Let $\mathbb{C} \mathbb{P}^1 $ the complex projective space and let consider the tautological bundle over $\mathbb{C} \mathbb{P}^1 $:
In the excerpt below (whole document: here ) the map belonging to tautological bundle of $\mathbb{C} \mathbb{P}^1 $ is defined as the canonical map $s: \mathbb{C}^2 \backslash \{0\} \to \mathbb{C} \mathbb{P}^1 $ which assoziates every $(x_0:x_1)$ in canonical way the fiber $s^{-1}(x_0:x_1) = \mathbb{C}*(x_0,x_1)$, therefore the corresponding one dimensional vector space.
My question is how does this constuction yields $\mathcal{O}(-1)$ as line bundle, therefore a twisted sheaf with empty global sections (because of $-1$)?