Indexes of generic Grassmannians for spin groups

Given integers d$d$ and m$m$ , satisfying 1⩽m⩽d/2$1\leqslant m\leqslant d/2$ , and an arbitrary base field, let Xm$X_m$ be the m$m$ th Grassmannian of a generic d$d$ ‐dimensional quadratic form of trivial discriminant and Clifford invariant. The index of Xm$X_m$ , defined as the g.c.d. of degrees of its closed points, is a 2‐power 2i(m)$2^{\mathrm{i}(m)}$ . We find a strong lower bound on the exponent i(m)$\mathrm{i}(m)$ which is its exact value for most d,m$d,m$ and which is always within 1 from the exact value.