An estimate for 𝐹-jumping numbers via the roots of the Bernstein-Sato polynomial

Mustaţă, Mircea

doi:10.1090/proc/16438

An estimate for $F$-jumping numbers via the roots of the Bernstein-Sato polynomial
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by Mircea Mustaţă;

Proc. Amer. Math. Soc. 151 (2023), 3751-3761

DOI: https://doi.org/10.1090/proc/16438

Published electronically: May 19, 2023

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Abstract:

Given a smooth, irreducible complex algebraic variety $X$ and a nonzero regular function $f$ on $X$, we give an effective estimate for the difference between the jumping numbers of $f$ and the $F$-jumping numbers of a reduction $f_p$ of $f$ to characteristic $p\gg 0$, in terms of the roots of the Bernstein-Sato polynomial $b_f$ of $f$. In particular, we get uniform estimates only depending on the dimension of $X$. As an application, we show that if $b_f$ has no roots of the form $-lct(f)-n$, with $n$ a positive integer, then the $F$-pure threshold of $f_p$ is equal to the log canonical threshold of $f$ for $p\gg 0$ with $(p-1)lct(f)\in {\mathbf Z}$.

References

I. N. Bernšteĭn, Analytic continuation of generalized functions with respect to a parameter, Funkcional. Anal. i Priložen. 6 (1972), no. 4, 26–40. MR 320735
Bhargav Bhatt, Karl Schwede, and Shunsuke Takagi, The weak ordinarity conjecture and $F$-singularities, Higher dimensional algebraic geometry—in honour of Professor Yujiro Kawamata’s sixtieth birthday, Adv. Stud. Pure Math., vol. 74, Math. Soc. Japan, Tokyo, 2017, pp. 11–39. MR 3791207, DOI 10.2969/aspm/07410011
Manuel Blickle, Mircea Mustaţǎ, and Karen E. Smith, Discreteness and rationality of $F$-thresholds, Michigan Math. J. 57 (2008), 43–61. Special volume in honor of Melvin Hochster. MR 2492440, DOI 10.1307/mmj/1220879396
Manuel Blickle, Mircea Mustaţă, and Karen E. Smith, $F$-thresholds of hypersurfaces, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6549–6565. MR 2538604, DOI 10.1090/S0002-9947-09-04719-9
Nero Budur, Mircea Mustaţǎ, and Morihiko Saito, Bernstein-Sato polynomials of arbitrary varieties, Compos. Math. 142 (2006), no. 3, 779–797. MR 2231202, DOI 10.1112/S0010437X06002193
Christopher Dodd, Differential operators, gauges, and mixed Hodge modules, arXiv:2210.12611 (2022).
Nobuo Hara and Shunsuke Takagi, On a generalization of test ideals, Nagoya Math. J. 175 (2004), 59–74. MR 2085311, DOI 10.1017/S0027763000008904
Nobuo Hara and Ken-Ichi Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), no. 8, 3143–3174. MR 1974679, DOI 10.1090/S0002-9947-03-03285-9
Daniel J. Hernández, $F$-invariants of diagonal hypersurfaces, Proc. Amer. Math. Soc. 143 (2015), no. 1, 87–104. MR 3272734, DOI 10.1090/S0002-9939-2014-12260-X
Masaki Kashiwara, $B$-functions and holonomic systems. Rationality of roots of $B$-functions, Invent. Math. 38 (1976/77), no. 1, 33–53. MR 430304, DOI 10.1007/BF01390168
Masaki Kashiwara, $D$-modules and microlocal calculus, Translations of Mathematical Monographs, vol. 217, American Mathematical Society, Providence, RI, 2003. Translated from the 2000 Japanese original by Mutsumi Saito; Iwanami Series in Modern Mathematics. MR 1943036, DOI 10.1090/mmono/217
János Kollár, Singularities of pairs, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221–287. MR 1492525, DOI 10.1090/pspum/062.1/1492525
Ernst Kunz, Characterizations of regular local rings of characteristic $p$, Amer. J. Math. 91 (1969), 772–784. MR 252389, DOI 10.2307/2373351
Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472, DOI 10.1007/978-3-642-18808-4
Mircea Mustaţă, Ordinary varieties and the comparison between multiplier ideals and test ideals II, Proc. Amer. Math. Soc. 140 (2012), no. 3, 805–810. MR 2869065, DOI 10.1090/S0002-9939-2011-11240-1
Mircea Mustaţă and Vasudevan Srinivas, Ordinary varieties and the comparison between multiplier ideals and test ideals, Nagoya Math. J. 204 (2011), 125–157. MR 2863367, DOI 10.1215/00277630-1431849
Mircea Mustaţǎ, Shunsuke Takagi, and Kei-ichi Watanabe, F-thresholds and Bernstein-Sato polynomials, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 341–364. MR 2185754
Mircea Mustaţă and Wenliang Zhang, Estimates for $F$-jumping numbers and bounds for Hartshorne-Speiser-Lyubeznik numbers, Nagoya Math. J. 210 (2013), 133–160. MR 3079277, DOI 10.1215/00277630-2077035
Morihiko Saito, On microlocal $b$-function, Bull. Soc. Math. France 122 (1994), no. 2, 163–184 (English, with English and French summaries). MR 1273899, DOI 10.24033/bsmf.2227
Shunsuke Takagi and Kei-ichi Watanabe, On F-pure thresholds, J. Algebra 282 (2004), no. 1, 278–297. MR 2097584, DOI 10.1016/j.jalgebra.2004.07.011
T. Yano, On the theory of $b$-functions, Publ. Res. Inst. Math. Sci. 14 (1978), no. 1, 111–202., DOI 10.2977/prims/1195189282