Rational curves on Fermat hypersurfaces

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 $N_0> p^r(p^r-1)$.