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 $e\ge N_0$ there is a degree $e$ very free rational curve on $X$, then $N_0>p^r(p^r-1)$.

4 pages

References in corpus (1)

• Fano Hypersurfaces in Positive Characteristic

Cited by in corpus (1)

• Free and Very Free Morphisms into a Fermat Hypersurface