Incompressibility of orthogonal grassmannians

TL;DR

This paper proves a conjecture that certain orthogonal grassmannians are 2-incompressible under specific algebraic conditions related to quadratic forms and their Witt indices.

Contribution

It establishes the 2-incompressibility of orthogonal grassmannians given divisibility and Witt index conditions, confirming a conjecture by Bryant Mathews.

Findings

Orthogonal grassmannians are 2-incompressible under the specified conditions.

The proof confirms the conjecture for all fields and applicable quadratic forms.

The result links algebraic properties of quadratic forms to geometric incompressibility.

Abstract

We prove the following conjecture due to Bryant Mathews (2008). Let Q be the orthogonal grassmannian of totally isotropic i-planes of a non-degenerate quadratic form q over an arbitrary field (where i is an integer in the interval [1, (\dim q)/2]). If the degree of each closed point on Q is divisible by 2^i and the Witt index of q over the function field of Q is equal to i, then the variety Q is 2-incompressible.