## Bertini irreducibility theorems over finite fields

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- by François Charles and Bjorn Poonen PDF
- J. Amer. Math. Soc.
**29**(2016), 81-94 Request permission

## Abstract:

Given a geometrically irreducible subscheme $X \subseteq \mathbb {P}^n_{\mathbb {F}_q}$ of dimension at least $2$, we prove that the fraction of degree $d$ hypersurfaces $H$ such that $H \cap X$ is geometrically irreducible tends to $1$ as $d \to \infty$. We also prove variants in which $X$ is over an extension of $\mathbb {F}_q$, and in which the immersion $X \to \mathbb {P}^n_{\mathbb {F}_q}$ is replaced by a more general morphism.## References

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## Additional Information

**François Charles**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307; and CNRS, Laboratoire de Mathématiques d’Orsay,Université Paris-Sud, 91405 Orsay CEDEX, France
- Email: francois.charles@math.u-psud.fr
**Bjorn Poonen**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
- MR Author ID: 250625
- ORCID: 0000-0002-8593-2792
- Email: poonen@math.mit.edu
- Received by editor(s): November 27, 2013
- Received by editor(s) in revised form: July 4, 2014, July 7, 2014, and September 15, 2014
- Published electronically: October 31, 2014
- Additional Notes: This material is based upon work supported by the National Science Foundation under grant number DMS-1069236. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
- © Copyright 2014 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**29**(2016), 81-94 - MSC (2010): Primary 14J70; Secondary 14N05
- DOI: https://doi.org/10.1090/S0894-0347-2014-00820-1
- MathSciNet review: 3402695