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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
MathSciNet review: 3402695
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

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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
Email: poonen@math.mit.edu

DOI: https://doi.org/10.1090/S0894-0347-2014-00820-1
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

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