On the behavior of singularities at the $F$-pure threshold

TL;DR

This paper explores the relationship between the $F$-pure threshold and the log canonical threshold, providing examples of their differences, analyzing the $F$-signature function, and examining the behavior of test ideals in specific cases.

Contribution

It introduces new examples where the $F$-pure and log canonical thresholds differ without $p$ dividing the denominator, and studies the $F$-signature function's behavior in these contexts.

Findings

$F$-pure threshold and log canonical threshold can differ without $p$ dividing the denominator.

$F$-signature function behaves similarly when these thresholds coincide or share certain properties.

Test ideals can behave unexpectedly even when thresholds coincide.

Abstract

We provide a family of examples where the $F$-pure threshold and the log canonical threshold of a polynomial are different, but where $p$ does not divide the denominator of the $F$-pure threshold (compare with an example of \mustata-Takagi-Watanabe). We then study the $F$-signature function in the case where either the $F$-pure threshold and log canonical threshold coincide or where $p$ does not divide the denominator of the $F$-pure threshold. We show that the $F$-signature function behaves similarly in those two cases. Finally, we include an appendix which shows that the test ideal can still behave in surprising ways even when the $F$-pure threshold and log canonical threshold coincide.