Rational points and rational curves over fields of cohomological dimension two -- Jason Starr, September 20, 2013
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., Fq(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 C2 (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.