The grassmanian $ \mathbf G(r,n)$ is the set of all $k$-dimensinal subspaces of a $n$-dimensional vector space. I understand how $ \mathbf G(r,n)$ can be embebbed in the projective space $\mathbb P^{{n \choose r }-1}$.Let $\theta$ be a full rank $d \times n$ matrix.We denote the submatrix with column indices $\sigma \subset \{1,..,n\}$ and $ |\sigma|=n$ by $\theta_\sigma$ , so the corresponding minor is $det(\theta_\sigma ).$ Then list $(det(\theta_\sigma ) | σ ⊆ [n])$ minors up to scale identifies the row span of Θ uniquely.So, each r-dimensional subspace can be thought of as a point in $\mathbb P^{{n \choose r }-1}$.
But, what is the set of homogenous polynomials in $K[x_1,x_2,...,x_{n \choose r}]$
that $ \mathbf G(r,n)$ satisfies?
