I'm studying Chow forms from the book Discriminants, Resultants, and Multidimensional Determinants of Andrei Zelevinsky and Izrail' Moiseevič Gel'fand. In order to introduce Chow forms the authors prove this proposition.
To be honest there is something I don't understand.
(1) When the authors define the degree of a surface in $G(k,n)$ they use the word generic pencil? What does this term mean?
(2) I know that the sheaf $\mathcal{O}(d)$ on $\mathbb{P}^n$ is the sheaf defined as $$\mathcal{O}(d)(U):=\lbrace f\colon p^{-1}(U)\longrightarrow \mathbb{C} \textit{ holomorphic and s.t. } f(tz)=t^df(z), \: \forall t\in\mathbb{C} \rbrace.$$
where $p$ is the usual projection $p\colon \mathbb{C}^{n+1}\setminus{0}\longrightarrow \mathbb{P}^N$.
How is defined the restriction of $\mathcal{O}(d)$ to $G(k,n)$? I would appreciate a lot if someone could explain me the proof of the equality $d=d'$ in the proposition below.
Thanks in advance.
- Proposition $1.6$
