Equations for polar grassmannians

Abstract

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