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An n-dimensional variety X over an algebraically closed field k is said to be uniruled if there exist an (n 1)-dimensional k-variety W and a dominant rational map f: P' x W -* X. X is called separably uniruled if f is a separable map (i.e. k(P' x W) is a finite separable extension of k(X) via f). If X is uniniled, then there exists a rational curve passing through a general k-valued point of X, and the converse is true (by the countability of the components of the Hilbert scheme) provided that the ground field k is uncountable. Uniruled varieties are of importance in the classification of varieties; it is conjectured that in characteristic 0 uniruledness is equivalent to K = Xc. This article gives a numerical criterion for a projective variety to be uniruled:
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Miyaoka et al. (1986) studied this question.
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