# Le théorème de Bertini en famille

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.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-02353223
Contributor : Olivier Benoist <>
Submitted on : Thursday, November 7, 2019 - 11:23:10 AM
Last modification on : Tuesday, November 12, 2019 - 10:50:04 AM

### Citation

Olivier Benoist. Le théorème de Bertini en famille. Bulletin de la société mathématique de France, Société Mathématique de France, 2011, 139 (4), pp.555-569. ⟨10.24033/bsmf.2619⟩. ⟨hal-02353223⟩

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