Bertini irreducibility theorems over finite fields

Abstract

Given a geometrically irreducible subscheme X in P^n over F_q of dimension at least 2, we prove that the fraction of degree d hypersurfaces H such that the intersection of H and X is geometrically irreducible tends to 1 as d tends to infinity. We also prove variants in which X is over an extension of F_q, and in which the immersion of X in P^n is replaced by a more general morphism.

14 pages. In the previous version, in the proof of Lemma 5.1 we forgot to reduce to the case of a normal variety before implicitly using what is called Lemma 3.6 in this version. We fixed this by inserting Lemma 3.6 and adjusting the proof of Lemma 5.1 (and the hypotheses in Lemma 5.2)

References in corpus (3)

• On Grothendieck-Serre conjecture concerning principal G-bundles over regular semi-local domains containing a finite field: I
• On Grothendieck-Serre conjecture concerning principal G-bundles over regular semi-local domains containing a finite field: II
• Proof of Grothendieck--Serre conjecture on principal G-bundles over regular local rings containing a finite field

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• Parameter spaces for algebraic equivalence
• Distinguished models of intermediate Jacobians
• On Grothendieck-Serre conjecture concerning principal G-bundles over regular semi-local domains containing a finite field: I
• Topological reconstruction theorems for varieties
• Holomorphic polylogarithms and Bloch complexes
• Nice triples and a moving lemma for motivic spaces
• Two purity theorems and the Grothendieck--Serre's conjecture concerning principal G-bundles
• Multiplicative independence of modular functions
• The $\theta$-density in Arakelov geometry
• Arithmetic Deformation of Line Bundles