## Bertini irreducibility theorems over finite fields

HTML articles powered by AMS MathViewer

- by François Charles and Bjorn Poonen
- J. Amer. Math. Soc.
**29**(2016), 81-94 - DOI: https://doi.org/10.1090/S0894-0347-2014-00820-1
- Published electronically: October 31, 2014
- PDF | Request permission

Erratum: J. Amer. Math. Soc.

**32**(2019), 605-607.

## 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

- Lior Bary-Soroker, 2013-09-16. Letter to Bjorn Poonen, available athttp://www.math.tau.ac.il/~barylior/files/let_poonen.pdf.
- Olivier Benoist,
*Le théorème de Bertini en famille*, Bull. Soc. Math. France**139**(2011), no. 4, 555–569 (French, with English and French summaries). MR**2869305**, DOI 10.24033/bsmf.2619 - Brian Conrad,
*Deligne’s notes on Nagata compactifications*, J. Ramanujan Math. Soc.**22**(2007), no. 3, 205–257. MR**2356346** - Alexander Duncan and Zinovy Reichstein,
*Pseudo-reflection groups and essential dimension*, 2014-02-28. Preprint, arXiv:1307.5724v2. - Jean-Pierre Jouanolou,
*Théorèmes de Bertini et applications*, Progress in Mathematics, vol. 42, Birkhäuser Boston, Inc., Boston, MA, 1983 (French). MR**725671** - Serge Lang,
*Sur les séries $L$ d’une variété algébrique*, Bull. Soc. Math. France**84**(1956), 385–407 (French). MR**88777**, DOI 10.24033/bsmf.1477 - Robert Lazarsfeld,
*Positivity in algebraic geometry. I*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR**2095471**, DOI 10.1007/978-3-642-18808-4 - David Mumford,
*Abelian varieties*, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR**0282985** - Konrad Neumann,
*Every finitely generated regular field extension has a stable transcendence base*, Israel J. Math.**104**(1998), 221–260. MR**1622303**, DOI 10.1007/BF02897065 - Ivan Panin,
*On Grothendieck-Serre conjecture concerning principal G-bundles over regular semi-local domains containing a finite field: I*, 2014-06-02. Preprint, arXiv:1406.0241v1. - Ivan Panin,
*On Grothendieck-Serre conjecture concerning principal G-bundles over regular semi-local domains containing a finite field: II*, 2014-06-02. Preprint, arXiv:1406.1129v1. - Ivan Panin,
*Proof of Grothendieck-Serre conjecture on principal G-bundles over regular local rings containing a finite field*, 2014-06-02. Preprint, arXiv:1406.0247v1. - Bjorn Poonen,
*Bertini theorems over finite fields*, Ann. of Math. (2)**160**(2004), no. 3, 1099–1127. MR**2144974**, DOI 10.4007/annals.2004.160.1099 - Bjorn Poonen,
*Smooth hypersurface sections containing a given subscheme over a finite field*, Math. Res. Lett.**15**(2008), no. 2, 265–271. MR**2385639**, DOI 10.4310/MRL.2008.v15.n2.a5

## Bibliographic 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