Algebraic Geometry Seminar
Jason Starr
Stony Brook
Rational points of varieties over global function fields
Abstract: There are several classical results asserting existence of rational points of smooth, projective varieties defined over global function fields, e.g., $F_p(t)$: the Tsen-Lang theorem about points on low degree complete intersections in projective space, the Brauer - Hasse - Noether theorem that "period equals index", equivalent to existence of points on twists of Grassmannians, and the "split" case of Serre's "Conjecture II" (proved also in both the split and non-split case by Harder), equivalent to existence of points on twists of projective homogeneous varieties with Picard rank 1. In joint work with Chenyang Xu, and using work of Esnault and of de Jong - He - Starr in an essential way, we find a new, uniform proof of these results, as well as some extensions, by studying rational curves on a lift of the variety to characteristic 0. This will be a down-to-earth lecture with examples; no prior knowledge of "global function fields", "Brauer groups", "rational connectedness" or "rational simple connectedness" will be needed.
Wednesday October 9, 2013 at 4:00 PM in SEO 427