I'm trying to understand the following extract:
A holomorphic foliation by curves $\mathcal{F}$ of degree $r$ on the projective space is a bundle map $\Omega : H_{-r+1} \rightarrow T\mathbb{P}^n$, from the line bundle of Chern class $-r+1$ to the tangent bundle of $\mathbb{P}^n$, defined up to multiplication by a non-zero scalar.
This means that $\mathcal{F}$ may be defined by a family of holomorphic vector fields $\{X_\alpha\}$ on an open cover $\{U_\alpha\}$ of $\mathbb{P}^n$, satisfying the conditions $$X_\alpha = f_{\alpha\beta}X_\beta$$ on $U_\alpha\cap U_\beta$, such that the line bundle $H_{r-1}$ is the one associated with the $1$-cocycle of holomorphic and non-vanishing functions $f_{\alpha\beta}\in \mathcal{O}^*(U_\alpha\cap U_\beta)$.
The extent of my knowledge is basically that $H_{r-1}$ is $\mathcal{O}(-1)^{\otimes(-r+1)}$ where $\mathcal{O}(-1)$ is the tautological line bundle on $\mathbb{P}^n$.
I can see that the cocycle associated with $H_{r-1}$ can be interpreted to be a collection of non-vanishing complex-valued functions on open subsets of $\mathbb{P}^n$ (instead of the usual isomorphism-valued functions) given that the fibers are $1$-dimensional. So an expression like $X_\alpha = f_{\alpha\beta}X_\beta$ makes sense.
However, I can't see how to define the $X_\alpha$ from the map $\Omega$, or $\Omega$ from the $X_\alpha$.