Abstract
A complex projective manifold is rationally connected, resp. rationally simply connected, if finite subsets are connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally connected. We prove that a projective scheme over a global function field with vanishing elementary obstruction has a rational point if it deforms to a rationally simply connected variety in characteristic 0. This gives new, uniform proofs over these fields of the Period-Index Theorem, the quasi-split case of Serre's Conjecture II, and Lang's $C_2$ property.
Highlights
Résumé. — Si une variété projective complexe est rationnellement connexe, chaque ensemble fini de points est contenu dans une courbe rationnelle ; si elle est rationnellement simplement connexe, les espaces paramétrant ces courbes rationnelles sont eux-mêmes rationnellement connexes
For a field F and a projective F -scheme XF, does XF have an F -rational point? Colliot-Thélène and Sansuc defined an obstruction to existence of an F -rational point, the elementary obstruction, [CTS87, Section 2.2]
For geometrically integral and smooth XF, the universal torsor is a torsor over XF for a multiplicative group F -scheme Q such that for every field extension E/F, for every multiplicative group F -scheme R, every R-torsor over XF ×Spec F Spec E arises from the universal torsor by a unique morphism of group E-schemes, Q ×Spec F Spec E → R
Summary
For a field F and a projective F -scheme XF , does XF have an F -rational point? Colliot-Thélène and Sansuc defined an obstruction to existence of an F -rational point, the elementary obstruction, [CTS87, Section 2.2]. If XF ×Spec F Spec F is integral and smooth, the elementary obstruction vanishes if and only if XF admits a universal torsor, cf [CTS87, Proposition 2.2.3], [Sko[01], Theorem 2.3.4]. It is important to extend these notions to a flat, projective scheme over a base scheme rather than a base field. This has been developed in [Pir[12], Definition, p. For a projective F(η)-variety fη : Xη → Spec F(η), our main theorem gives an F(η)-rational point of Xη provided that there exists a lift to characteristic 0 of a projective model that has vanishing elementary obstruction and that is a rationally connected fibration. A key role is played by Esnault’s theory of rational points of specializations over finite fields of varieties of coniveau 1, [Esn[07], Corollary 1.2]
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