Assume that $A$ is a graded $k$-algebra with $A_d=k$ in every degree $d$. Let $M$ be an $A$-module with finite free presentation $F_1\to F_0\to M\to 0$. I want to undertand the collection of finitely presented submodules $N\subseteq M$ of prescribed dimension.
So let $G_1\to G_0\to N\to 0$ be the presentation of $N$. An inclusion $N\subseteq M$ is determined by a commutative square
$$\begin{array}{} G_1 & \xrightarrow{g}& G_0\\\llap{\scriptstyle i_1}\downarrow &&\downarrow\rlap{\scriptstyle i_0}\\F_1&\xrightarrow[f]{}&F_0\end{array}$$
with both vertical maps injections. Since $A$ is one-dimensional in every degree, these maps can be represented by $k$-valued matrices (where the grading might determine certain entries to be zero).
So I end up with a quadratic equation $i_0 g=fi_1$ in the entries of these matrices. The solution set $Z(i_0 g-fi_1)$ is an algebraic variety.
What would be a good strategy to find the solutions of this equation system, provided that I know it only has finitely many solutions (up to a $GL_\text{something}$-action)?