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

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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".                                                                                                                 
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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.                                                   
    
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I will explain joint work with Chenyang Xu extending de
Jong-He-Starr over global function fields, e.g.,              
F<sub>q</sub>(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<sub>2</sub>
(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.