Let $ \nu = \nu_{n,d} : \mathbb{P}^{n} \to \mathbb{P}^N $ be the Veronese map : $ [Z_0 , \dots , Z_n ] \to [ \ \dots , Z^I , \dots \ ] $ where : $ N = \begin{pmatrix} n+d \\ d \end{pmatrix} - 1 $, and $ Z^I $ ranges over monomials of degree $ d $ in $ n+1 $ variables. The image $ \Phi = \Phi_{n,d} \subset \mathbb{P}^n $ of the Veronese map $ \nu = \nu_{n,d} $ is called the $ d $ -th Veronese variety of $ \mathbb{P}^n $, as is any subvariety of $ \mathbb{P}^N $ projectively equivalent to it.
- Theorem : The degree of $ \Phi_{n,d} $ is $ d^n $
- Proof : The degree of $ \Phi $ is the cardinality of its intersection with $ n$ general hyperplanes $ H_1 , \dots ,H_n \subset \mathbb{P}^n $ ; since the map $ \nu $ is one-to-one, this is in turn the cardinality of the intersection $ \nu^{-1} (H_1 ) \cap \dots \cap \nu^{-1} (H_n ) \subset \mathbb{P}^n $. The preimages of the hyperplanes $ H_i $ are $ n $ general hypersurfaces of degree $d$ in $ \mathbb{P}^n $. By Bezout's theorem, the cardinality of their intersection is $d^n $.
- Question : In the proof above about how to find the degree of a Veronese variety $ \Phi_{n,d} $ which i found in a free text book on the net, i would like to ask you why is the degree of $ \Phi $ the cardinality of its intersection with $ n $ general hyperplanes $ H_1 , \dots ,H_n \subset \mathbb{P}^n $ ? Why do we introduce this machinery of $ n $ general hyperplanes $ H_1 , \dots ,H_n \subset \mathbb{P}^n $ in this story ? What does this machinery of $ n $ general hyperplanes $ H_1 , \dots ,H_n \subset \mathbb{P}^n $ represent when we talk about $ \Phi_{n,d} $ as a image of $ \mathbb{P}^n $ by the Veronese map $ \nu $ ?
Thanks in advance for your help.