A numerical criterion for uniruledness
type: Item
instance of: scholarly article
title: A numerical criterion for uniruledness
zbMATH Open document ID: 0606.14030
DOI: 10.2307/1971387
author: Yoichi Miyaoka
author: Shigefumi Mori
published in: Annals of Mathematics. Second Series
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 3979249
zbMATH Keywords: Uniruled varieties
zbMATH Keywords: ample divisors
zbMATH Keywords: Fano varieties
Wikidata QID: Q101086597
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971387
OpenAlex ID: W2325783356
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description: scientific article; zbMATH DE number 3979249