The only maps $\mathbb A^1 \rightarrow C$ are constant if $C$ is a curve with genus $\geq 1$

Question

My professor claimed that if $C$ is a curve over $k$ of genus $\geq 1$ the only morphisms $\mathbb A_k^1 \rightarrow C$ are the constant morphisms, or something along those lines I can't exactly remember. What is the relevant theorem here? Does $C$ also have to be smooth?

Answer 1

A nonconstant map $\Bbb A^1\to C$ gives an injection of function fields $k(C)\to k(t)$ with $t$ an indeterminant. By Luroth's theorem, such a subextension $k\subset k(C)\subset k(t)$ is of the form $k(f(t))$ for some rational function $f$. In particular, this cannot be the function field of a curve of positive genus: every such field is a nontrivial algebraic extension of $k(t)$. So our assumption that $\Bbb A^1\to C$ was nonconstant must have been incorrect.

(If you're looking for an elementary proof of Luroth's theorem, Wikimedia has an example.)