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
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.
Here we mean that the fibers are normal after any base change, including inseparable ones.
A \(\mathbb {Q}\)-divisor in which no denominators contain p.
Meaning outside a countable union of proper closed subsets of \({{\,\mathrm{{Spec}}\,}}A\).
See for instance [25] (and references therein) where these properties are used systematically in the study of Hilbert–Kunz multiplicities and F-signatures.
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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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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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DOI: https://doi.org/10.1007/s00209-021-02712-y