Geodesic
Bertini irreducibility theorems over finite fields
Charles, François 1   ; Poonen, Bjorn 2
1 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
2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
Journal of the American Mathematical Society, Tome 29 (2016) no. 1, pp. 81-94 Cet article a éte moissonné depuis la source American Mathematical Society

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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.
DOI : 10.1090/S0894-0347-2014-00820-1

Charles, François  1   ; Poonen, Bjorn  2

1 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
2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307 
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Charles, François; Poonen, Bjorn. Bertini irreducibility theorems over finite fields. Journal of the American Mathematical Society, Tome 29 (2016) no. 1, pp. 81-94. doi: 10.1090/S0894-0347-2014-00820-1

[1] Bary-Soroker, Lior 2013-09-16

[2] Benoist, Olivier Le théorème de Bertini en famille Bull. Soc. Math. France 2011 555 569

[3] Conrad, Brian Deligne’s notes on Nagata compactifications J. Ramanujan Math. Soc. 2007 205 257

[4] Duncan, Alexander, Reichstein, Zinovy Pseudo-reflection groups and essential dimension 2014-02-28

[5] Jouanolou, Jean-Pierre Théorèmes de Bertini et applications 1983

[6] Lang, Serge Sur les séries 𝐿 d’une variété algébrique Bull. Soc. Math. France 1956 385 407

[7] Lazarsfeld, Robert Positivity in algebraic geometry. I 2004

[8] Mumford, David Abelian varieties 1970

[9] Neumann, Konrad Every finitely generated regular field extension has a stable transcendence base Israel J. Math. 1998 221 260

[10] Panin, Ivan On Grothendieck-Serre conjecture concerning principal G-bundles over regular semi-local domains containing a finite field: I 2014-06-02

[11] Panin, Ivan On Grothendieck-Serre conjecture concerning principal G-bundles over regular semi-local domains containing a finite field: II 2014-06-02

[12] Panin, Ivan Proof of Grothendieck-Serre conjecture on principal G-bundles over regular local rings containing a finite field 2014-06-02

[13] Poonen, Bjorn Bertini theorems over finite fields Ann. of Math. (2) 2004 1099 1127

[14] Poonen, Bjorn Smooth hypersurface sections containing a given subscheme over a finite field Math. Res. Lett. 2008 265 271

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