On the Chow groups of a biquaternion Severi--Brauer variety

TL;DR

This paper offers a new geometric proof that the Chow group of 1-cycles on a Severi--Brauer variety linked to a biquaternion division algebra is torsion-free, differing from previous K-theory-based proofs.

Contribution

It introduces a geometric approach using degenerations of quartic elliptic normal curves, providing an alternative to existing algebraic proofs.

Findings

Chow group of 1-cycles is torsion-free for the given variety

New geometric proof complements existing K-theory methods

Enhances understanding of algebraic cycles on Severi--Brauer varieties

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.