# For all schemes w/Hilbert polynomial P, exists $m_P$ s.t. no higher cohomology, $I(k)$ generated by globally sections, multiplication is surjective

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?
• 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.
– gsvr
Aug 26, 2015 at 11:45