Let $k$ be an algebraic closed field (say, $\text{char}(k)\neq 2$), $n \in \mathbb N\setminus \{0\}$ and $G(m, n) = G(m, \mathbb P^n(k))$ the variety of Grassmmanian of $m$-dimensional linear subspaces of $\mathbb P^n(k)$ ($m= 0, 1, \dots n$). For every $0\leq a\leq b \leq n, (a, b) \in \mathbb N^2$, define $$I_{a, b} = \{(V, W) \in G(a, n) \times G(b, n)| V\subseteq W).$$
I would like to prove that $I_{a, b}$ is a non-singular closed sub-variety of $G(a, n) \times G(b, n)$.
I tought to "separate" the subset I'm interested in with a morphism of variety $G(a, n) \times G(b, n) \to \mathbb P^1(k)$, sending my set to $[1:0]$ and the complementary to $[0:1]$. I have few ideas about the regularity of $I_{a, b}$.
Can it be done avoiding Plücker embeding?