On the Chow groups of certain geometrically rational 5-folds

TL;DR

This paper constructs explicit models for certain quadric fibrations, providing counterexamples to the integral Tate conjecture over large finite fields, and analyzes their etale cohomology to show these counterexamples are non-torsion.

Contribution

It offers an explicit regular model for a specific quadric fibration and demonstrates its use in producing counterexamples to the integral Tate conjecture in odd characteristic.

Findings

Counterexamples to the integral Tate conjecture in odd characteristic

The constructed models are non-torsion in etale cohomology

Provides explicit geometric models for complex fibrations

Abstract

We give an explicit regular model for the quadric fibration studied in Pirutka (2011). As an application we show that this construction furnishes a counterexample for the integral Tate conjecture in any odd characteristic for some sufficiently large finite field. We study the etale cohomology of this regular model, and as a consequence we derive that these counterexamples are not torsion.