Pushforward from Flag variety

Question

Consider a vector space $V$ and some $n,n+1\leqslant\dim V$. Let $F\subset G(n,V)\times G(n+1,V)$ be the Flag variety (where $G(n,V)$ is the Grassmannian of $n$-dimensional subspaces of $V$), i.e. the subvariety of couples $(W,W')$ such that $W\subset W'$. We have projections from $F$ to both $G(n,V)$ and $G(n+1,V)$. I can show that the projection to $G(n,V)$ is just the projectivization $P(Q)$, where $Q$ is the universal quotient bundle on $G(n,V)$. Is there a way to describe the second projection, i.e. $\pi:F\rightarrow G(n+1,V)$, and more specifically, what is the pushforward $\pi_{\ast}(\mathcal{O})$ of the structure sheaf? Is $\pi$ just the projectivization $P(S)$, where $S$ is the universal subbundle on $G(n+1,V)$ so $\pi_{\ast}(O)=O$?

Answer 1

Concretely, a point of $G(n+1, V)$ is a choice of $W' \subset V$ of dimension $n+1$. We can then think of the choice of $W$ as being "a choice of $n$-plane in $W'$". Or, dually, a choice of one-dimensional subspace of $(W')^*$.

(Phrasing it the latter way is sort of nice since it happens to make the description become in terms of a "projective space (of lines)".)

So, we want to parametrize all the 1-dimensional subspaces of $(W')^*$, i.e. we want $\mathbb{P}((W')^*)$. This is the fiber of the projective bundle $\mathbb{P}(\mathcal{S}^*)$, almost what you guessed.

Comment: dualizing isn't really necessary if you already believe in Grassmannians. There are "Grassmann bundles of vector bundles" also, just like projective bundles. The initial description was "an $n$-dimensional subspace of $W'$". Since $W'$ is the fiber of $\mathcal{S}$, this description means the Grassmann bundle $\mathbf{G}(n, \mathcal{S})$.

(edit: and $\mathbf{G}(n, \mathcal{S}) \cong \mathbb{P}(\mathcal{S}^*)$. More generally, $\mathbf{G}(k, \mathcal{E}) \cong \mathbf{G}(\mathrm{rank}(\mathcal{E}) - k, \mathcal{E}^*)$, just like for vector spaces.)