By Graber-Harris-Starr and de Jong-Starr, a smooth, projective variety defined over the function field k(B) of a curve B over an algebraically closed field k has a k(B)-point if it is (geometrically, separably) rationally connected, i.e., every pair of (geometric) points is connected by a rational curve. Moreover, this is also true for degenerations of rationally connected varieties. By work of Esnault, Esnault-Xu, etc., this is also known over finite fields, i.e., the other main class of fields of "cohomological dimension one".

What about fields of cohomological dimension two? By de Jong-He-Starr in the case of Picard number 1, and by Yi Zhu for higher Picard number, over the function field k(B) of a surface B, every "rationally simply connected" variety with vanishing "elementary obstruction" has a k(B)-point.

I will explain joint work with Chenyang Xu extending de
Jong-He-Starr over global function fields, e.g.,
F_{q}(t), as well as over some other fields of
cohomological dimension two. This gives new, uniform
proofs, as well as extensions, of some classical theorems over
these fields: period equals index for division
algebras (Brauer-Hasse-Noether), these fields are C_{2}
(S. Lang), and the split case of Serre's "Conjecture II"
(G. Harder). Time permitting, I will also explain recent
progress and challenges in the classification of
rationally simply connected varieties.