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Bertini theorems for F-signature and Hilbert–Kunz multiplicity

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Abstract

We show that Bertini theorems hold for F-signature and Hilbert–Kunz multiplicity. In particular, if \(X \subseteq {\mathbb {P}}^n\) is normal and quasi-projective with F-signature greater than \(\lambda \) (respectively the Hilbert–Kunz multiplicity is less than \(\lambda \)) at all points \(x \in X\), then for a general hyperplane \(H \subseteq {\mathbb {P}}^n\) the F-signature (respectively Hilbert–Kunz multiplicity) of \(X \cap H\) is greater than \(\lambda \) (respectively less than \(\lambda \)) at all points \(x \in X \cap H\).

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Notes

  1. Note that upper semi-continuity of the Hilbert–Kunz multiplicity is also known to hold for a ring which is essentially of finite type over an excellent local ring.

  2. Here we mean that the fibers are normal after any base change, including inseparable ones.

  3. A \(\mathbb {Q}\)-divisor in which no denominators contain p.

  4. Meaning outside a countable union of proper closed subsets of \({{\,\mathrm{{Spec}}\,}}A\).

  5. See for instance [25] (and references therein) where these properties are used systematically in the study of Hilbert–Kunz multiplicities and F-signatures.

References

  1. Aberbach, I.M., Leuschke, G.J.: The \(F\)-signature and strong \(F\)-regularity. Math. Res. Lett. 10(1), 51–56 (2003)

    Article  MathSciNet  Google Scholar 

  2. Blickle, M., Schwede, K., Tucker, K.: \(F\)-signature of pairs and the asymptotic behavior of Frobenius splittings. Adv. Math. 231(6), 3232–3258 (2012)

    Article  MathSciNet  Google Scholar 

  3. Bydlon, A.: Counterexamples to Bertini theorems for test ideals. J. Algebra 501, 150–165 (2018)

    Article  MathSciNet  Google Scholar 

  4. Cumino, C., Greco, S., Manaresi, M.: An axiomatic approach to the second theorem of Bertini. J. Algebra 98(1), 171–182 (1986)

    Article  MathSciNet  Google Scholar 

  5. Datta, R., Simpson, A.: Hilbert–Kunz multiplicity of fibers and Bertini theorems (2020). arXiv:1908.04819

  6. De Stefani, A., Polstra, T., Yao, Y.: Globalizing \(F\)-invariants. Adv. Math. 350, 359–395 (2019)

    Article  MathSciNet  Google Scholar 

  7. Gabber, O.: Notes on some \(t\)-structures. In: Geometric Aspects of Dwork Theory, vol. I, II, pp. 711–734. Walter de Gruyter GmbH & Co. KG, Berlin (2004). https://doi.org/10.1515/9783110198133

  8. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)

  9. Hochster, M., Huneke, C.: \(F\)-regularity, test elements, and smooth base change. Trans. Am. Math. Soc. 346(1), 1–62 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Hochster, M., Huneke, C.: Applications of the existence of big Cohen–Macaulay algebras. Adv. Math. 113(1), 45–117 (1995)

    Article  MathSciNet  Google Scholar 

  11. Huneke, C., Leuschke, G.J.: Two theorems about maximal Cohen–Macaulay modules. Math. Ann. 324(2), 391–404 (2002)

    Article  MathSciNet  Google Scholar 

  12. Hochster, M., Roberts, J.L.: The purity of the Frobenius and local cohomology. Adv. Math. 21(2), 117–172 (1976)

    Article  MathSciNet  Google Scholar 

  13. Hara, N., Watanabe, K.-I.: F-regular and F-pure rings vs. log terminal and log canonical singularities. J. Algebraic Geom. 11(2), 363–392 (2002)

    Article  MathSciNet  Google Scholar 

  14. Kleiman, S.L.: Bertini and his two fundamental theorems. Studies in the history of modern mathematics, III. Rend. Circ. Mat. Palermo 2(Suppl. 55), 9–37 (1998)

    Google Scholar 

  15. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, vol. 134. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1998). With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original

