Bertini irreducibility theorems over finite fields
Algebraic Geometry
2017-06-08 v3 Number Theory
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.
Cite
@article{arxiv.1311.4960,
title = {Bertini irreducibility theorems over finite fields},
author = {François Charles and Bjorn Poonen},
journal= {arXiv preprint arXiv:1311.4960},
year = {2017}
}
Comments
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)