Separable rational connectedness and $k$-plane sections of hypersurfaces

TL;DR

This paper investigates conditions under which smooth hypersurfaces over algebraically closed fields are separably rationally connected, establishing new results in positive characteristic and generalizing properties of k-planes and linear sections.

Contribution

It proves separable rational connectedness and existence of free lines in hypersurfaces under new characteristic-related conditions and extends results on k-planes to positive characteristic.

Findings

Hypersurfaces are separably rationally connected if p ≥ d.

Existence of free lines in hypersurfaces under certain partial derivative conditions.

Generalization of k-plane space results to characteristic p.

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.