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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?

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up vote 1 down vote accepted

For any non-constant $x$ in the function field $k(X)$ of $X$ (i.e. $x$ is not algebraic over $k$), there exists an $y\in k(X)$ such that $k(X)=k(x)[y]$ (primitive element theorem). The minimal polynomial of $y$ over $k(x)$ (after multiplying by a suitable non-zero polynomial in $x$) will give you a plane model of $X$ ! So there is no hope to say something on $f$ in general.

You should at least require some minimality conditions.

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