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Rational Curve

A rational curve is an integral algebraic curve birational to the projective line over the stated field. Over C, this means that its function field is isomorphic to C(t), and its smooth projective normalization has genus zero. The curve itself can have singularities. For example, the nodal cubic has a rational parametrization, whereas a smooth projective cubic has genus one and is not rational.

A nonconstant morphism from P^1 to a complex variety has a rational curve as its image. Rational connectedness asks that two general points of a smooth projective variety lie on such an image.

See also

Algebraic Curve, Birational Transformation, Projective Line, Rationally Connected Variety

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References

Fujino, O. "Notes on Rational Chain Connectedness." Version 0.08, 5 Mar 2026. https://www.math.kyoto-u.ac.jp/~fujino/rationallychainconnectedness4.pdf.The Stacks Project Authors. "Curves and Function Fields." The Stacks Project, Tag 0BY1. https://stacks.math.columbia.edu/tag/0BY1.The Stacks Project Authors. "Curves of Genus Zero." The Stacks Project, Tag 0C6L. https://stacks.math.columbia.edu/tag/0C6L.

Cite this as:

Weisstein, Eric W. "Rational Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RationalCurve.html

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