Let $X$ be a smooth projective variety and $F \to X$ be a free algebraic vector bundle of rank $n$. I know that there exists a correspondence between
- Vector subbundles $E \subset F$ of rank $k$.
- Morphisms $X \to G$, where $G$ is the Grassmannian of $k$-planes in $\mathbb A^n$.
Now suppose that I am interested in
- Flags of vector subbundles $E_1 \subset ... \subset E_r \subset F$, where each $E_i$ is of rank $k_i$.
Is there a correspondence between these and
- Commutative diagrams of morphisms $X \to G_i$ and $G_i \to G_{i+1}$, where each $G_i$ is the Grassmannian of $k_i$-planes in $\mathbb A^n$
...?