  16. Kunz, E.: Characterizations of regular local rings for characteristic \(p\). Am. J. Math. 91, 772–784 (1969)

    Article  MathSciNet  Google Scholar 

  17. Kunz, E.: On Noetherian rings of characteristic \(p\). Am. J. Math. 98(4), 999–1013 (1976)

    Article  MathSciNet  Google Scholar 

  18. Lazarsfeld, R.: Positivity in Algebraic Geometry. II, vol. 49. 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]. Springer, Berlin (2004). Positivity for vector bundles, and multiplier ideals

  19. Matsumura, H.: Commutative Algebra, vol. 56, 2nd edn. Mathematics Lecture Note Series. Benjamin/Cummings Publishing Co., Inc., Reading, 1980

  20. Matsumura, H.: Commutative Ring Theory, vol. 8, 2nd edn. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1989). Translated from the Japanese by M. Reid

  21. Monsky, P.: The Hilbert–Kunz function. Math. Ann. 263(1), 43–49 (1983)

    Article  MathSciNet  Google Scholar 

  22. Monsky, P.: Hilbert–Kunz functions in a family: point-\(S_4\) quartics. J. Algebra 208(1), 343–358 (1998)

    Article  MathSciNet  Google Scholar 

  23. Polstra, T.: Uniform bounds in F-finite rings and lower semi-continuity of the F-signature. Trans. Am. Math. Soc. 370(5), 3147–3169 (2018)

    Article  MathSciNet  Google Scholar 

  24. Patakfalvi, Z., Schwede, K., Zhang, W.: \(F\)-singularities in families. Algebraic Geom. 5(3), 264–327 (2018)

    Article  MathSciNet  Google Scholar 

  25. Polstra, T., Tucker, K.: \(F\)-signature and Hilbert–Kunz multiplicity: a combined approach and comparison. Algebra Number Theory 12(1), 61–97 (2018)

    Article  MathSciNet  Google Scholar 

  26. Pérez, F., Tucker, K., Yao, Y.: Uniformity in reduction to characteristic p (2020) (in preparation)

  27. Smirnov, I.: Upper semi-continuity of the Hilbert–Kunz multiplicity. Compos. Math. 152(3), 477–488 (2016)

    Article  MathSciNet  Google Scholar 

  28. Schwede, K., Zhang, W.: Bertini theorems for \(F\)-singularities. Proc. Lond. Math. Soc. (3) 107(4), 851–874 (2013)

    Article  MathSciNet  Google Scholar 

  29. The Stacks Project Authors: Stacks project (2020). http://stacks.math.columbia.edu

  30. Tucker, K.: \(F\)-signature exists. Invent. Math. 190(3), 743–765 (2012)

    Article  MathSciNet  Google Scholar 

  31. Watanabe, K., Yoshida, K.: Hilbert–Kunz multiplicity and an inequality between multiplicity and colength. J. Algebra 230(1), 295–317 (2000)

    Article  MathSciNet  Google Scholar 

  32. Yao, Y.: Modules with finite \(F\)-representation type. J. Lond. Math. Soc. (2) 72(1), 53–72 (2005)

    Article  MathSciNet  Google Scholar 

  33. Yao, Y.: Observations on the \(F\)-signature of local rings of characteristic \(p\). J. Algebra 299(1), 198–218 (2006)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank Patrick Graf and Yongwei Yao for stimulating discussions. Work on this project was conducted in CIRM (Luminy) and Oberwolfach. They also thank the anonymous referee for very valuable comments and suggestions.

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Correspondence to Karl Schwede.

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J. Carvajal-Rojas was supported in part by the NSF FRG Grant DMS #1265261/1501115 and NSF CAREER Grant DMS #1252860/1501102 and by the ERC-STG #804334. K. Schwede was supported in part by the NSF FRG Grant DMS #1265261/1501115, NSF CAREER Grant DMS #1252860/1501102 and NSF Grant DMS #1801849. K. Tucker was supported in part by NSF Grants DMS #1602070 and #1707661 and a fellowship from the Sloan foundation.

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Carvajal-Rojas, J., Schwede, K. & Tucker, K. Bertini theorems for F-signature and Hilbert–Kunz multiplicity. Math. Z. 299, 1131–1153 (2021). https://doi.org/10.1007/s00209-021-02712-y

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