Illinois Journal of Mathematics

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

Abstract

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

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

Dates
Revised: 15 March 2017
First available in Project Euclid: 22 September 2017

https://projecteuclid.org/euclid.ijm/1506067286

Digital Object Identifier
doi:10.1215/ijm/1506067286

Mathematical Reviews number (MathSciNet)
MR3705442

Zentralblatt MATH identifier
06790322

Citation

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

References

• I. M. Aberbach and F. Enescu, The structure of $F$-pure rings, Math. Z. 250 (2005), no. 4, 791–806.
• I. M. Aberbach and G. J. Leuschke, The $F$-signature and strong $F$-regularity, Math. Res. Lett. 10 (2003), no. 1, 51–56.
• A. Benito, E. Faber and K. E. Smith, Measuring singularities with Frobenius: The basics, Commutative algebra, Springer, New York, 2013, pp. 57–97.
• B. Bhatt and A. K. Singh, The $F$-pure threshold of a Calabi–Yau hypersurface, Math. Ann. 362 (2015), no. 1–2, 551–567.
• M. Blickle, M. Mustaţǎ and K. E. Smith, Discreteness and rationality of $F$-thresholds, Michigan Math. J. 57 (2008), 43–61. Special volume in honor of Melvin Hochster.
• M. Blickle, M. Mustaţă and K. E. Smith, $F$-thresholds of hypersurfaces, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6549–6565.
• M. Blickle and K. Schwede, $p^{-1}$-linear maps in algebra and geometry, Commutative algebra, Springer, New York, 2013, pp. 123–205.
• M. Blickle, K. Schwede and K. Tucker, $F$-signature of pairs and the asymptotic behavior of Frobenius splittings, Adv. Math. 231 (2012), no. 6, 3232–3258.
• M. Blickle, K. Schwede and K. Tucker, $F$-signature of pairs: Continuity, $p$-fractals and minimal log discrepancies, J. Lond. Math. Soc. (2) 87 (2013), no. 3, 802–818.
• E. Canton, Relating $F$-signature and $F$-splitting ratio of pairs using left-derivatives, 2012; available at \arxivurlarXiv:1206.4293.
• R. Fedder and K. Watanabe, A characterization of $F$-regularity in terms of $F$-purity, Commutative algebra (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 227–245.
• N. Hara and K.-I. Watanabe, $F$-regular and $F$-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom. 11 (2002), no. 2, 363–392.
• N. Hara and K.-I. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), no. 8, 3143–3174 (electronic).
• D. J. Hernández, $F$-purity of hypersurfaces, Math. Res. Lett. 19 (2012), no. 2, 389–401.
• D. J. Hernández, $F$-pure thresholds of binomial hypersurfaces, Proc. Amer. Math. Soc. 142 (2014), no. 7, 2227–2242.
• D. J. Hernández, $F$-invariants of diagonal hypersurfaces, Proc. Amer. Math. Soc. 143 (2015), no. 1, 87–104.
• D. J. Hernández, L. Núñez-Betancourt, E. E. Witt and W. Zhang, $F$-pure thresholds of homogeneous polynomials, Michigan Math. J. 65 (2016), no. 1, 57–87.
• D. J. Hernández and P. Teixeira, $F$-threshold functions: Syzygy gap fractals and the two-variable homogeneous case, J. Symbolic Comput. 80 (2017), no. 2, 451–483.
• M. Hochster and C. Huneke, Tight closure in equal characteristic zero, Preprint, 2006.
• J. Kollár, Singularities of pairs, Algebraic geometry–-Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221–287.
• J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original.
• P. Monsky, The Hilbert–Kunz function, Math. Ann. 263 (1983), no. 1, 43–49.
• P. Monsky and P. Teixeira, $p$-fractals and power series. I. Some 2 variable results, J. Algebra 280 (2004), no. 2, 505–536.
• P. Monsky and P. Teixeira, $p$-fractals and power series. II. Some applications to Hilbert–Kunz theory, J. Algebra 304 (2006), no. 1, 237–255.
• M. Mustaţă and V. Srinivas, Ordinary varieties and the comparison between multiplier ideals and test ideals, Nagoya Math. J. 204 (2011), 125–157; available at http://projecteuclid.org/euclid.nmj/1323107839.
• M. Mustaţǎ, S. Takagi and K. Watanabe, $F$-Thresholds and Bernstein–Sato polynomials, European congress of mathematics, Eur. Math. Soc., Zürich, 2005, pp. 341–364.
• M. Mustaţă and K.-I. Yoshida, Test ideals vs. multiplier ideals, Nagoya Math. J. 193 (2009), 111–128.
• K. Schwede, Generalized test ideals, sharp $F$-purity, and sharp test elements, Math. Res. Lett. 15 (2008), no. 6, 1251–1261.
• K. Schwede, Centers of $F$-purity, Math. Z. 265 (2010), no. 3, 687–714.
• S. Takagi and K. Watanabe, On $F$-pure thresholds, J. Algebra 282 (2004), no. 1, 278–297.
• J. C. Vassilev, Test ideals in quotients of $F$-finite regular local rings, Trans. Amer. Math. Soc. 350 (1998), no. 10, 4041–4051.
• Z. Zhu, Log canonical thresholds in positive characteristic, 2013; available at \arxivurlarXiv:1308.5445.