Projective homogeneous varieties birational to quadrics

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 $G=\text{Aut}(J)$. 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 $X(J)$.