Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), and let $C\subset \mathbb P_k^2$ be an irreducible plane cubic. Suppose that $C$ has a node or a cusp at a point $p\in C$. Let $\Lambda_p$ denote the pencil of projective lines in $\mathbb P_k^2$ passing through $p$, identified with $\mathbb P_k^1$. Then the rational projection map

\begin{align*} \pi_p:C\dashrightarrow \Lambda_p\cong \mathbb P_k^1 \end{align*}

which sends a point $q\in C\setminus\{p\}$ to the line joining $p$ and $q$, is birational.

Conversely, if $C$ is an irreducible plane curve over $k$ and there is a birational rational parametrisation $\varphi:\mathbb P_k^1\dashrightarrow C$, meaning a dominant rational map admitting a rational inverse on dense open subsets, then pullback along $\varphi$ induces an isomorphism of function fields

\begin{align*} k(C)\cong k(t). \end{align*}