Also I realized that my first sentence "This is only the case for $m$ large" is misleading: I believe it is open for $d$ from $n$ to $\frac{3n}{2}$ that the degrees of rational curves on hypersurfaces are bounded.

(continued) There is a calculational error in this answer that gives you the wrong expectation (this heuristic should work, at least for large n): For $dm+1>(n+1)(m+1)$ to hold, you need $d>\frac{nm+n+m}{m}=\frac{n+1}+\frac{n}{m}$, so for $m=1$ you get $d>2n+1.$ I'm pretty sure there's some error here in the constant term, but I'm not sure what; in any case, you will get something with a $2n$ term. Also I realized that I forgot to say that everything I said was for a generic hypersurface; none of this works at all if your hypersurface is not generic.

(continued) rational curves in $P^n$ and hypersurfaces. For example you don't even know the dimension of this incidence correspondence. The way Clemens's result is proven is by looking at your hypersurfaces as hyperplane sections of higher-dimensional hypersurfaces, arguing that the rational curves from these sections must cover this higher dimensional hypersurface, and then using that rational curves cannot cover a Calabi-Yau variety. The proof (in "Curves on generic hypersurfaces") is relatively short but not completely trivial.

This is only the case for $m$ large. In fact for $d < 2n-1$ there will always be lines on your hyper surface. It is nontrivial that for $d \geq 2n-1$ there are no rational curves on hypersurfaces. This was first shown by Clemens, see for example Claire Voisin's paper "On a conjecture of Clemens on rational curves on hypersurfaces." (There are later contributions trying to bound the degree of rational curves that can appear for hypersurfaces with $d<2n-1.$) The reason why the naive argument fails is that it is very hard to control the behavior of the natural incidence correspondence between

See math.stackexchange.com/questions/348105/…

What are you denoting by $W$?

As far as I can tell, this is the same question as mathoverflow.net/questions/102839/….

I usually take $\phi_*(K_X)=(K_Y)$ as part of the G-R vanishing theorem statement. Actually though, I think G-R usually has stronger conditions than the ones you assume...

I think the key words here are "Grauert-Reimenschneider vanishing theorem."

The splitting basically comes from the theory of Dedekind domains: the function field of an affine curve over an algebraically closed field is a Dedekind domain, and you have such a splitting for any Dedekind domain. The case of a projective curve reduces to the case of an affine curve because the problem is local.