Exceptional loci in Lefschetz theory

Sam Raskin; Geoffrey Smith

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arXiv · 2106.10332

Exceptional loci in Lefschetz theory

Sam Raskin·Geoffrey Smith

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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.

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Sam Raskin, Geoffrey Smith. 2021-06-18. Exceptional loci in Lefschetz theory. https://arxiv.org/abs/2106.10332

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