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15 January 2015 Semiample Bertini theorems over finite fields
Daniel Erman, Melanie Matchett Wood
Duke Math. J. 164(1): 1-38 (15 January 2015). DOI: 10.1215/00127094-2838327

Abstract

We prove a semiample generalization of Poonen’s Bertini theorem over a finite field that implies the existence of smooth sections for wide new classes of divisors. The probability of smoothness is computed as a product of local probabilities taken over the fibers of the morphism determined by the relevant divisor. We give several applications including a negative answer to a question of Baker and Poonen by constructing a variety (in fact one of each dimension) which provides a counterexample to Bertini over finite fields in arbitrarily large projective spaces. As another application, we determine the probability of smoothness for curves in Hirzebruch surfaces and the distribution of points on those smooth curves.

Citation

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Daniel Erman. Melanie Matchett Wood. "Semiample Bertini theorems over finite fields." Duke Math. J. 164 (1) 1 - 38, 15 January 2015. https://doi.org/10.1215/00127094-2838327

Information

Published: 15 January 2015
First available in Project Euclid: 9 January 2015

zbMATH: 1349.14092
MathSciNet: MR3299101
Digital Object Identifier: 10.1215/00127094-2838327

Subjects:
Primary: 14G15
Secondary: 11G25

Rights: Copyright © 2015 Duke University Press

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Vol.164 • No. 1 • 15 January 2015
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