An upper bound on the number of rational points of arbitrary projective varieties over finite fields
Alain Couvreur · Proceedings of the American Mathematical Society · 2015
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 \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.