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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:

  1. They fit into the exact sequence $0 \to \mathcal{E} \to \mathcal{F} \to \mathcal{G} \to 0$.
  2. 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.

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    $\begingroup$ Do you really mean $\mathbb P \mathcal F = A_n(j) \times A_n(k)$, or do you just want it to map to $A_n(j)$? $\endgroup$
    – Will Sawin
    Jan 22, 2021 at 18:24
  • $\begingroup$ Take $V$ of dimension $n+1$ and $W \subset V$ of dimension $r$. Consider the vector bundle over $\mathbb{P}(V/W)$ given by $E:=\mathcal{O}(-1) \oplus (W \otimes \mathcal{O})$ and consider the projective bundle $\mathbb{P}(E)$. This has an obvious projection towards $\mathbb{P}(V/W)$. On the other hand it has a projection towards $\mathbb{P}(V)$ as well, which is naturally the blow up of $\mathbb{P}(V)$ with center $\mathbb{P}(W)$. Is this an example of what you are looking for? $\endgroup$
    – Enrico
    Jan 22, 2021 at 18:24
  • $\begingroup$ @WillSawin you right, I've edited the question $\endgroup$
    – Bobech
    Jan 23, 2021 at 12:00
  • $\begingroup$ @Enrico no, because I want the projective bundle \mathbb{P}E to have two projections to some Grassmannians $\endgroup$
    – Bobech
    Jan 23, 2021 at 12:02
  • $\begingroup$ Of course projective spaces are Grassmannians, so maybe you should have specified that (also, as I am sure you will know, $Gr(\mathbb{P}^{k-1}, \mathbb{P}^{n-1}) \cong Gr(k,n)$). However, if you want an example with $k \neq 1$, you can consider the relative Grassmannian $\mathbb{Gr}(k, \mathcal{S} \oplus \mathcal{O})$ (which is a projective bundle by duality) over $Gr(k,n-1)$. There is a second projection to $Gr(k,n)$ which blows up a copy of $Gr(k-1,n-1)$, embedded as the zero locus of a general section of $\mathcal{Q}$. $\endgroup$
    – Enrico
    Jan 23, 2021 at 13:49

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