# Rational curve

A one-dimensional algebraic variety, defined over an algebraically closed field $k$, whose field of rational functions is a purely transcendental extension of degree 1 of $k$. Every non-singular complete rational curve is isomorphic to the projective line $\mathbf P^1$. A complete singular curve $X$ is rational if and only if its geometric genus $g$ is zero, that is, when there are no regular differential forms on $X$.

When $k$ is the field $\mathbf C$ of complex numbers, the (only) non-singular complete rational curve $X$ is the Riemann sphere $\mathbf C\cup\{\infty\}$.

If $X$ is defined over a not necessarily algebraically closed field $k$ and $X$ is birationally equivalent to $P_k^1$ over $k$, $X$ is said to be a $k$-rational curve.