A Lefschetz theorem for intersections with 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 étale fundamental groups ${\pi }_1(H\cap U)\to {\pi }_1(U)$ is surjective. We prove a generalization, replacing the hyperplane by a general ${\mathrm{PGL}}_{n+1}$-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 ${\mathrm{PGL}}_{n+1}$-translate of $X$, then the map ${\pi }_1(Y\cap U)\to {\pi }_1(U)$ is surjective.

References in corpus (1)

• Hilbert's irreducibility theorem and the larger sieve