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Consider the following theorem.

For every polynomial $P$, there exists an integer $m_P$ such that for all ideal subsheaves $I \subset \mathcal{O}_{\mathbb{P}^n}$ with Hilbert polynomial $P$ and every $k \ge m_P$, we have the following three things.

  1. There is no higher cohomology, i.e., $h^i(\mathbb{P}^n, I(k)) = 0$ for all $i > 0$.
  2. $I(k)$ is generated by global sections.
  3. The multiplication map$$H^0(\mathbb{P}^n, I(k)) \otimes H^0(\mathcal{O}_{\mathbb{P}^n}(1)) \to H^0(\mathbb{P}^n, I(k+1))$$is surjective.

I have two questions.

  1. What is the intuition behind this theorem?
  2. What are some key examples to have in mind when thinking about this theorem?
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    $\begingroup$ It says that all varieties $X$ with a fixed Hilbert polynomial $P$ have an ideal generated in degrees bounded by a number depending only on $P$. In other words, you can recover $X$ by knowing how finitely many vector spaces $H^0(I_X(d))$ embed into $H^0(P^n,O(d))$. This sets up an embedding the Hilbert scheme of $P$ into a flag variety. $\endgroup$ – gsvr Aug 26 '15 at 11:45

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