Section: New Results

A new bound on the number of rational points of arbitrary projective varieties

In [38] , the authors asked for a general upper bound on the number of rational points of a (possibly reducible) equidimensional variety $X\subseteq {𝐏}^n$ of dimension $d$ and degree $\delta$. They conjectured that

where for all positive integer $\ell$, ${\pi }_{\ell }$ is defined as the number of rational points of the projective space of dimension $\ell$ over ${𝐅}_q$. That is to say, ${\pi }_{\ell }=\frac{q^{\ell +1}-1}{q-1}.$

By combining algebraic geometric methods with a combinatorial method of double counting, A. Couvreur proved this conjecture [32] and got a more general upper bound on the number of rational points of arbitrary varieties (possibly non-equidimensional). In addition, he proved that (1 ) is sharp by providing examples of varieties reaching this bound.