Étale cohomology, Lefschetz Theorems and Number of Points of Singular Varieties over Finite Fields

Details Étale cohomology, Lefschetz Theorems and Number of Points of Singular Varieties over Finite Fields Étale cohomology, Lefschetz Theorems and Number of Points of Singular Varieties over Finite Fields

2008-08-15

arxiv.org/abs/0808.2169v1

Mosc. Math. J. 2 (2002), 589--631 and Mosc. Math. J. 9 (2009), 431-438.

Mathematics

Algebraic Geometry

42 pages; corrected, revised and updated version of a paper published earlier

Scientific paper

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.

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