Setup: Let $V$ be a $(n+1)$-dimensional vector space. We define $\mathbb{P}V=\mathbb{P}^n$ as follows: points of $\mathbb{P}V$ correspond to 1-dimensional vector subspaces of $V^\vee$. Moreover, $$ A_n(k):=\{\mathbb{P}W \subset \mathbb{P}V : \dim W=k\} $$ is the Grassmannian of $\mathbb{P}^{k-1}$ into $\mathbb{P}^n$. Once we define $\mathcal{Q}^\vee$ to be the tautological bundle over $A_n(k)$ (that is the bundle such that, over a point $\mathbb{P}W \in A_n(k)$, it puts $W \subset V$ as a subspace), we have the following short exact sequence: $$ 0 \to \mathcal{S}^\vee \to V \otimes \mathcal{O}_{A_n(k)} \to \mathcal{Q} \to 0. $$ If we apply the functor $\mathbb{P}():=Proj(Sym())$ we obtain a map $$ \mathbb{P}\mathcal{Q} \to \mathbb{P}V \times A_n(k) $$ which has two canonical projections to $\mathbb{P}V$ and to $A_n(k)$. By definition $\mathbb{P}V=A_n(1)$ and so we define $\mathbb{P}\mathcal{Q}:=A_n(1,k)$.
Question: Are there generalizations of this construction? In the sense, if we suppose that we have vector bundles $\mathcal{E},\mathcal{F},\mathcal{G}$ over $A_n(k)$ such that:
- They fit into the exact sequence $0 \to \mathcal{E} \to \mathcal{F} \to \mathcal{G} \to 0$.
- There is a map $\mathbb{P}\mathcal{F} \to A_n(j) \times A_n(k)$ for some $j \in \{1,\ldots,n\}$.
What can we say about these bundles? I think that $\mathcal{F}$ is the tensor product of $\mathcal{O}_{A_n(k)}$ and something else, that gives the projection to $A_n(j)$, but I cannot figure out how it would look like.
Any idea, hint or reference will be much appreciate.