I'm reading now the book "Vector Bundles on Complex Projective Spaces" from Okonek, Schneider and Spindler and I have an understanding problem with the interpretation of a section in following excerpt (see also page 18):
Here we take an open set $U \subset \mathbb{C}^2$ and consider the "$\sigma$-process" for $U$ in $0$, better known as a blow up in $0$:
$$V := \{(u:v,x,y) \in \mathbb{P}^1 \times U \vert xv = yu\} \to U $$
Then the problem:
In the excerpt the authors introduce the section
$s \in H^0(V, \mathcal{O}_V ^{\oplus 2})$ by strange notation
$$s(u:v,x,y) = (u:v, x,x, (x,y))$$
My question is just what does $ (u:v, x,y, (x,y))$ here mean?
By definition, if $\mathcal{O}_V$ denotes the sheaf of holomorphic functions on $V$ then a section $s \in H^0(V, \mathcal{O}_V ^{\oplus 2})$ is a tuple $(s_1, s_2)$ where $s_i$ are holomorphic functions on $V$.
But I don't really understand the notation $s(u:v,x,y) = (u:v, x,y, (x,y))$.