# 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 $\mathbf C\cup\{\infty\}$.

#### Comments

In classic literature a rational curve is also called a unicursal curve.

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.

#### References

[a1] | W. Fulton, "Algebraic curves" , Benjamin (1969) pp. 66 MR0313252 MR0260752 Zbl 0194.21901 Zbl 0181.23901 |

[a2] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |

**How to Cite This Entry:**

Rational curve.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Rational_curve&oldid=34209