Let $G := G(k,V)$ denote the Grassmannian of $k$-linear subspaces in a $\mathbb{C}$-vector space $V$ of dimension $n$. Let $S$ denote the tautological bundle over $G$. There is a canonical map on sheaf of sections $$S \otimes S^\vee \to \mathcal{O}_G$$ by evaluative pairing of sections of $S$ and $S^\vee$. What can be said about the kernel of this map? Certainly it is of rank $k^2-1$.