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 Mustaţȃ-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.