Illinois Journal of Mathematics
- Illinois J. Math.
- Volume 60, Number 3-4 (2016), 669-685.
On the behavior of singularities at the $F$-pure threshold
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.
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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. https://projecteuclid.org/euclid.ijm/1506067286