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

Abstract

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.

Ascending chain condition for -pure thresholds on a fixed strongly -regular germ

“…This problem has been considered by several authors [BMS09, HNWZ16], and [HNW17]. We give an affirmative answer to this conjecture.…”

Continuity of Hilbert–kunz Multiplicity and F-Signature

“…As the F-pure threshold function is continuous with respect to the m-adic topology, it follows that if the ACC conjecture holds then for each f ∈ R there exists N ∈ N such that for each ∈ m N , fpt(f + ) fpt(f ). Hernández, Núñez-Betancourt, and Witt [HNnW17] recently investigated this particular implication of the ACC conjecture. Their techniques establish that if the Jacobian ideal of an element f ∈ R is m-primary, then fpt(f ) = fpt(g) for all g sufficiently close to f .…”

Continuity of Hilbert-Kunz multiplicity and F-signature

“…As the F-pure threshold function is continuous with respect to the m-adic topology, it follows that if the ACC conjecture holds then for each f ∈ R there exists N ∈ N such that for each ǫ ∈ m N , fpt(f + ǫ) fpt(f ). Hernández, Núñez-Betancourt, and Witt ( [HNnW17]) recently investigated this particular implication of the ACC conjecture. Their techniques establish that if the Jacobian ideal of an element f ∈ R is m-primary, then fpt(f ) = fpt(g) for all g sufficiently close to f .…”