Permutation modules and Chow motives of geometrically rational surfaces

TL;DR

This paper characterizes when the Chow motive of a geometrically rational surface over a perfect field is zero dimensional, linking it to the Picard group structure and the existence of a zero cycle of degree one.

Contribution

It establishes a precise criterion connecting Chow motives, Picard groups, and zero cycles for geometrically rational surfaces over perfect fields.

Findings

Chow motive of the surface is zero dimensional iff Picard group is a summand of a permutation module and a zero cycle of degree one exists.

Zero cycle of degree one is equivalent to torsion-free zero cycles over the function field.

Provides a criterion linking algebraic cycles, Galois modules, and motives for rational surfaces.

Abstract

We prove that the Chow motive with integral coefficient of a geometrically rational surfaces~$S$ over a perfect field~$k$ is zero dimensional if and only if the Picard group of~$\bar{k}\times_{k}S$, where~$\bar{k}$ is an algebraic closure of~$k$, is a direct summand of a $\Gal (\bar{k}/k)$-permutation module, and~$S$ possesses a zero cycle of degree one. As shown by Colliot-Th\'el\`ene in a letter to the author (which we have reproduced in the appendix) this is in turn equivalent to~$S$ having a zero cycle of degree~$1$ and $\CH_{0}(k(S)\times_{k}S)$ being torsion free.