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

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14 Citations (Scopus)

Résumé

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.

langue originale	Anglais
Pages (de - à)	3671-3685
Nombre de pages	15
journal	Proceedings of the American Mathematical Society
Volume	144
Numéro de publication	9
Les DOIs	https://doi.org/10.1090/proc/13015
état	Publié - 1 janv. 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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