Say $F$ is a vector bundle of rank $r+1$ on some scheme $X$ with transition maps (cocycle) $\psi_{ij}$ (with respect to some open cover $U_i, U_j,\ldots, U_k$ of $X$). We denote by $\mathbb{P}(F)$ the associated projective bundle. Locally, $$\mathbb{P}(F)|_{U_i}\simeq U_i\times\mathbb{P}^r$$ and these local patches are glued together by the same cocycle $\psi_{ij}$.
I want to understand the line bundles $O_{\mathbb{P}(F)}(d)$ on $\mathbb{P}(F)$. (i.e. its cocycle and sections)
For simplicity, say $d=-1$. Over $\mathbb{P}^r$, the line bundle $O(-1)$ has cocycle $g_{ij}$ given by $g_{ij}(x)=x_i/x_j$, where $[x]=[x_0,\ldots,x_r]\in\mathbb{P}^r$ are homogeneous coordinates.
Let $V_i=U_i\times\mathbb{P}^r$ denote the local patches of $\mathbb{P}(F)$. Then, locally over $V_i$ we have
$$O_{\mathbb{P}(F)}(-1)|_{V_i} \simeq V_i\times\mathbb{A}^1$$
and we want to find the transition functions $\xi_{ij}$ which give the glueing
$$(V_i\cap V_j)\times \mathbb{A}^1 \to (V_i\cap V_j)\times \mathbb{A}^1$$ $$(p,t) \mapsto (p, \xi_{ij}(p) t)$$
So we have $\xi_{ij} \colon (U_i\cap U_j)\times\mathbb{P}^r \to k^\times$ sending $(u,[x])\mapsto \xi_{ij}(u,[x])$.
What is the right choice for $\xi_{ij}$?
My very naive guess would be $\xi_{ij}(u,[x]) = g_{ij}(\psi_{ij}(u)\cdot[x])$. But I guess this is not correct.
I am in fact reading Gathmann's notes of Algebraic Geometry (p.189). He says
On the overlaps $U_i\cap U_j$ these line bundles [ $O_{\mathbb{P}(F)}(d)$ ] are glued by $\varphi\mapsto\varphi\circ\psi_{ij}$, where $\varphi = \frac{f}{g}$ is (locally) a quotient of homogeneous polynomials $f, g \in k[x_1 ,\ldots , x_r]$ with $\deg f − \deg g = d$
I do not clearly understand what it means. And how about sections? I would highly appreciate if somebody could elaborate a bit on this construction.