Normal bundles of rational curves and separably rationally connected varieties
In positive characteristic, there are two different notions of rational connectedness: a variety can be rationally connected or separably rationally connected (SRC). SRC varieties share many of the nice properties that rationally connected varieties have in characteristic 0. But, while it is conjectured that smooth Fano varieties are SRC, it is only known that they are rationally connected. In the last decade, several mathematicians have come up with different ways to show that general Fano complete intersections are SRC. In this talk, I'll explain this story, and then discuss an approach Izzet Coskun and I are using to show that other sorts of varieties are SRC by comparing the normal bundle of a rational curve on a variety and its normal bundle to some subvariety containing it. For instance, I'll show that a Fano complete intersection of hypersurfaces each of degree at least 3 on a Grassmannian is SRC.
The discussion for Geoff Smith’s talk is taking place not in zoom-chat, but at https://tinyurl.com/2021-05-07-gs (and will be deleted after ~3-7 days).