An upper bound on the number of rational points of arbitrary projective varieties over finite fields (Q2814392)

The paper under review is concerned with a bound of the number of rational points of a projective variety defined over a finite field. Let \(\mathbb F_{q}\) be a finite field with \(q\)-elements and \(X \subset \mathbb P^{n}\) be a projective variety defined over \(\mathbb F_{q}\). Let \(X =X_{1} \cup \cdots \cup X_{r}\) be the decomposition of X where \(X_{i}\) is an irreducible component of \(X\) of dimension \(d_{i}\) and degree \(\delta _{i}\). We set \(\pi _{m} = | P^{m}(\mathbb F_{q})| = \frac{q^{m+1}-1}{q-1}\) and \(\pi_{m}=0\) if \(m<0\). The main result of this paper is following.NEWLINENEWLINETheorem 3.1. \(|X(\mathbb F_{q})| \leq \sum _{i=1}^{r} \delta (\pi _{d_{i} - \pi _{2d_{i}-n}}) + \pi _{2D_{X} -n}\) where \(D_{X} = \max(d_{1} , \cdots , d_{r})\).NEWLINENEWLINEThe complete intersection case of this theorem answers to the conjecture of \textit{S. R. Ghorpade} and \textit{G. Lachaud} [Mosc. Math. J. 2, No. 3, 589--631 (2002; Zbl 1101.14017)].