Question

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.

Answer 1

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.