An effective Bertini theorem and the number of rational points of a normal complete intersection over a finite field
Antonio Cafure, Guillermo Matera · Acta Arithmetica · 2007
Buenos Aires) 1. Introduction.Let F q be the nite eld of q elements and let F q be the algebraic closure of F q .We denote the n-dimensional projective and ane spaces dened over F q and F q by P n (F q ), P n := P n (F q ), A n (F q ) and A n := A n (F q ) respectively.Let V be an ane or a projective variety dened over F q (an F q -variety for short).Counting or estimating the numberIn [19℄ (see also [15℄), S. Lang and A. Weil establish a prototype estimate on |V (F q )| for absolutely irreducible F q -varieties.They prove that for an absolutely irreducible F q -variety V ⊂ P n of dimension r and degree δ,and C(n, r, δ) is a constant independent of q.We remark that [19℄ does not provide an explicit expression for C(n, r, δ).From the point of view of practical applications, it is usually necessary to provide explicit expressions of the constant C := C(n, r, δ) (see, e.g., [14℄, [16℄, [24℄, [2℄).Further, particular families of varieties for which better estimates hold are also of interest (see, e.g., [32℄, [33℄, [21℄, [25℄).S. Ghorpade and G. Lachaud ([10℄, [9℄) show that one can take C = 9 • 2 s (sd + 3) n+1 in (1), provided that the variety V is dened by s equations of degree at most d.The proof of this result relies on the Grothendieck Lefschetz trace formula and estimates of the Betti numbers of suitable spaces of étale ℓ-adic cohomology.