Illinois Journal of Mathematics

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

Eric Canton, Daniel J. Hernández, Karl Schwede, and Emily E. Witt

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We provide a family of examples for which the $F$-pure threshold and the log canonical threshold of a polynomial are different, but such that the characteristic $p$ does not divide the denominator of the $F$-pure threshold (compare with an example of Mustaţă–Takagi–Watanabe). We then study the $F$-signature function in the case that either the $F$-pure threshold and log canonical threshold coincide, or that $p$ 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 that shows that the test ideal can still behave in surprising ways even when the $F$-pure threshold and log canonical threshold coincide.

Article information

Illinois J. Math., Volume 60, Number 3-4 (2016), 669-685.

Received: 11 May 2016
Revised: 15 March 2017
First available in Project Euclid: 22 September 2017

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Zentralblatt MATH identifier

Primary: 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22] 14J17: Singularities [See also 14B05, 14E15] 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]


Canton, Eric; Hernández, Daniel J.; Schwede, Karl; Witt, Emily E. On the behavior of singularities at the $F$-pure threshold. Illinois J. Math. 60 (2016), no. 3-4, 669--685. doi:10.1215/ijm/1506067286.

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