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

TL;DR

This paper investigates the isomorphism properties of a canonical map related to maximal orthogonal grassmannians of quadratic forms, showing it fails for dimensions 13 to 22, thus identifying the smallest dimension where the rings differ.

Contribution

It extends previous results by proving the non-isomorphism of the map for all dimensions from 13 to 22, including the smallest such case at dimension 13.

Findings

The map is not an isomorphism for 13 ≤ n ≤ 22.

The case n=13 is the smallest dimension with non-isomorphic graded rings.

Previous conjectures for all n are challenged by these results.

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 $\phi$ from the Chow ring of $X$ to the associated graded ring of the coniveau filtration on the Grothendieck ring of $X$. In \cite{Kar2018} Karpenko proved that $\phi$ is an isomorphism for all $n\leq 12$ (conjecturally for all $n$). Recently, in \cite{Yagita} Yagita showed that $\phi$ 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 $\phi$ is not an isomorphism for all $13\leq n\leq 22$. In particualr, the case $n=13$ gives the smallest dimensional maximal orthogonal grassmannian whose two graded rings are not isomorphic.