A criterion for uniruledness in positive characteristic
type: Item
instance of: scholarly article
title: A criterion for uniruledness in positive characteristic
zbMATH Open document ID: 0838.14030
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1996-06-09T00:00:00Z
review text: Let \(X\) be an \(n\)-dimensional variety defined over an algebraically closed field \(k\) which is uncountable. \(X\) is said to be uniruled if there exist an \((n - 1)\)-dimensional \(k\)-variety \(W\) and a dominant rational map \(f : \mathbb{P}^1 \times W \to X\). \(X\) is said to be separably uniruled if the morphism \(f\) can be chosen to be separable. \textit{Y. Miyaoka} and \textit{S. Mori} [Ann. Math., II. Ser. 124, 65-69 (1986; Zbl 0606.14030)] gave the following numerical criterion for uniruledness: Theorem: Let \(X\) be a smooth projective variety over the complex number field \(\mathbb{C}\). Then the following two condition are equivalent: (UR) \(X\) is uniruled. (NC) There exists a nonempty open subset \(U \subset X\) such that for every \(x \in U\), there is an irreducible curve \(C\) through \(x\) with \((K_X \cdot C) < 0\). In positive characteristic, the implication (NC)\(\Rightarrow\)(UR) holds but the converse (UR)\(\Rightarrow \) (NC) does not hold as was pointed out by Miyaoka and Mori (loc. cit.). They asked if (NC) is equivalent to: (SUR) \(X\) is separably uniruled. In this paper we first consider a criterion for a variety to be separably uniruled and show that a separable uniruled and smooth projective variety has the property (NC). Next we show the existence of a smooth projective uniruled but not separably uniruled variety in positive characteristic satisfying the numerical condition of Miyaoka and Mori. It is a counterexample to the problem which Miyaoka and Mori posed.
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 549616
zbMATH Keywords: positive characteristic
zbMATH Keywords: criterion for a variety to be separably uniruled
author: Ei-ichi Sato
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225839
OpenAlex ID: W2019754092
cites work: Toroidal embeddings. I
cites work: A numerical criterion for uniruledness
DOI: 10.2748/TMJ/1178225839
description: scientific article; zbMATH DE number 549616