Families of rationally simply connected varieties over surfaces and torsors for semisimple groups

TL;DR

This paper proves that certain rationally simply connected varieties over surfaces have rational points, leading to the completion of Serre's Conjecture II for function fields over algebraically closed fields.

Contribution

It introduces an algebro-geometric analogue of simple connectedness to establish the existence of rational points, advancing the understanding of rationality in algebraic geometry.

Findings

Proves forms of projective homogeneous varieties over surface function fields have rational points.

Completes the proof of Serre's Conjecture II in Galois cohomology for these fields.

Develops a new method replacing the unit interval with the projective line in geometric arguments.

Abstract

Under suitable hypotheses, we prove that a form of a projective homogeneous variety $G/P$ defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of simple connectedness replacing the unit interval by the projective line. As a consequence, we complete the proof of Serre's Conjecture II in Galois cohomology for function fields over an algebraically closed field.