A Lefschetz theorem for intersections with projective varieties

TL;DR

This paper generalizes the Lefschetz hyperplane theorem by showing that for a normal quasi-projective variety, a general projective translate of a given variety induces a surjective map on étale fundamental groups, extending classical results.

Contribution

It extends the classical Lefschetz theorem by replacing hyperplanes with general projective translates of arbitrary varieties, broadening the scope of fundamental group surjectivity results.

Findings

Surjective map on étale fundamental groups for general PGL-translates

Generalization from hyperplanes to arbitrary projective translates

Applicable to normal quasi-projective varieties

Abstract

One version of the classical Lefschetz hyperplane theorem states that for $U \subset \mathbb P^n$ a smooth quasi-projective variety of dimension at least $2$, and $H \cap U$ a general hyperplane section, the resulting map on \'etale fundamental groups $\pi_1(H \cap U) \rightarrow \pi_1(U)$ is surjective. We prove a generalization, replacing the hyperplane by a general $\operatorname{PGL}_{n+1}$-translate of an arbitrary projective variety: If $U \subset \mathbb P^n$ is a normal quasi-projective variety, $X$ is a geometrically irreducible projective variety of dimension at least $n + 1 - \dim U$, and $Y$ is a general $\operatorname{PGL}_{n+1}$-translate of $X$, then the map $\pi_1(Y \cap U) \rightarrow \pi_1(U)$ is surjective.