Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Bertini irreducibility theorems over finite fields
HTML articles powered by AMS MathViewer

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
  • 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
Similar Articles
  • Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 14J70, 14N05
  • Retrieve articles in all journals with MSC (2010): 14J70, 14N05
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