Projective homogeneous varieties birational to quadrics

TL;DR

This paper constructs an explicit birational map between quadrics and certain homogeneous varieties derived from Jordan algebras, enabling a motivic decomposition using blow-up formulas and Vishik’s work.

Contribution

It introduces a new explicit birational map between quadrics and Jordan algebra varieties, facilitating the motivic analysis of these homogeneous spaces.

Findings

The birational map is a blow-up followed by a blow-down.

The motivic decomposition of X(J) is achieved using the blow-up formula.

The approach connects Jordan algebra varieties with classical quadrics through explicit birational transformations.

Abstract

We will consider an explicit birational map between a quadric and the projective variety X(J) of traceless rank one elements in a simple reduced Jordan algebra J. X(J) is a homogeneous G-variety for the automorphism group G=Aut(J). We will show that the birational map is a blow up followed by a blow down. This will allow us to use the blow up formula for motives together with Vishik's work on the motives of quadrics to give a motivic decomposition of X(J).