Free and very free morphisms into a Fermat hypersurface

Abstract

This paper studies the existence of free and very free curves on the degree $5$ Fermat hypersurface in ${\mathbb{P}}^5$ over an algebraically closed field of characteristic $2$. We explicitly compute a free curve in degree $8$, and a very free curve in degree $9$. We also prove that free and very free curves cannot exist in lower degrees.