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In mathematics, a '''ruled variety''' is a variety [[birational]] to a product of the projective line and another variety, and a '''uniruled variety''' is a variety that is dominated by a ruled variety. This concept is a generalisation (not too remote) of the [[ruled surface]]s of classical differential geometry.

A variety is uniruled if and only if there is a rational curve passing though every point.

Any uniruled variety has [[Kodaira dimension]] −∞. In dimension at most 3, and conjecturally in all dimensions, the converse is true: a variety of Kodaira dimension −∞ is uniruled.

==Consequences of the Miyaoka-Mori theorem for smooth varieties==

Let ''X'' be a smooth projective variety over an algebraically closed field and <math> \mathcal K_X</math> its [[canonical divisor]]. Then if there exists a curve ''C'' in ''X'' such that <math>C . \mathcal K_X < 0</math>, the variety ''X'' is ruled.

In particular, if ''X'' has [[nef line bundle|nef]] [[anticanonical|anticanonical divisor]], then for ''X'' to be ruled, it suffices for the anticanonical divisor to not be numerically trivial.

==References==
{{Citation | last1=Clemens | first1=Herbert | last2=Kollár | first2=János | last3=Mori | first3=Shigefumi | title=Higher-dimensional complex geometry | id={{MathSciNet | id = 1004926}} | year=1988 | journal=Astérisque | issn=0303-1179 | issue=166 | pages=144 pp. (1989)}}

[[Category:Algebraic geometry]]

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