On the Chow groups of a biquaternion Severi–Brauer variety

Eoin Mackall

Vol. 321 (2022), No. 2, 359–374

DOI: 10.2140/pjm.2022.321.359

Abstract

We provide an alternative proof that the Chow group of $1$-cycles on a Severi–Brauer variety associated to a biquaternion division algebra is torsion-free. There are three proofs of this result in the literature, all of which are due to Karpenko and rely on a clever use of $K$-theory. The proof that we give here, by contrast, is geometric and uses degenerations of quartic elliptic normal curves.

Keywords

Chow groups, Severi–Brauer variety

Mathematical Subject Classification

Primary: 14C25

Milestones

Received: 21 November 2021

Revised: 10 August 2022

Accepted: 25 October 2022

Published: 21 March 2023