Let $X$ be a (smooth projective geometrically connected) curve over $k$, where $k$ is a field of characteristic zero. We assume $X$ has genus at least $2$.
I know that $X$ has a plane model. More precisely, the curve $X$ is (birational to) some curve $f(x,y) =0$, where $f(x,y)\in k[x,y]$.
I'm pretty sure the polynomial is not unique.
Have people ever studied the set of planar representations of a curve $X$?
I'm especially interested in knowing what the possible degrees of $f$ are.
That is, consider the subset of $k[x,y]$ consisting of polynomials $f(x,y)$ such that $X$ is birational to the curve $f(x,y) =0$. Which values do $\deg_x f$ and $\deg_y f$ take?