Exceptional loci in Lefschetz theory

TL;DR

This paper investigates the conditions under which the Gysin map induced by hyperplane sections in Lefschetz theory is an isomorphism, extending previous results using Beilinson's singular support theory.

Contribution

It generalizes existing Lefschetz theorems by establishing new degree conditions for cohomology isomorphisms via singular support techniques.

Findings

Identifies degree conditions ensuring Gysin map is an isomorphism

Extends Lefschetz theory to broader classes of morphisms

Utilizes Beilinson's singular support for étale sheaves

Abstract

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.