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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 $X$,

  1. the sheaf $F(n):=F\otimes \mathcal{O}_X(n)$ is generated by its sections and
  2. 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.

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    I'm not sure how "new" (or suitable) this question is for the site. In any case it's very useful to browse the long list of questions you'll find if you search for "ample line bundle". The notion itself is somewhat abstract but comes up in a lot of specific settings such as abelian varieties and flag varieties. –  Dec 04 '13 at 20:47
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    Your statement of Serre's theorem isn't quite right: the correct version says that given $F$, then for sufficiently large $n$ (depending on $F$), your statements 1 and 2 are true. Unfortunately I don't have anything to say about intuition for these facts. –  Dec 05 '13 at 12:45

1 Answers1

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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 $\mathcal F = m_{p_1}^{\otimes d_1} \otimes m_{p_2}^{\otimes d_2} \otimes \cdots \otimes m_{p_k}^{\otimes d_k}$, 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 (Degree of the polynomial has to satisfy $n\ge d_1+d_2\dots+d_k$).

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.

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