Le th\'eor\`eme de Bertini en famille

TL;DR

This paper establishes optimal upper bounds on the dimension of hypersurface sets in projective space that fail to intersect an integral variety integrally, and constructs explicit examples of hypersurfaces with integral intersections across families.

Contribution

It provides the first optimal bounds for the dimension of hypersurfaces with non-integral intersections and constructs explicit hypersurfaces with integral intersections in families.

Findings

Upper bounds are proven to be optimal.

Explicit construction of hypersurfaces with integral intersections.

Applications to families of integral projective varieties.

Abstract

We give upper bounds for the dimension of the set of hypersurfaces of $\mathbb{P}^N$ whose intersection with a fixed integral projective variety is not integral. Our upper bounds are optimal. As an application, we construct, when possible, hypersurfaces whose intersections with all the varieties of a family of integral projective varieties are integral. The degree of the hypersurfaces we construct is explicit. ----- On majore la dimension de l'ensemble des hypersurfaces de $\mathbb{P}^N$ dont l'intersection avec une vari\'et\'e projective int\`egre fix\'ee n'est pas int\`egre. Les majorations obtenues sont optimales. Comme application, on construit, quand c'est possible, des hypersurfaces dont les intersections avec toutes les vari\'et\'es d'une famille de vari\'et\'es projectives int\`egres sont int\`egres. Le degr\'e des hypersurfaces construites est explicite.