Exceptional loci in Lefschetz theory

Abstract

Let $\ddot{I}•:X\to {\mathbb{P}}^n$ be a morphism of varieties. Given a hyperplane $H$ in ${\mathbb{P}}^n$, there is a Gysin map from the compactly supported cohomology of $\ddot{I}{•}^{-1}(H)$ to that of $X$. We give conditions on the degree of the cohomology under which this map is an isomorphism for all but a low-dimensional set of hyperplanes, generalizing results due to Skorobogatov, Benoist, and Poonen-Slavov. Our argument is based on Beilinson's theory of singular supports for étale sheaves.

6 pages