= Solution
In a local trivialization, identify the lines in each fibre with $U_i\times\mathbb P^1$. On overlaps glue by
$$
(x,[v])\longmapsto(x,[g_{ij}(x)v]).
$$
This 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> $S\subset p^*E$ 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 therefore
$$
\boxed{L=S^*,\qquad L|_{p^{-1}(x)}\cong\mathcal O_{\mathbb P^1}(1).}
$$
This fixes the lines convention for projectivization and the sign of the fibre degree.
Back to article page
