Projective homogeneous varieties birational to quadrics (Q1016998)

An important problem in the theory of quadratic forms over fields is the classification of quadrics up to birational equivalence. \textit{B. Totaro} [Bull. Soc. Math. Fr. 137, No. 2, 253--276 (2009; Zbl 1221.14014)] has shown that over a field of characteristic not \(2\), if \(\phi\) is an \(r\)-fold Pfister form and \(b=\langle b_1,\dots, b_n\rangle\) (\(n\geq 2\)) is a nondegenerate quadratic form, then up to birational equivalence, the quadric defined by the quadratic form \(q=\phi\otimes\langle b_1,\dots, b_{n-1}\rangle\perp \langle b_n\rangle\) only depends on the isometry class of \(\phi\) and \(\phi\otimes b\), but not on the actual diagonalization of \(b\). The Sarkisov program [see, e.g., \textit{A. Corti}, J. Algebr. Geom. 4, No. 2, 223--254; appendix: 248--254 (1995; Zbl 0866.14007)] predicts that birational maps between quadrics (or more generally, between any two Mori fibre spaces) factors as a chain of composites of so-called ``elementary links''. In the present paper, the author gives explicit such factorizations for many of Totaro's birational maps for fields of characteristic \(0\). In particular, it is proved that if \(0\leq r\leq 2\) and \(n\geq 3\), or if \(r=n=3\), then for the birational equivalences for quadrics defined by quadratic forms \(q\) as above, there is a birational map that factors into two elementary links, each of which is a blow up of a reduced subscheme followed by a blow down. Furthermore, if \(r\neq 1\) and \(\phi\) is not hyperbolic, then the intermediate Mori fibre space of this factorization is given by the projective homogeneous variety \(X(J)\) of traceless rank one elements in a Jordan algebra \(J\). Using results by \textit{A. Vishik} [Lect. Notes Math. 1835, 25--101 (2004; Zbl 1047.11033)] on motivic decomposition of quadrics, it is shown that the motives of projective quadrics defined by quadratic forms \(q\) as above decompose into the sum, up to Tate twists, of Rost motives and higher forms of Rost motives. This in turn is used to show that the motives of non-generically split projective homogeneous \(G\)-varieties \(X(J)\) as above (where the groups \(G\) are automorphism groups of simple reduced Jordan algebras of degree at least \(3\)) decompose into the direct sum of a higher form of a Rost motive and several copies of Tate twists of the Rost motive.