7
$\begingroup$

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?
Improve this question
$\endgroup$
1
  • 2
    $\begingroup$ 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, 2015 at 11:45

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.