Based on a recent joint work with Rostislav Devyatov and Alexander Merkurjev, the sharp upper bounds on indexes of twisted flag varieties under the spin groups Spin(n) are being established.
Equivalently, for every integer m of the interval [1,n/2], we are looking for the sharp upper bound on the greatest common divisor of degrees of the finite base field extensions over which the Witt index of an n-dimensional quadratic form of trivial discriminant and Clifford invariant becomes at least m.
The case of m=[n/2] is done by Burt Totaro in 2005.
