Bertini theorems for F-singularities

Abstract

We prove that strongly F-regular and F-pure singularities satisfy Bertini-type theorems (including in the context of pairs) by building upon a framework of Cumino, Greco and Manaresi (compare with the work of Jouanolou and Spreafico). We also prove that F-injective singularities fail to satisfy even the most basic Bertini-type results.

Typos corrected and other minor changes. To appear in the Proceedings of the London Mathematical Society

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• The dualizing complex of $F$-injective and Du Bois singularities
• F-thresholds of graded rings
• Semiample perturbations for log canonical varieties over an F-finite field containing an infinite perfect field
• F-injectivity and Buchsbaum singularities
• Upper bound of multiplicity of F-rational rings and F-pure rings
• Centers of perfectoid purity
• Rational points on 3-folds with nef anti-canonical class over finite fields
• On Rational Connectedness of Globally F-Regular Threefolds
• Multiplicity bounds in prime characteristic
• Canonical singularities of dimension three in characteristic 2 which do not follow Reid's rules