Maximal orthogonal grassmannians of quadratic forms of dimensions up to $22$

Abstract

Let X be a connected component of the maximal orthogonal grassmannian of a generic n-dimensional quadratic form q with trivial Clifford invariant. Consider the canonical epimorphism φ from the Chow ring of X to the associated graded ring of the coniveau filtration on the Grothendieck ring of X. In [6] Karpenko proved that φ is an isomorphism for all n ≤ 12 (conjecturally for all n). Recently, in [10] Yagita showed that φ is not an isomorphism for n = 17, 18. In the present paper, together with Yagita's results for n = 17, 18, we show that the map φ is not an isomorphism for all 13 ≤ n ≤ 22. In particualr, the case n = 13 gives the smallest dimensional maximal orthogonal grassmannian whose two graded rings are not isomorphic.