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

Ruled surface

In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the curved surface of a cylinder or cone...

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

Kodaira dimension

In algebraic geometry, the Kodaira dimension κ measures the size of the canonical model of a projective variety V.The definition of Kodaira dimension, named for Kunihiko Kodaira, and the notation κ were introduced in the seminar.-The plurigenera:...

 −∞. 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 its canonical divisor. Then if there exists a curve C in X such that , the variety X is ruled.

In particular, if X has nef anticanonical divisor, then for X to be ruled, it suffices for the anticanonical divisor to not be numerically trivial.