An upper bound on the number of rational points of arbitrary projective varieties over finite fields

Alain Couvreur

doi:10.1090/proc/13015

An upper bound on the number of rational points of arbitrary projective varieties over finite fields

Alain Couvreur

Research output: Contribution to journal › Article › peer-review

Abstract

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field F_q. This bound depends only on the dimensions and degrees of the irreducible components and holds for very general projective varieties, even reducible and nonequidimensional. As a consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal number of rational points of an equidimensional projective variety.

Original language	English
Pages (from-to)	3671-3685
Number of pages	15
Journal	Proceedings of the American Mathematical Society
Volume	144
Issue number	9
DOIs	https://doi.org/10.1090/proc/13015
Publication status	Published - 1 Jan 2016

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abstract = "We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field Fq. This bound depends only on the dimensions and degrees of the irreducible components and holds for very general projective varieties, even reducible and nonequidimensional. As a consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal number of rational points of an equidimensional projective variety.",

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AB - We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field Fq. This bound depends only on the dimensions and degrees of the irreducible components and holds for very general projective varieties, even reducible and nonequidimensional. As a consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal number of rational points of an equidimensional projective variety.

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An upper bound on the number of rational points of arbitrary projective varieties over finite fields

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