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It is well-known that there exist rational curves on Fano manifold. The only existing proof is due to Mori. His proof uses geometry of characteristic p.

My question is: For a hypersurface of degree $\leq n$ in $CP^n$, it is Fano by the adjunction formula. Does there exist an "elementary" proof of existence of rational curves on such hypersurface? In this direction, the only result I know is Cayley and Salmon's theorem about 27 lines on cubic surface.

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It is elementary to see that these hypersurfaces all contain lines; see for instance Harris' "First course" book. The interesting part of Mori's result is that they are actually covered by rational curves. I don't know of a proof other than the Frobenius trick, but I'm certainly not up to date on this. – Jack Huizenga Aug 10 '11 at 9:58
Thank you for the comments. – Jun Li Aug 10 '11 at 10:19
Jack -- It is possible to give an elementary proof that a hypersurface in $\mathbb{P}^n$ of degree $d < n$ is uniruled by lines, cf. Exercise V.4.4.3, p. 269, of Koll&aacute;r's "Rational curves on algebraic varieties". – Jason Starr Aug 10 '11 at 13:28
Good to know, thanks Jason! – Jack Huizenga Aug 10 '11 at 14:04
up vote 3 down vote accepted

I don't know if there is an elementary proof out there which works for all hypersurfaces of degree $d\ge n$, but if $X$ is general, you can see quite easily that they contain projective lines - or equivalently, the Fano variety of lines is of positive (expected) dimension. This is proved in Joe Harris' algebraic geometry book.

In fact, the following theorem is proved in M. Hochster and D. Laksov, The Linear Syzygies of Generic Forms, Comm. Algebra 15 (1987) and the proof is elementary. Here $F_k(X)$ denotes the Fano variety of $k-$planes in $X$ (thus if $F_1(X)$ is non-empty then your $X$ contains lines.

Theorem. For $n\ge k\ge 0$ and $d \ge 3$, let $\phi= (k + 1)(n − k)-{k+d > \choose d}$.

a) if $\phi <0$ the subvariety of hypersurfaces that contain a $k$-plane has codimension $−\phi$.

b) For $\phi=0$ every hypersurface of degree $d$ in $P^n$ contains a k-plane, and a general hypersurface contains a positive number of k-planes.

c) For $\phi>0$, a hypersurface $X$ has $\dim(F_k(X))$ \ge \phi$ with equality for general X.

See also Joe Harris' algebraic geometry book and Alex Waldron's thesis and the references given there.

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Thank you very much – Jun Li Aug 10 '11 at 10:19
As Ottem says, many more hypersurfaces than simply Fano hypersurfaces contain lines. But only Fano hypersurfaces are uniruled (in fact rationally connected). There are techniques for proving this for Fano hypersurfaces without using positive characteristic. Also uniruledness is a closed property in families of smooth, projective varieties. So once you prove this for a "generic" Fano hypersurface, you have also proved uniruledness for every Fano hypersurface. – Jason Starr Aug 10 '11 at 13:24

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