Rational curve

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

When is the field of complex numbers, the (only) non-singular complete rational curve is the Riemann sphere .


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

If is defined over a not necessarily algebraically closed field and is birationally equivalent to over , is said to be a -rational curve.


[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:
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article