For a variety defined over a global function field K, e.g., F _{p}(t),
the simplest cohomological obstruction is the "elementary obstruction"
of ColliotThélène and Sansuc. Assuming this vanishes, what additional
geometric hypotheses imply existence of a rational point? Combining
"rational simple connectedness" in characteristic 0 with work of Esnault
on rational points over finite fields, Chenyang Xu and I give new,
uniform proofs of three classical results: Lang's proof that K is C _{2},
a theorem of BrauerHasseNoether that twisted forms of Grassmannians
over K admit rational points, and the split case of Harder's theorem /
Serre's "Conjecture II" over K. I will focus on the geometry of
rational curves over C, and how one
"transports" this to results over a field such as K. Pretalk (background, aimed at graduate students):
