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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 smooth projective variety X over a field of characteristic zero is uniruled if and only if the canonical bundle of X is not pseudo-effective (that is, not in the closed convex cone spanned by effective divisors in the Néron-Severi group tensored with the real numbers).[1] 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. (In fact, a smooth hypersurface of degree d ≤ n in Pn is a Fano variety and hence is rationally connected, which is stronger than being uniruled.)
A variety X over an uncountable algebraically closed field k 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 closure k of a finite field which are not uniruled but have a rational curve through every k-point. (The Kummer variety of any non-supersingular abelian surface over p with p odd has these properties.[2]) It is not known whether varieties with these properties exist over the algebraic closure of the rational numbers.
Uniruledness is a geometric property (it is unchanged under field extensions), whereas ruledness is not. For example, the conic x2 + y2 + z2 = 0 in P2 over the real numbers R is uniruled but not ruled. (The associated curve over the complex numbers C is isomorphic to P1 and hence is ruled.) In the positive direction, 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.
Uniruledness behaves very differently in positive characteristic. In particular, there are uniruled (and even unirational) 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 number p ≥ 5.[3] 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 dominant separable rational map Y × P1 – → X which does not factor through the projection to Y. ("Separable" means that the derivative is surjective at some point; this would be automatic for a dominant rational map in characteristic zero.) 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.[4] It is not known whether every smooth Fano variety in positive characteristic is separably uniruled.