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

Couvreur, Alain

doi:10.1090/proc/13015

An upper bound on the number of rational points of arbitrary projective varieties over finite fields
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Proc. Amer. Math. Soc. 144 (2016), 3671-3685 Request permission

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 $\mathbf {F}_q$. 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.

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