Rationally connected homogeneous spaces
The central object of study in this paper are homogeneous spaces \(X=G/H\), where \(G\) is a connected affine \(k\)-group, \(k\) algebraically closed, and \(H\) a closed subgroup. The first part of the paper discusses in detail rational chains (i.e., certain copies of \({\mathbb P}^1(k))\) connecting points. For precise definitions see section 0 of the paper. One of the main results in section 2 is the following: Theorem. Let \(X=G/H\). Then there is a closed subgroup \(H'\supset H\) of \(G\) with the following properties, where \(p:G/H\rightarrow G/H'\) is the canonical projection: (i) If \(f:\mathbb P^1\rightarrow X\) is a morphism, then \(f(\mathbb P^1)\) is contained in a fiber of \(p\). (ii) Every fiber of \(p\) is generically rationally connected. And as a consequence: Proposition. Assume \(k\) to be of characteristic zero. Suppose \(X=G/H\) is rationally chain connected, then \(X\) is rationally connected. Section 3 is devoted to show that the projection \(G/H\rightarrow G/H'\) preserves the coherence of \(G\)-sheaves and section 4 and 5 contain a large number of applications. Section 6 is devoted to open problems.
scientific article; zbMATH DE number 866086