For large $M$, the set of subvarieties of $\mathbb{P}^n$ with fixed Hilbert polynomial $P$ embeds naturally into $\operatorname{Gr}(K, N)$ for $K, N$ sufficiently large. The image is Zariski closed and independent of the choice of $M$ (for $M$ large enough). The resulting projective variety is the **Hilbert scheme** $\operatorname{Hilb}^P(\mathbb{P}^n)$.