Projective Homogeneous Varieties Birational to Quadrics

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

2000 Mathematics Subject Classification: Primary 11E04; Secondary 14E05, 14L30, 14C15

Keywords and Phrases: Motivic decompositions, Sarkisov links, Jordan algebras

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