A superelliptic curve is defined by the equation \begin{equation} y^n - P(x) = 0 \end{equation} for some $n \geq 2$, where $P(x)$ is a polynomial of order $m$ in $x$. This equation can be thought of as defining a one (complex-) dimensional hypersurface in $\mathbb{CP}^1 \times \mathbb{CP}^1$.
Now, one can think of a more general equation $Q(x,y) = 0$, where $Q$ is a polynomial of order $m(n)$ in $x(y)$, respectively. This equation also defines a one (complex-) dimensional hypersurface in $\mathbb{CP}^1 \times \mathbb{CP}^1$. Have such curves been studied? Or is there a straightforward way of reducing $Q(x)$ to the defining equation of the superellptic curves?
In particular, I have a matrix-valued function $M \colon \mathbb{CP}^1 \to \text{GL}(n, \mathbb{C})$, such that the entries of $M(x)$ are polynomials in $x$. I am interested in the topological properties (the genus, for instance) of the complex curve defined by the corresponding characteristic polynomial $Q(x,y) = \det[y - M(x)]$. Has anyone come across this setup by any chance?
