I was reading about the Hirzebruch surfaces construction to understand a few examples of Delzant construction, and need some help. I am using this reference Hirzebruck surfaces and can't understand the followinga few things about the final step: $P(L_{-n} \oplus \bar{\mathbb{C}}) $.
I understand the construction of $L_{-n}$ but is that a usual direct sum or what? Can't understand what is $\bar{\mathbb{C}}=X \times \mathbb{C}$ the "trivial complex line bundle", of what space? If someone can answer this questions perfect, or also a reference for me to read on my own would also be nice. Thanks!