Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 17 / 2 / b / Solution

In a local trivialization, identify the lines in each fibre with $U_i\times {\mathbb{P}}^1$. On overlaps glue byThis action is well defined because $g_{ij}(x)$ is invertible and scaling $v$ does not change the resulting line. It is holomorphic: in affine projective coordinates it is a ratio of holomorphic functions on the open set where its denominator is nonzero. The transition maps satisfy the cocycle condition and have holomorphic inverses. They give the holomorphic projectivization by lines its complex-manifold atlas, of dimension ${\dim }_{\mathbb{C}}X+1$, on the usual projective-bundle topology. Points over different base points are separated by base neighborhoods; points over the same base point are separated inside a common local product chart. A countable trivializing cover and the standard charts of the complex projective line give second countability.

The local projections agree, defining a holomorphic map $p:\mathbb{P}(E)\to X$ whose fibre is $\mathbb{P}(E_x)\cong {\mathbb{P}}^1$.

Define the relative tautological line bundle $S\subset p^{\ast }E$ by $S_l=l$. Its local construction is holomorphic, and its restriction to a fibre is ${\mathcal{O}}_{{\mathbb{P}}^1}(-1)$. The required relative hyperplane line bundle is thereforeThis fixes the lines convention for projectivization and the sign of the fibre degree.