Corrigenda and Addenda: Étale Cohomology, Lefschetz The­o­rems and Number of Points of Singular Varieties over Finite Fields

Abstract

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 andétale 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 theétale 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.

Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field

“…Let V ⊂ P n be a variety of pure dimension r and let Σ be its singular locus. Let L ⊂ P n be the linear variety of dimension n − s − 2 defined in (14) and let π and π x be defined as in (15) and (16). Then:…”

“…A more precise variant asserts that, if V ⊂ P n is a projective variety with singular locus of dimension at most s, then a section of V defined by a generic linear space of P n of codimension at least s + 1 is nonsingular (see, e.g., [16,Proposition 1.3]). In this section we consider the existence of nonsingular linear sections of codimension s +2 of a complete intersection having a singular locus of dimension at most s. Identifying each section of this type with a point in the multiprojective space (P n ) s+2 , we show the existence of a hypersurface of (P n ) s+2 containing all the linear subvarieties of codimension s + 2 of (P n ) s+2 which yield singular sections of V .…”

“…Now we can state and prove a sufficient condition for the nonsingularity of the linear section V y . To this end, for a given x ∈ V \ E we consider as in (16) the linear mapping π x : T x V P s+1 defined by Y 0 , . .…”

“…where b r (n, d) is the rth primitive Betti number of any nonsingular complete intersection of P n of dimension r and multidegree d (see, e.g., [16,Theorem 4.1] for an explicit expression of b r (n, d) in terms of n, r and d). This result has been extended by C. Hooley and N. Katz to singular complete intersections.…”

“…In [16] (see also [17]), S. Ghorpade and G. Lachaud have obtained the following explicit version of (2):…”