In algebraic geometry, a variety over a fieldk is ruled if it is birational to the product of the projective line with some variety over k. A variety is uniruled if it is covered by a family of rational curves. The concept arose from the ruled surfaces of 19th-century geometry, meaning surfaces in affine space or projective space which are covered by lines. Uniruled varieties can be considered to be relatively simple among all varieties, although there are many of them.
Properties
Every uniruled variety over a field of characteristic zero has Kodaira dimension −∞. The converse is a conjecture which is known in dimension at most 3: a variety of Kodaira dimension −∞ over a field of characteristic zero should be uniruled. A related statement is known in all dimensions: Boucksom, Demailly, Păun and Peternell showed that a smoothprojective varietyX over a field of characteristic zero is uniruled if and only if the canonical bundle of X is not pseudo-effective. As a very special case, a smooth hypersurface of degree d in Pn over a field of characteristic zero is uniruled if and only if d ≤ n, by the adjunction formula. A variety X over an uncountablealgebraically closed fieldk is uniruled if and only if there is a rational curve passing through every k-point of X. By contrast, there are varieties over the algebraic closurek of a finite field which are not uniruled but have a rational curve through every k-point. It is not known whether varieties with these properties exist over the algebraic closure of the rational numbers. Uniruledness is a geometric property, whereas ruledness is not. For example, the conic x2 + y2 + z2 = 0 in P2 over the real numbersR is uniruled but not ruled. In the positivedirection, every uniruled variety of dimension at most 2 over an algebraically closed field of characteristic zero is ruled. Smooth cubic 3-folds and smooth quartic 3-folds in P4 over C are uniruled but not ruled.
Positive characteristic
Uniruledness behaves very differently in positive characteristic. In particular, there are uniruled surfaces of general type: an example is the surface xp+1 + yp+1 + zp+1 + wp+1 = 0 in P3 over p, for any prime numberp ≥ 5. So uniruledness does not imply that the Kodaira dimension is −∞ in positive characteristic. A variety X is separably uniruled if there is a variety Y with a dominantseparablerational mapY × P1 – → X which does not factor through the projection to Y. A separably uniruled variety has Kodaira dimension −∞. The converse is true in dimension 2, but not in higher dimensions. For example, there is a smooth projective 3-fold over 2 which has Kodaira dimension −∞ but is not separably uniruled. It is not known whether every smooth Fano variety in positive characteristic is separably uniruled.
