Equations for polar grassmannians

Given an $N$-dimensional vector space $V$ over a field $\mathbb{F}$ and a trace-valued $(\sigma ,\epsilon )$-sesquilinear form $f:V\times V\rightarrow $\mathbb{F}$$,with$\varepsilon = \pm 1$and$\sigma^2 = $\mathrm{i}\mathrm{d}$_{$\mathbb{F}$}$,let${\cal S}$bethepolarspaceoftotally$f$-isotropicsubspacesof$V$andlet$n$betherankof${\cal S}$\mathrm{.}Assuming$n \geq 2$,let$2 \leq k \leq n$,let${\cal G}_k$the$k$-grassmannianof$$\mathrm{P}\mathrm{G}$(V)$,embeddedin$$\mathrm{P}\mathrm{G}$(\wedge^kV)$ 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 $\mathrm{P}\mathrm{G}({\wedge }^{k}V)$.