Indexes of generic Grassmannians for spin groups

• Published in: Proceedings of the London Mathematical Society Vol. 125; no. 4; pp. 825 - 840
• Main Authors: Karpenko, Nikita A., Merkurjev, Alexander S.
• Format: Journal Article
• Language: English
• Published: 01-10-2022
• ISSN: 0024-6115; 1460-244X
• DOI: 10.1112/plms.12471

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.