nLab
hyperplane line bundle

Let ${\mathbb{P}}^n={\mathbb{P}}_k^n$ be the $n$-dimensional projective space over a field $k$, whose points are the equivalence classes $[z_0,\dots ,z_n]$ of $(n+1)$-tuples $(z_0,\dots ,z_n)\in k^{n+1}\backslash \{0\}$; the (co)domains of usual open charts in the sense of manifolds, which are Zariski-open subsets for general $k$, have $U_i=\{[z_0,\dots ,z_n]\mid z_i\ne 0\}$.

The hyperplane line bundle $𝒪(1)$ on ${\mathbb{P}}^n$ is a line bundle given by the transition functions $g_{\mathrm{ij}}([z_0,\dots ,z_n])=z_j/z_i$ on $U_i\cap U_j$. Its dual bundle $𝒪(1{)}^{*}$ is the tautological bundle (or universal bundle) usually denoted by $𝒪(-1)$ and the tensor powers are $𝒪(n)=𝒪(1{)}^{\otimes n}$, $𝒪(-n)=𝒪(-1{)}^{\otimes n}$ for $n\ge 0$. The total space of the tautological line bundle can be identified with ${ℂ}^{n+1}$ and the projection is exactly $(z_0,\dots ,z_m)\mapsto [z_0,\dots ,z_m]$, i.e. the fiber over $[z_0,\dots ,z_m]$ is the line $\{(\lambda z_0,\dots ,\lambda z_n),0\ne \lambda \in ℂ\}$. The canonical line bundle $K={\Lambda }^n{T}^{*}{\mathbb{P}}^n$ equals $𝒪(-n-1)$.

The bundles $𝒪(n)$ are holomorphic if $k=ℂ$. The sheaves of (regular or holomorphic) sections are also denoted as $𝒪(n)$ and are said to be the twists of the structure sheaf $𝒪$; they restrict to the equally denoted sheaves on any projective subvariety and these restrictions up to an isomorphism do not depend on a particular embedding into a particular projective space.