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Bertini irreducibility theorems over finite fields


Authors: François Charles and Bjorn Poonen
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
Published electronically: October 31, 2014
Erratum: J. Amer. Math. Soc. 32 (2019), 605-607.
MathSciNet review: 3402695
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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.


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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

Keywords: Bertini irreducibility theorem, finite field
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.
Article copyright: © Copyright 2014 American Mathematical Society