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Projective homogeneous varieties birational to quadrics

Research output: Contribution to journalJournal article

Published

Journal publication date2009
JournalDocumenta Mathematica
Volume14
Number of pages20
Pages47-66
Original languageEnglish

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