# A question about a definition of the Hilbert scheme $\mathcal{H}_P ( \mathbb{P}^n )$.

First of all, i'm sorry for my bad english. I'm from a foreign country. :-)

I have some questions about a paragraph appearing in page : $205$ of the following electronic textbook : https://scholar.harvard.edu/files/joeharris/files/000-final-3264.pdf

• The paragraph says :

Let $S = K[x_0 , \dots , x_n]$ be the homogeneous coordinates ring of : $X = \mathbb{P}^n$.

The Hilbert scheme $\mathcal{H}_P (X)$ is constructed as a subscheme of the Grassmannian of $P(d)$ - dimentional subspaces of $S_d$, the space of homogeneous forms of degree $d$, for suitably large $d$.

• Question :

I don't understand properly this definition above of $\mathcal{H}_P (X)$, for $X = \mathbb{P}^n$. if we believe in this definition, does it-mean that $\mathcal{H}_P (X)$ is possibly constructed on the one hand as a subscheme of the Grassmannian of $P(d_1 )$ - dimentional subspaces of $S_{d_{1}}$, the space of homogeneous forms of degree $d_1$, for suitably large $d_1$, and in the same time, as a subscheme of the Grassmannian of $P(d_{2})$ - dimentional subspaces of $S_{d_{2}}$, the space of homogeneous forms of degree $d_{2}$, for suitably large $d_{2}$, such that : $d_1 \neq d_2$ ?

• If we assume that $\mathcal{H}_P (\mathbb{P}^n ) \subset G(P(d_1 ) , S_{d_{1}} )$ and $\mathcal{H}_P (\mathbb{P}^n ) \subset G(P(d_2 ) , S_{d_{2}} )$, and since : $G(P(d_1 ) , S_{d_{1}} ) \cap G(P(d_2 ) , S_{d_{2}} ) = \emptyset$, so $\mathcal{H}_P ( \mathbb{P}^n )$ doesn't exist in this case, no ? – YoYo Apr 21 '18 at 11:33