Bertini theorems for \(F\)-singularities (Q2856118)

The article under review studies the Bertini type theorems for varieties with mild singularities in characteristic \(p>0\). In the case of characteristic zero, it is known that the `mild' singularities remain `mild' after cutting by general hyperplanes. Therefore it is natural to ask the same question in characteristic \(p>0\). In this article, the authors prove that it is true for certains \(F\)-singularities.NEWLINENEWLINERecall that the notion strongly \(F\)-regular (resp. sharply \(F\)-pure singularities) is the moral equivalent of log terminal singularities (resp. log canonical singularities). Here is an important special case of the main theorem in this article: for a projective variety \(X\) over an algebraically closed field \(k\), if \(X\) is \(F\)-pure (respectively, strongly \(F\)-regular), then so is a general hyperplane section of a very ample line bundle.NEWLINENEWLINEThe authors also prove that \(F\)-injective singularities doesnot satify Bertini's theorem. In fact, based on a work of Cumino, Greco and Manaresi [\textit{C. Cumino} et al., Proc. Am. Math. Soc. 106, No. 1, 37--42 (1989; Zbl 0699.14063)], the authors proved that there exists a \(F\)-injective projective surface whose general hyperplane section is not \(F\)-injective.