Is the dimension of the projective space $\mathbb{P}^{{n \choose k} -1}$ into which we embed the Grassmannian $G(k,n)$ of $k$-planes in $n$-space minimal? In other words, is the Grassmannian variety (ever) contained in a Plücker hyperplane section?


Sorry, I wasn't able to find a reference for this.



Is the Grassmannian contained in a Plücker hyperplane?
The Grassmannian in its Plücker embedding spans the space. The space $\mathbb{P}^{\binom{n}{k}-1}$ of alternating tensors is spanned by the simple wedges (also called decomposable) $v\_1 \wedge \dotsb \wedge v\_k$, i.e., elements of the Grassmannian. So, no, in this embedding the Grassmannian is not contained in any hyperplane.


But that does not mean it's the minimum possible ambient dimension in which the Grassmannian can be embedded. You can project from a point outside of the secant variety of the Grassmannian to get an embedding in a smaller dimension.


I don't know what is the smallest dimension a Grassmannian can be embedded into.


