Geodesic
Projective homogeneous varieties birational to quadrics
Documenta mathematica, Tome 14 (2009), pp. 47-66.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $X(J)$.
Classification : 11E04, 14E05, 14L30, 14C15
Keywords: motivic decompositions, sarkisov links, Jordan algebras 
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     author = {MacDonald, Mark L.},
     title = {Projective homogeneous varieties birational to quadrics},
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     pages = {47--66},
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     volume = {14},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2009__14__a25/}
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MacDonald, Mark L. Projective homogeneous varieties birational to quadrics. Documenta mathematica, Tome 14 (2009), pp. 47-66. http://geodesic.mathdoc.fr/item/DOCMA_2009__14__a25/