ETALE COHOMOLOGY, LEFSCHETZ THEOREMS AND NUMBER OF POINTS OF SINGULAR VARIETIES OVER FINITE FIELDS

We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this inequality generalizes also the classical Lang-Weil inequality. Moreover, we prove the Lang-Weil inequality for affine, as well as projective, varieties with an explicit description and a bound for the constant appearing therein. We also prove a conjecture of Lang and Weil concerning the Picard varieties and etale cohomology spaces of projective varieties. The general inequality for complete intersections may be viewed as a more precise version of the estimates given by Hooley and Katz. The proof is primarily based on a suitable generalization of the Weak Lefschetz Theorem to singular varieties together with some Bertini-type arguments and the Grothendieck-Lefschetz Trace Formula. We also describe some auxiliary results concerning the etale cohomology spaces and Betti numbers of projective varieties over finite fields, and a conjecture along with some partial results concerning the number of points of projective algebraic sets over finite fields.