Chow Groups of Quadrics in Characteristic Two

TL;DR

This paper investigates the torsion subgroup of Chow groups of quadrics over fields of characteristic two, establishing bounds and conditions for nontrivial torsion in specific codimensions, extending known results from other characteristics.

Contribution

It extends Karpenko's results on Chow groups of quadrics to characteristic two, providing new bounds and exact conditions for torsion in codimension 2 and 3.

Findings

Torsion subgroup in codimension 2 has at most two elements.

Precisely characterizes when torsion in codimension 2 is nontrivial.

No torsion in codimension 3 if dimension of X is at least 11.

Abstract

Let $X$ be a smooth projective quadric defined over a field of characteristic 2. We prove that in the Chow group of codimension 2 or 3 of $X$ the torsion subgroup has at most two elements. In codimension 2, we determine precisely when this torsion subgroup is nontrivial. In codimension 3, we show that there is no torsion if $\dim X\ge 11$. This extends the analogous results in characteristic different from 2, obtained by Karpenko in the nineteen-nineties.