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**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 KollĂˇr. Further results on the structure of uniruled varieties have been recently established by the first author.

(*) 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 KollĂˇr. Further results on the structure of uniruled varieties have been recently established by the first author.

Reviewer: A.Lanteri