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arXiv · 1508.05427

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

Eric Canton·Daniel Hernández·Karl Schwede·Emily Witt·Alessandro De Stefani·Jack Jeffries·Zhibek Kadyrsizova·Robert Walker·George Whelan
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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.

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Eric Canton, Daniel Hernández, Karl Schwede, Emily Witt, Alessandro De Stefani, Jack Jeffries, Zhibek Kadyrsizova, Robert Walker, George Whelan. 2016-05-10. On the behavior of singularities at the $F$-pure threshold. https://doi.org/10.1215/ijm%2F1506067286

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