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

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.

Typos corrected, other improvements to the exposition

References in corpus (3)

• F-thresholds and Bernstein-Sato polynomials
• Log canonical thresholds in positive characteristic
• Relating F-Signature and F-Splitting Ratio of Pairs Using Left-Derivatives

Cited by in corpus (3)

• Seshadri Constants and Fujita's Conjecture via Positive Characteristic Methods
• Inversion of adjunction for $F$-signature
• $F$-thresholds $c^I(\mathbf{m})$ for projective curves