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Ascending chain condition for $F$-pure thresholds on a fixed strongly $F$-regular germ

Part of: Algebraic groups General commutative ring theory Local theory

Published online by Cambridge University Press:  28 May 2019

Kenta Sato
Kenta Sato*
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan email ktsato@ms.u-tokyo.ac.jp
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Abstract

In this paper, we prove that the set of all $F$-pure thresholds on a fixed germ of a strongly $F$-regular pair satisfies the ascending chain condition. As a corollary, we verify the ascending chain condition for the set of all $F$-pure thresholds on smooth varieties or, more generally, on varieties with tame quotient singularities, which is an affirmative answer to a conjecture given by Blickle, Mustaţǎ and Smith.

Keywords

ascending chain conditionF-jumping numberF-pure thresholdtame quotient singularitiesnon-standard extension

MSC classification

Primary: 14B05: Singularities
Secondary: 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure 14L30: Group actions on varieties or schemes (quotients)

Information

Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 6 , June 2019 , pp. 1194 - 1223
Copyright
© The Author 2019 

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References

André, M., Localisation de la lissité formelle , Manuscripta Math. 13 (1974), 297–307.Google Scholar
Aoyama, Y., Some basic results on canonical modules , J. Math. Kyoto Univ. 23 (1983), 85–94.Google Scholar
Benson, D. J., Polynomial invariants of finite groups, London Mathematical Society Lecture Note Series, vol. 190 (Cambridge University Press, Cambridge, 1993).Google Scholar
Blickle, M., Mustaţă, M. and Smith, K. E., Discreteness and rationality of F-thresholds , Michigan Math. J. 57 (2008), 43–61.Google Scholar
Blickle, M., Mustaţă, M. and Smith, K. E., F-thresholds of hypersurfaces , Trans. Amer. Math. Soc. 361 (2009), 6549–6565.Google Scholar
Blickle, M., Schwede, K., Takagi, S. and Zhang, W., Discreteness and rationality of F-jumping numbers on singular varieties , Math. Ann. 347 (2010), 917–949.Google Scholar
de Fernex, T., Ein, L. and Mustaţă, M., Shokurov’s ACC conjecture for log canonical thresholds on smooth varieties , Duke Math. J. 152 (2010), 93–114.Google Scholar
de Fernex, T., Ein, L. and Mustaţă, M., Log canonical thresholds on varieties with bounded singularities , in Classification of algebraic varieties, EMS Series of Congress Reports, vol. 3 (European Mathematical Society, Zürich, 2011), 221–257.Google Scholar
Goldblatt, R., Lectures on the hyperreals , in An introduction to nonstandard analysis, Graduate Texts in Mathematics, vol. 188 (Springer, New York, NY, 1998).Google Scholar
Hacon, C. D., McKernan, J. and Xu, C., ACC for log canonical thresholds , Ann. of Math. (2) 180 (2014), 523–571.Google Scholar
Hara, N. and Takagi, S., On a generalization of test ideals , Nagoya Math. J. 175 (2004), 59–74.Google Scholar
Hernández, D. J., Núñez-Betancourt, L. and Witt, E. E., Local m-adic constancy of F-pure thresholds and test ideals , Math. Proc. Cambridge Philos. Soc. 64 (2017), 1–11.Google Scholar
Hernández, D. J., Núñez-Betancourt, L., Witt, E. E. and Zhang, W., F-pure thresholds of homogeneous polynomials , Michigan Math. J. 65 (2016), 57–87.Google Scholar
Kunz, E., On Noetherian rings of characteristic p , Amer. J. Math. 98 (1976), 999–1013.Google Scholar
Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, second edition (Cambridge University Press, Cambridge, 1989).Google Scholar
Pérez, F., On the constancy regions for mixed test ideals , J. Algebra 396 (2013), 82–97.Google Scholar
Sato, K. and Takagi, S., General hyperplane sections of threefolds in positive characteristic , J. Inst. Math. Jussieu, doi:10.1017/S1474748018000166.Google Scholar
Schoutens, H., The use of ultraproducts in commutative algebra, Lecture Notes in Mathematics, vol. 1999 (Springer, Berlin, 2010).Google Scholar
Schwede, K., F-adjunction , Algebra Number Theory 3 (2009), 907–950.Google Scholar
Schwede, K., Centers of F-purity , Math. Z. 265 (2010), 687–714.Google Scholar
Schwede, K. and Tucker, K., Test ideals of non-principal ideals: computations, jumping numbers, alterations and division theorems , J. Math. Pure Appl. 102 (2014), 891–929.Google Scholar
Shokurov, V., Three-dimensional log perestroikas , Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), 105–203. With an appendix by Yujiro Kawamata.Google Scholar
The Stacks Project Authors, Stacks Project, https://stacks.math.columbia.edu(2018).Google Scholar
Takagi, S., Formulas for multiplier ideals on singular varieties , Amer. J. Math. 128 (2006), 1345–1362.Google Scholar