Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.