Q_0}$ defining a subrepresentation of $V$.

The submodule Grassmannian is in fact a Zariski-closed subset of $\prod_{i\in Q_0} {\mbox{{\rm Gr}}_{e_i}}(V_i)$, so it is a projective variety.

\[exam::Grassmannians\]

1.  If the quiver $Q$ has only one vertex and no arrows, then representations of $Q$ are just vector spaces, and their submodule Grassmannians are just usual Grassmannians.

2.  \[exam::grassKron\] Let $Q =  \xymatrix{ 1\ar@<.3ex>[r] \ar@<-.3ex>[r]  & 2}$ be the Kronecker quiver. Consider the representation $$V=\xymatrix{ k^2\ar@<.3ex>[r]^{\footnotesize\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}} \ar@<-.3ex>[r]_{\footnotesize\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}}  & k^2.}$$ Then there are six dimension vectors for which the submodule Grassmannian of $V$ is non-empty. The table below lists those dimension vectors and gives a variety to which the corresponding submodule Grassmannian is isomorphic. $$\begin{tabular}{|c||c|c|c|c|c|c|} 
              \hline
              ${\mathbf{e}}$   & $(0,0)$ & $(0,1)$ & $(0,2)$ & $(1,1)$ & $(1,2)$ & $(2,2)$ \\
              \hline 
              ${\mbox{{\rm Gr}}_{{\mathbf{e}}}}(V)$   & point & ${\mathbb{P}}^1$  & point & point & ${\mathbb{P}}^1$ & point \\
              \hline
            \end{tabular}$$

$F$-polynomials of modules
--------------------------

We now define the $F$-polynomial of a representation of a quiver with relations $(Q,I)$ (or, equivalently, of a module over $A=kQ/I$). Roughly, the $F$-polynomial can be seen as a generating function for counting submodules of 