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 $ ?
Thanks in advance for your help.