$k$ is algebraically closed. For any finite-dimensional associative $k$-algebra $A$, there is a unique quiver $Q_A$ and a (non-unique) ideal $I$ of $kQ_A$ such that $A$ and $kQ_A/I$ are Morita equivalent, and ${\mathfrak{m}}^r \subset I \subset {\mathfrak{m}}^2$ for some $r\geq 2$.

An ideal $I$ satisfying ${\mathfrak{m}}^r \subset I \subset {\mathfrak{m}}^2$ is called an *admissible ideal*.

Submodule Grassmannian
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Let $Q$ be a finite quiver, $I$ be an admissible ideal, and $V$ be a representation of $(Q,I)$.

A *subrepresentation* of $V$ is a tuple $(W_i)_{i\in Q_0}$, where

-   each $W_i$ is a subspace of $V_i$, and

-   for each arrow $a$ in $Q$, we have that $V_{a}(W_{s(a)}) \subset W_{t(a)}$.

In that case, $W= (W_i, V_a|_{W_{s(a)}})_{i\in Q_0, a\in Q_1}$ is a representation of $(Q,I)$, and the canonical inclusion into $V$ is a morphism of representations.

Grassmannians of vector spaces are projective varieties whose points parametrize subvector spaces of a given dimension. Submodule Granssmannians of modules generalize this notion: they are projective varieties whose points parametrize submodules of a given dimension vector.

\[defi::grassmannian\] Let ${\mathbf{e}}\in {\mathbb{N}}^{Q_0}$ be a dimension vector. The *submodule Grassmannian of $V$ with dimension vector ${\mathbf{e}}$* is the subset ${\mbox{{\rm Gr}}_{{\mathbf{e}}}}(V)$ of $\prod_{i\in Q_0} {\mbox{{\rm Gr}}_{e_i}}(V_i)$ of all points $(W_i)_{i\in 