let $(R, \mathfrak{m}, k)$ be a local $F$-finite ring of prime characteristic $p>0$, and $\mathfrak{b}\subseteq R$ be an $\mathfrak{m}$-primary ideal. Then, where $\mathfrak{n}$ is the maximal homogeneous ideal of the associated graded ring $\mathrm{gr}_{\mathfrak{m}}(R)$ and $I=\mathrm{gr}_\mathfrak{m}(\mathfrak{b})$.

Theorems & Definitions (38)

  • Theorem A: \ref{['20']}
  • Corollary B: \ref{['21']}
  • Definition 1
  • Definition 2: AE
  • Definition 3: TW
  • Definition 4: HMTWDSNBP
  • Theorem 2.1: MTW
  • Definition 5
  • Remark 1
  • Remark 2
  • ...and 28 more