Intuition for ample line bundles Let $X\subset \mathbb{P}^N$ be a smooth projective variety over $\mathbb{C}$. We let $\mathcal{O}_X(n)$ denote the bundle induced by $\mathcal{O}_{\mathbb{P}^N}(n)$. For a coherent sheaf $F$ on $X$, we write $F(n):=F\otimes \mathcal{O}_X(n)$. A fundamental theorem of Serre says that for any coherent sheaf $F$ on $X$, 


*

*the sheaf $F(n):=F\otimes \mathcal{O}_X(n)$ is generated by its sections and

*we have $H^p(X,F(n))=0$ for $p>0$. 


I try to understand these claims by following proofs, but they are still beyond my intuition. Are there any good ways to understand these claims? I am aware that each claim characterizes ample line bundles. 
 A: Here's how I always think about 1).  Whether it's a good intuition is another question.
For simplicity, let's think of $\mathcal F$ as being the ideal sheaf of some closed subscheme.  The idea here isn't so far from the general case, I think.  In fact, for concreteness, let's take $X = \mathbb P^2$ and $\mathcal F = m_{p_1}^{\otimes d_1} \otimes m_{p_2}^{\otimes d_2} \otimes \cdots \otimes m_{p_n}^{\otimes d_n}$, where $m_{p_i}$ is the sheaf corresponding to some closed point.  Sections of $\mathcal F$ correspond to functions which vanish to order at least $d_i$ at each point $p_i$.
Now, $H^0(X,F(n))$ is given by polynomials of degree $n$ that have at least the order of vanishing prescribed by the sheaf $F$, i.e. polynomials that vanish to order $d_i$ at the points.  If $n$ is small, there aren't going to be any such things; vanishing at the points imposes a lot of conditions on polynomials, and unless the polynomial has large degree they can't all be satisfied.
On the other hand, once $n$ is big, there do exist polynomials satisfying the prescribed vanishing conditions, so that $H^0(X,F(n))$ has some sections.  Global generation means that sections generate in every stalk; for us, this means that if $n$ is large enough, for every $p \in \mathbb P^2$, there exists a global section (i.e. a polynomial of degree $n$ satisfying the vanishing conditions), which doesn't vanish to order at $p$ to any higher than it's required to.  But if you think about this in terms of polynomial interpolation, of course that can be arranged.
There should be a similar story for 2), though I haven't worked it out in concrete terms.  I'll think about it.
