Topics on algebraic varieties in characteristic p

Abstract

In this thesis, which consists of two parts, we study some questions relating to the geometry of algebraic varieties in characteristic $p$, where $p$ is 0 or a prime number. The two parts may be read independently of one other; we give a separate introduction and background section for each one.

In the first part, we prove the Noether--Lefschetz theorem for divisor class groups of normal varieties in arbitrary characteristic. Our proof does not use any Hodge theory, monodromy, or cohomological arguments. The main ingredients are based on techniques and ideas that were available during M. Noether's lifetime. As an addendum to the first part, we give an alternative argument for showing the injectivity statement in Noether--Lefschetz using alterations, following ideas of Ravindra and Srinivas.

The second part is joint work with Joe Waldron and is in positive characteristic $p>0$. We study pathologies that can arise from the failure of Bertini's theorem, in particular geometric non-reducedness, and show a structural result that has applications to Fano varieties including Mori fiber spaces.