Rational curves on Fermat hypersurfaces

TL;DR

This paper investigates the behavior of rational curves on Fermat hypersurfaces in positive characteristic, revealing that Frobenius morphism significantly alters their properties compared to characteristic zero.

Contribution

It establishes a lower bound on the degree of very free rational curves on Fermat hypersurfaces in characteristic p, highlighting the impact of Frobenius morphism.

Findings

Existence of very free rational curves is constrained by degree in positive characteristic.

Frobenius morphism causes deviations from characteristic zero behavior.

Lower bound for degrees of rational curves depends on p^r.

Abstract

In this note we study rational curves on degree $p^r+1$ Fermat hypersurface in $\PP^{p^r+1}_k$, where $k$ is an algebraically closed field of characteristic $p$. The key point is that the presence of Frobenius morphism makes the behavior of rational curves to be very different from that of charateristic 0. We show that if there exists $N_0$ such that for all $e\geq N_0$ there is a degree $e$ very free rational curve on $X$, then $N_0> p^r(p^r-1)$.