Equations for polar grassmannians

Abstract

Given an $N$-dimensional vector space $V$ over a field $\mathbb{F}$ and a trace-valued $(\sigma ,\epsilon )$-sesquilinear form $f:V\times V\to \mathbb{F}$, with $\epsilon =\pm 1$ and ${\sigma }^2={\mathrm{id}}_{\mathbb{F}}$, let $\mathcal{S}$ be the polar space of totally $f$-isotropic subspaces of $V$ and let $n$ be the rank of $\mathcal{S}$. Assuming $n\ge 2$, let $2\le k\le n$, let ${\mathcal{G}}_k$ the $k$-grassmannian of $\mathrm{PG}(V)$, embedded in $\mathrm{PG}({\wedge }^{k}V)$ as a projective variety and ${\mathcal{S}}_k$ the $k$-grassmannian of $\mathcal{S}$. In this paper we find one simple equation that, jointly with the equations of ${\mathcal{G}}_k$, describe ${\mathcal{S}}_k$ as a subset of $\mathrm{PG}({\wedge }^{k}V)$.

8 pages