Exceptional loci in Lefschetz theory

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