Bertini’s theorem for F-rationality (Austyn Simpson, Bates College)

A classical principle in algebraic geometry is that a general hyperplane section of a variety should have singularities no worse than those of the variety itself. Over the complex numbers, this principle is known to hold for rational singularities. Rational singularities are related via reduction mod p to a notion called F-rationality, which is defined in terms of the Frobenius endomorphism. In this talk, I will explore whether F-rationality is inherited by a general hyperplane section of a quasi-projective variety. I will explain why this question has a negative answer in general, but an affirmative one provided that the variety X admits a Frobenius splitting. This is based on joint work with Alessandro De Stefani and Thomas Polstra.