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On the Tate conjectures for divisors on a fibred variety and on its generic scheme fibre in the case of finite characteristic
T. V. Prokhorova A. G. and N. G. Stoletov Vladimir State University,
87 Gorky str., Vladimir 600000, Russia
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 $\mathbb{F}_q$ 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,
$\operatorname{NS}(V)$ is the Néron–Severi group of classes of divisors on the variety $V$
modulo algebraic equivalence, and assume that the following conditions hold:
$H^1(V\otimes\overline k,\mathcal O_{V\otimes\,\overline k})=0$, $\operatorname{NS}(V)=\operatorname{NS}(V\otimes\overline k)$.
If, for a prime number $l$ not dividing ${\operatorname{Card}}([\operatorname{NS}(V)]_{\operatorname{tors}})$ and different from the characteristic of the field
$\mathbb{F}_q$, the following relation holds $\operatorname{NS}(V)\otimes\mathbb{Q}_l\,\,\widetilde{\rightarrow}\,\,[H^2(V\otimes
k^{\operatorname{s}},\mathbb{Q}_l(1))]^{\operatorname{Gal}( k^{\operatorname{s}}/k)}
$ (in other words, if the Tate conjecture for divisors on $V$ holds), then for
any prime number $l\neq\operatorname{char}(\mathbb{F}_q)$ the Tate conjecture holds for divisors on $X$:
$\operatorname{NS}(X)\otimes\mathbb{Q}_l\,\,\widetilde{\rightarrow}
\,\,[H^2(X\otimes\overline{\mathbb{F}}_q,\mathbb{Q}_l(1))]^{\operatorname{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)}$. In particular, it follows from this result that the Tate conjecture for divisors on an
arithmetic model of a $\operatorname{K}3$ surface over a sufficiently large global field
of finite characteristic different from $2$ holds as well.
Keywords:
Tate conjecture, global field, Brauer group, arithmetic model, $\operatorname{K}3$ surface.
Received: 12.12.2016
Citation:
T. V. Prokhorova, “On the Tate conjectures for divisors on a fibred variety and on its generic scheme fibre in the case of finite characteristic”, Model. Anal. Inform. Sist., 24:2 (2017), 205–214
Linking options:
https://www.mathnet.ru/eng/mais558 https://www.mathnet.ru/eng/mais/v24/i2/p205
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Abstract page: | 157 | Full-text PDF : | 71 | References: | 32 |
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