## A numerical criterion for uniruledness.(English)Zbl 0606.14030

Let X be an n-dimensional algebraic variety defined over an algebraically closed field k. X is said to be uniruled if there exist an (n-1)- dimensional variety W and a dominant rational map $$f: {\mathbb{P}}^ 1\times W\to X.$$ Uniruled varieties play an important role in the classification theory of algebraic varieties, particularly in connection with varieties of Kodaira dimension -$$\infty$$. They have the remarkable property that through the general k-valued point there passes a rational curve. The authors prove the following theorem. Assume that X is projective; then X is uniruled if
(*) there exists a non-empty open subset $$U\subset X$$ such that for every $$x\in U$$ there passes an irreducible curve C satisfying $$K_ X\cdot C<0.$$
If $$k={\mathbb{C}}$$ also the converse is true, while, if Char k$$>0$$, the authors conjecture that (*) is equivalent to the fact that X is separably uniruled (i.e. f is separable). As a consequence of their criterion the authors get that if X is projective with Codim Sing(X)$$\geq 2$$ and contains ample divisors $$H_ 1,...,H_{n-1}$$ such that $$K_ XH_ 1...H_{n-1}<0$$, then X is uniruled. In particular $${\mathbb{Q}}$$-Fano varieties are uniruled; this generalizes the corresponding result for Fano manifolds due to Kollár. Further results on the structure of uniruled varieties have been recently established by the first author.
Reviewer: A.Lanteri

### MSC:

 14J10 Families, moduli, classification: algebraic theory 14E05 Rational and birational maps

### Keywords:

Uniruled varieties; ample divisors; Fano varieties
Full Text: