Here's the question:
R.Vakil, Exercise 18.4.L: Suppose $\mathcal{L}$ is a base-point free invertible sheaf on a proper variety $X$, and hence induces some morphism $\phi: X\rightarrow\mathbb{P}^n$. Then $\mathcal{L}$ is ample if and only if $\phi$ is finite.
So suppose that $\phi$ is finite, then by the hint in Ex 16.6.G, $\phi^{*}\mathcal{O}_{\mathbb{P}^n}(1)$ is ample and this is $\mathcal{L}$ (hope I am correct here).
But for the converse here is the hint:
If $\phi$ is not finite, show that there is a curve $C$ contracted by $\pi$, using theorem 18.1.9. Show that $\mathcal{L}$ has degree 0 on $C$.
I am completely lost here. What is $\pi$? So I looked up at the theorem and it says
Theorem 18.1.9 Suppose $\pi:X\rightarrow Y$ with $Y$ noetherian. Then $\pi$ is projective and has finite fibers if and only if it is finite. Equivalently $\pi$ is projective and quasifinite if and only if it is finite.
Question 1: How should I make sense of the ''contractible'' part and how do I use the theorem to prove it?
Question 2: Assume that the contractible part is done. I need to show that $\mathcal{L}$ has degree zero on $C$. Why? I tried Riemann-Roch but at least I can't get anything out of it. Any hints?
(I think the $\pi$ in the hint is just a typo for $\phi$.)
Note that for the direction you want to prove, we're assuming that $L$ is ample, which in particular implies that $X$ is projective.
Question 1: Since $X$ is projective, so is the morphism $\phi$. If $\phi$ isn't finite, Theorem 18.1.9 says it isn't quasifinite, so there must be some fibre $F$ of dimension $>0$. Then choose any curve $C \subset F$: this $C$ gets contracted by $\phi$.
Question 2: As you correctly observed, $L=\phi^* O(1)$. By definition of the pullback of line bundles, this means that $L$ is trivial when restricted to any fibre of $\phi$. Since $C$ is contained in a fibre, $L_{|C}$ is trivial.