A criterion for uniruledness in positive characteristic
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.
criterion for a variety to be separably uniruled
scientific article; zbMATH DE number 549616