NOTICE: This is not a question about research.
Hi guys. 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.
There is something I don't understand and that of course some algebraic geometers can explain me.
(i) The authors use the term generic pencil when they define the degree. What is the precise meaning of the word generic? Where can I find more about the definition of degree of hypersurfaces in grasmannians?
(ii) To prove that the restriction of the sheaf $\mathcal{O}_{G(k,n)}(1)$ to $P_{NM}$ is isomorphic to $\mathcal{O}_{\mathbb{P}^1}(1)$ the author claim that it suffices to prove that any section of $\mathcal{O}_{G(k,n)}(1)$ has a exactly a zero on $P_{NM}$. Why?