Measuring Singularities with Frobenius: The Basics

In this note, we consider a corollary of the ACC conjecture for F -pure thresholds. Specifically, we show that the F -pure threshold (and more generally, the test ideals) associated to a polynomial with an isolated singularity are locally constant in the m-adic topology of the corresponding local ring. As a by-product of our methods, we also describe a simple algorithm for computing all of the Fjumping numbers and test ideals associated to an arbitrary polynomial over an F -finite field.

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“…The same is true after passing to the localisation S, and it now follows from a well-known consequence 2…”

Section: Remark 4•1 Ifmentioning

confidence: 55%

Local 𝔪-adic constancy of F-pure thresholds and test ideals

Hernández¹,

Núñez-Betancourt²,

Witt³

2017

Math. Proc. Camb. Phil. Soc.

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“…Proof. This is immediate from the definition (1.0.1) and from [1,Prop 3.26] which implies that for any Proof. The first statement is immediate since any monomial of f N is of degree dN .…”

Section: Proofmentioning

confidence: 92%

“…This fact is the culmination of a series of papers, going back to [15], [26], [8], [10], [11], [27], [9], [28]. See the survey [1] for a gentle introduction.…”

Section: Introductionmentioning

confidence: 99%

An elementary computation of the F-pure threshold of an elliptic curve

Pagi¹

2018

Journal of Algebra

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“…Furthermore, if , then . For justification of these facts, and for more references to the theory of log canonical thresholds (and related topics), we refer the reader to the surveys [BL04, EM06, BFS13]. In what follows, we summarize all of the properties of log canonical thresholds that we need in this article.…”

Section: Log Canonical Singularities and (Dense/open) -Pure Typementioning

confidence: 99%

-Purity Versus Log Canonicity For polynomials

Hernández¹

2016

Nagoya Math. J.

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In this article, we consider the conjectured relationship between F -purity and log canonicity for polynomials over C. We associate to a collection M of n monomials a rational polytope P contained in [0, 1] n . Using P and the Newton polyhedron associated to M , we define a non-degeneracy condition under which log canonicity and dense F -pure type are equivalent for all C * -linear combinations of the monomials in M . We also show that log canonicity corresponds to F -purity for very general polynomials. Our methods rely on showing that the F -pure and log canonical threshold agree for infinitely primes, and we accomplish this by comparing these thresholds with the thresholds associated to their monomial ideals.

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Measuring Singularities with Frobenius: The Basics

Cited by 18 publications

References 60 publications

Local 𝔪-adic constancy of F-pure thresholds and test ideals

Local 𝔪-adic constancy of F-pure thresholds and test ideals

An elementary computation of the F-pure threshold of an elliptic curve

-Purity Versus Log Canonicity For polynomials

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