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

TL;DR

This paper establishes a universal upper bound on the number of rational points for any projective variety over finite fields, based solely on dimensions and degrees, confirming a conjecture for equidimensional cases.

Contribution

It introduces a general upper bound applicable to all projective varieties over finite fields, including reducible and non-equidimensional ones, and proves a conjecture for equidimensional varieties.

Findings

Derived a bound depending only on dimensions and degrees

Confirmed Ghorpade and Lachaud's conjecture for equidimensional varieties

Applicable to reducible and non-equidimensional varieties

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. This bound depends only on the dimensions and degrees of the irreducible components and holds for very general varieties, even reducible and non equidimensional. As a consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal number of rational points of an equidimensional projective variety.