Exercise II7.9 of Hartshorne's Algebraic Geometry

Let $X$ be a regular Noetherian scheme, and $\mathcal E$ a locally free coherent sheaf of rank $\geq 2$ on $X$.

(a) Show that $\text{Pic } \mathbb P(\mathcal E) \cong \text{Pic } X \times \mathbb Z$.

(b) If $\mathcal E'$ is another locally free coherent sheaf on $X$, show that $\mathbb P(\mathcal E) \cong \mathbb P(\mathcal E')$ (over $X$) if and only if there is an invertible sheaf $\mathscr L$ on $X$ such that $\mathcal E' \cong \mathcal E \otimes \mathscr L$.

I wanted to mark my problems in color so I didn't type everything but made pictures. My problems are:

1. Is this part (in the red brackets, from "Pick" to "$\text{Pic } \mathbb P(\mathcal E)$") necessary in the proof? As far as I see, the next two chapters really prove the injectivity and surjectivity of $\alpha$, showing that it is indeed an isomorphism.
2. Why does $\text{rank } \mathcal E \geq 2$ imply $n=0$?
3. How can I get $\mathbb P(\mathcal E|_{U_i}) \cong U_i \times \mathbb P^{r-1}$ from the fact that $\mathcal E|_{U_i}$ is trivial on $U_i$?
4. Ex II6.1 only gives the isomorphism of Cl-groups. In order to get $\text{Pic }V_i \cong \text{Cl }V_i$, we need $V_i$ to be locally factorial (II Cor6.16). (Similar for $U_i$.) Why are they locally factorial? Or how can we get the isomorphism without requiring them to be locally factorial?
5. Why does the restriction of $\mathscr L$ have the form $\mathcal O_i(n_i)\otimes \pi_i^*\mathscr L_i$?
6. Still about ranks. How does $n_i=n_j$ come from the consideration about ranks?
7. Is there any detailed explanation about the process from $$\mathcal O_{ij} \otimes \pi_i^* \mathscr L_i|_{V_{ij}} \cong \mathcal O_{ij} \otimes \pi_j^*\mathscr L_j|_{V_{ij}}$$ to $$\mathscr L_i|_{U_{ij}} \cong \mathscr L_j|_{U_{ij}} ?$$
8. Why is it $\mathcal O(1)$ here (right hand side of the equation)? Is it still about the ranks?
9. Why is it true that $\pi_*'(\mathcal O'(1)) = \pi_*(\mathcal O(1) \otimes \pi^* \mathscr L)$?

I am still have hard times reading this book and doing the exercises. So thanks for any help.

• This is a pretty long list of questions... – Cantlog Nov 2 '13 at 16:10
• @Cantlog I am sorry, but it is really not easy to split it up. – sunkist Nov 3 '13 at 7:27