Separable rational connectedness and $k$-plane sections of hypersurfaces
Roya Beheshti, Shibashis Mukhopadhyay, Eric Riedl
TL;DR
The authors develop a framework to study separable rational connectedness of smooth Fano hypersurfaces in P^n over fields of positive characteristic by focusing on free lines and higher-dimensional linear sections. They derive precise dimension and irreducibility conditions for spaces of lines and k-planes, and connect linear sections to the RC-property via Starr-type dominance results, with Fermat hypersurfaces illustrating sharp characteristic-related phenomena. In particular, they prove that in any characteristic, quartic hypersurfaces with sufficiently large ambient dimension are separably rationally connected, and they establish broader criteria involving kernel bundles that may extend RC-ness to wider degrees in large dimensions. The paper also extends the analysis to higher-degree rational curves, providing dimension bounds for moduli spaces of maps and highlighting how Bend-and-Break arguments yield control over nonfree maps. These results offer new tools to relate linear sections to RC properties and suggest conjectures about kernel bundles that could unlock RC-ness for a wide class of hypersurfaces in large dimension.
Abstract
Let X be a smooth hypersurface of degree d in P^n over an algebraically closed field of characteristic p. We show that X must be separably rationally connected and must contain a free line if either p is at least d or if p is at least d-1 and the defining equation has some partial derivative that is not too singular. We also show that X must be separably rationally connected in any characteristic if d = 4 and n is sufficiently large. Along the way, we generalize results on the spaces of k-planes in X to characteristic p and connect some of these questions to the spaces of linear sections of X.