Algebraic Geometry Seminar

For a variety $X$ over a global function field, e.g.,
$K=\mathbb{F}_q(t)$, one obstruction to existence of a $K$-point is the
"elementary obstruction" $e(X)$.  Assuming $e(X)$ vanishes, what "geometric" conditions guarantee existence of a $K$-point?  Building on earlier work with A. J. de Jong and Xuhua He proving a 2-dimensional version of the theorem of Graber, Harris and myself, and using work of H. Esnault in an essential way, Chenyang Xu and I prove that "rationally simply connected"varieties, and specializations thereof, have $K$-points if $e(X)$ vanishes.  Using this, we give uniform proofs and some extensions of results of Tsen-Lang ($K$ is $C_2$), Brauer-Hasse-Noether (period equals index for division algebras over $K$) and Harder (the split case of Harder's general proof of Serre's "Conjecture II" for $K$).  No background will be assumed, and I will emphasize examples.