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arXiv · 0911.1118

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

Olivier Benoist
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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.

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Olivier Benoist. 2009-11-05. Le th\'eor\`eme de Bertini en famille. https://doi.org/10.24033/bsmf.2619

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