Local-global principle and integral Tate conjecture for certain varieties

TL;DR

This paper establishes geometric criteria for the integral Tate conjecture and the local-global principle for zero-cycles on certain rationally connected varieties over global function fields, proving these principles for specific classes of surfaces.

Contribution

It introduces a geometric criterion for the integral Tate conjecture and confirms the Brauer-Manin obstruction as the only barrier for zero-cycles on certain rationally connected varieties over global function fields.

Findings

Brauer-Manin obstruction is the only obstruction for zero-cycles on rational surfaces.

Proved the Hasse principle for rational points on degree four del Pezzo surfaces in odd characteristic.

Established equality of coniveau and strong coniveau filtrations on degree 3 homology.

Abstract

We give a geometric criterion to check the validity of the integral Tate conjecture for one-cycles on a smooth projective variety that is separably rationally connected in codimension one, and to check that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on a separably rationally connected variety defined over a global function field. We prove that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on all geometrically rational surfaces defined over a global function field, and to the Hasse principle for rational points on del Pezzo surfaces of degree four defined over a global function field of odd characteristic. Along the way, we also prove some results about the space of one-cycles on a smooth projective variety that is separably rationally connected in codimension one, which leads to the equality of the coniveau filtration and the strong coniveau filtration on degree $3$ homology of such varieties.