Rationally Connected Varieties over Local Fields

LetX be a proper variety defined over a fieldK. Following [Manin72], two points x, x ∈ X(K) are called R-equivalent if they can be connected by a chain of rational curves defined over K, cf. (4.1). In essence, two points are R-equivalent if they are “obviously” rationally equivalent. Several authors have proved finiteness results over local and global fields (cubic hypersurfaces [Manin72, Swinnerton-Dyer81], linear algebraic groups [CT-Sa77, Voskresenskǐı79, Gille97, Voskresenskǐı98], rational surfaces [CT-Co79, Colliot-Thélène83, CT-Sk87], quadric bundles and intersections of two quadrics [CT-Sa-SD87, Parimala-Suresh95]). R-equivalence is only interesting if there are plenty of rational curves on X, at least over K̄. Such varieties have been studied in the series of papers [Ko-Mi-Mo92a, Ko-Mi-Mo92b, Ko-Mi-Mo92c], see also [Kollár96]. There are many a priori different ways of defining what “plenty” of rational curves should mean. Fortunately many of these turn out to be equivalent and this leads to the notion of rationally connected varieties. See [Ko-Mi-Mo92b], [Kollár96, IV.3], [Kollár98, 4.1.2].