I am stuck at solving 20.2.I from Vakil's notes.
The set up is as follows: suppose $X$ is a scheme, $\mathcal{V}$ is a rank 2 locally free sheaf.
Then given a short exact sequence $$0 \to \mathcal{S} \to \mathcal{V} \to \mathcal{Q} \to 0 $$ with $\mathcal{S}, \mathcal{Q}$ being invertible sheaves, we obtain a section $$\sigma: X \cong \mathbb{P}(\mathcal{Q}) \to \mathbb{P}(\mathcal{V})$$ by projectivizing the sequence.
Remark. Vakil''s convention for projectivization is $$\mathbb{P}(\mathcal{V}) = \mathcal{Proj} (Sym^{\bullet} \mathcal{V})$$ without taking the dual (17.2.3).
The task is to show that the normal bundle $\mathcal{N}_{\sigma(X)/\mathbb{P}(\mathcal{V})}$ is isomorphic to $\mathcal{Q} \otimes \mathcal{S}^{\vee}$.
Following the hint, we can assume $\mathcal{S} \cong \mathcal{O}_X$.
To get a trivialization for $\mathcal{N}_{\sigma(X)/\mathbb{P}(\mathcal{V})}$, take set $U \subset X$ where $\mathcal{Q}$ is trivial. Then the exact sequence splits $$0 \to \mathcal{O}_U \to \mathcal{O}_{U}^{\oplus 2} \to \mathcal{O}_U \to 0,$$where the last map is just the projection onto the second factor and the section thus identifies with $$U \to U \times \mathbb{P}^1$$ $$p \mapsto (p,[0,1]).$$
We see that therefore the image is the locus $V(x_0)$ with $x_0 \in \Gamma(U, \mathcal{O}_{U \times \mathbb{P}^1}(1))$. Moreover the section factors through $U \times U_1$ ($U_1$ being the standard chart for projective line), and we can identify the conormal ideal as $$\frac{x_{0/1}\mathcal{O}_{U}[x_{0/1}]}{x_{0/1}^2\mathcal{O}_{U}[x_{0/1}]} \cong \mathcal{O}_{U}.$$
The last step in the hint is to take trivializing cover $\{U_i\}$ for $\mathcal{Q}$ with cocycle $g_{ij}$ and show that it also serves as a cocylce for the normal bundle.
\begin{array} 00 & \stackrel{}{\longrightarrow} & \mathcal{O}_{U_{ij}} & \stackrel{}{\longrightarrow} & \mathcal{O}_{U_{ij}}^{\oplus 2} & \stackrel{}{\longrightarrow} & \mathcal{O}_{U_{ij}} & \stackrel{}{\longrightarrow} 0\\ & & \uparrow{id} && \uparrow{} && \uparrow\\ 0 & \stackrel{}{\longrightarrow} & \mathcal{O}_{U_{ij}} & \stackrel{}{\longrightarrow} & \mathcal{V}|_{U_{ij}} & \stackrel{}{\longrightarrow} & \mathcal{Q}|_{U_{ij}} & \stackrel{}{\longrightarrow} 0\\ & & \downarrow{id} && \downarrow{} && \downarrow\\ 0 & \stackrel{}{\longrightarrow} & \mathcal{O}_{U_{ij}} & \stackrel{}{\longrightarrow} & \mathcal{O}_{U_{ij}}^{\oplus 2} & \stackrel{}{\longrightarrow} & \mathcal{O}_{U_{ij}} & \stackrel{}{\longrightarrow} 0. \end{array}
The outer right square is \begin{array} \mathcal{O}_{U_{ij}}^{\oplus 2} & \stackrel{}{\longrightarrow} & \mathcal{O}_{U_{ij}} \\ \uparrow{_{\begin{bmatrix} 1 & a_{ij} \\ 0 & g_{ij} \end{bmatrix}}} && \uparrow{g_{ij}} \\ \mathcal{O}_{U_{ij}}^{\oplus 2} & \stackrel{}{\longrightarrow} & \mathcal{O}_{U_{ij}} \end{array}
So it seems like when I take the corresponding projectivization
\begin{array} UU_{ij} \times \mathbb{P}^1 & \stackrel{}{\longleftarrow} & X \\ \downarrow{} && \downarrow{id} \\ U_{ij} \times \mathbb{P}^1 & \stackrel{}{\longleftarrow} & X \end{array} section $x_0 \in \Gamma(U_{ij} \times \mathbb{P}^1, \mathcal{O}_{U_{ij} \times \mathbb{P}^1}(1))$ downstairs pulls back to $x_0$ upstairs again, thus not producing interesting cocycle in the trivialization for the normal bundle. (And as far as I can tell, in order to get the correct cocylcle, we need $x_0$ to pullback to $g_{ij}^{-1} x_0$.)
Can someone point to a mistake in the argument above?
Typically, this is done much more directly as follows. If we have a non-zero section $\mathcal{O}_X\to V$, this gives a non-zero section of $\mathcal{O}_{P(V)}(1)$, using the surjection $\pi^*V\to\mathcal{O}_{P(V)}(1)$, where $\pi:P(V)\to X$ is the natural projection. If $Y\subset P(V)$ is the divisor where this section vanishes, it is clear that $N_{Y/P(V)}=\mathcal{O}_{P(V)}(1)|Y$. In your case, chasing diagrams, it is immediate that $\mathcal{O}_{P(V)}(1)|Y=\pi^*Q|Y$, where you have started with $0\to\mathcal{O}_X\to V\to Q\to 0$.