On the Tate Conjectures for Divisors on a Fibred Variety and on its Generic Scheme Fibre in the Case of Finite Characteristic

Abstract

We investigate interrelations between the Tate conjecture for divisors on a fibred variety over a finite field and the Tate conjecture for divisors on the generic scheme fibre under the condition that the generic scheme fibre has zero irregularity. Let \(\pi:X\to C\) be a surjective morphism of smooth projective varieties over a finite field \(F_q\) of characteristic \(p\), \(C\) is a curve and the generic scheme fibre of \(\pi\) is a smooth variety \(V\) over the field \(k=\kappa(C)\) of rational functions of the curve \(C\), \(\overline k\) is an algebraic closure of the field \(k\), \(k^s\) is its separable closure, \(NS(V)\) is the N\'eron - Severi group of classes of divisors on the variety \(V\) modulo algebraic equivalence, and assume that the following conditions hold: \(H^1(V\otimes\overline k,\mathcal O_{V\otimes\,\overline k})=0,\) \(NS(V)=NS(V\otimes\overline k).\) If, for a prime number \(l\) not dividing \({Card}([NS(V)]_{tors})\) and different from the characteristic of the field \(F_q\), the following relation holds \(NS(V)\otimes\Bbb Q_l\,\,\widetilde{\rightarrow}\,\,[H^2(V\otimes k^{sep},Q_l(1))]^{Gal( k^{sep}/k)} \) \((\)in other words, if the Tate conjecture for divisors on \(V\) holds\()\), then for any prime number \(l\neq charr(F_q)\) the Tate conjecture holds for divisors on \(X\): \(NS(X)\otimes Q_l\,\,\widetilde{\rightarrow} \,\,[H^2(X\otimes\overline F_q,Q_l(1))]^{Gal(\overline F_q/F_q)}.\) In  particular, it follows from this result that the Tate conjecture for divisors on an arithmetic model of a \(K3\) surface over a sufficiently large global field of finite characteristic different from 2 holds as well.