Exceptional loci in Lefschetz theory

Let phi:X -> Pn$\phi :X\rightarrow \mathbb {P}<^>n$ be a morphism of varieties. Given a hyperplane H$H$ in Pn$\mathbb {P}<^>n$, there is a Gysin map from the compactly supported cohomology of phi-1(H)$\phi <^>{-1}(H)$ to that of X$X$. 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 etale sheaves.