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.

- There is no higher cohomology, i.e., $h^i(\mathbb{P}^n, I(k)) = 0$ for all $i > 0$.
- $I(k)$ is generated by global sections.
- 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.

- What is the intuition behind this theorem?
- What are some key examples to have in mind when thinking about this theorem?