Free curves in Fano hypersurfaces must have high degree

TL;DR

This paper demonstrates that in positive characteristic, the minimal degree of free rational curves in smooth Fano hypersurfaces cannot be linearly bounded by the dimension, using Fermat hypersurfaces as examples.

Contribution

It establishes a super-linear lower bound on free curve degrees in Fano hypersurfaces in positive characteristic, challenging previous linear bounds.

Findings

Minimal degree of free curves cannot be linearly bounded in positive characteristic.

Super-linear bounds are achieved for Fermat hypersurfaces.

Linear bounds do not hold universally in positive characteristic settings.

Abstract

The purpose of this note is to show that the minimal $e$ for which every smooth Fano hypersurface of dimension $n$ contains a free rational curve of degree at most $e$ cannot be bounded by a linear function in $n$ when the base field has positive characteristic. This is done by providing a super-linear bound on the minimal possible degree of a free curve in certain Fermat hypersurfaces.