Bertini's theorem for $F$-rational $F$-pure singularities

Alessandro De Stefani; Thomas Polstra; Austyn Simpson

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arXiv · 2509.04433

Bertini's theorem for $F$-rational $F$-pure singularities

Alessandro De Stefani·Thomas Polstra·Austyn Simpson

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Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $X\subseteq\mathbb{P}^n_k$ be a quasi-projective variety that is $F$-rational and $F$-pure. We prove that if $H \subseteq \mathbb{P}^n_k$ is a general hyperplane, then $X \cap H$ is also $F$-rational and $F$-pure. Of related but independent interest, we present a relationship between the characteristic and index of a $\mathbb{Q}$-Gorenstein variety with isolated non-$F$-regular locus which is $F$-pure but not $F$-regular.

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Alessandro De Stefani, Thomas Polstra, Austyn Simpson. 2025-09-04. Bertini's theorem for $F$-rational $F$-pure singularities. https://arxiv.org/abs/2509.04433

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