As we know, the projective space $\mathbb{P}^n$ represents the functor sending $X$ to the set of line bundles $L$ on $X$ together with a surjection from the trivial vector bundle to $L$.
My question is, what functor does the Grassmannian $Gr(d,n)$ represent? Sending $X$ to the set of rank-$n$ vector bundles together with a sub-bundle of rank $d$?
More generally, what functors do flag varieties represent?
Best Answer
Let me elaborate on some of the other answers.
On the Grassmannian X = Gr(k,n) (I am using this notation to mean k-dimensional subspaces of an n-dimensional vector space), we have the trivial bundle $X \times K^n$ (here K is our field of definition), and the tautological subbundle R (naively, this is the subset {$(x,v) \in X \times K^n \mid v \in x$} which is a locally free sheaf of rank k, and its quotient Q is also a locally free sheaf of rank n-k. So we write
$0 \to R \to X \times K^n \to Q \to 0$,
which is the tautological sequence. Given a map to $f \colon Y \to X$, where Y is a k-scheme, we can pull back this sequence to get
$0 \to f^*R \to Y \times K^n \to f^*Q \to 0$.
Conversely, given such a sequence, there is a unique map to X which gives this pullback. Naively, over a point y, the fiber of $f^*R$ is a subspace of $K^n$, so we send it to that closed point. So X represents the functor which sends Y to the set of short exact sequences as above. The quotient being locally free implies that the subsheaf is also locally free, so it really represents the functor which sends Y to the set of quotients $\mathcal{O}_Y^n \to F \to 0$ where the rank of F is n-k.
For general flag varieties, we have similar tautological sequences (but there are more subbundles to consider). This kind of analysis also makes sense for symplectic and orthogonal Grassmannians and flag varieties.
We can also do something similar if we're talking about the Grassmannian of a vector bundle instead of a vector space. Then we just work in the category of S-schemes instead of k-schemes where S is whatever the base space is.