I have a little problem to fully understand the next thing:

Let $F:\mathbb{P}^2 \dashrightarrow \mathbb{P}$ a rational map. Let $\sigma:\bar{\mathbb{P}}^2 \rightarrow \mathbb{P}^2$ be a composition of $\sigma$-processes resolving the indeterminacy points of the rational map $F$, so that $\bar{F} = F\circ \sigma$ is a morphism.

We denote by $\bar{Y}_z$ the fibers of the morphism $\bar{F}$. Let $\sigma^{-1}(L_{\infty}) = \bar{S}_{\infty} \cup \bar{Y_{\infty}} \cup(\bigcup L_i)$, where $\infty = \mathbb{P}^1 \setminus \mathbb{C}^1$, the curves $L_i$ are components of the fibers of $\bar{F}$, and the morphism $\bar{F}$ maps each of the irreducible components of the curve $\bar{S}_{\infty}$ onto $\mathbb{P}^1$.

My question is. What are exactly the sets $\bar{S}_{\infty}$, $\bar{Y_{\infty}}$ and $L_i$?.

[Edited] I forgot to define $L_{\infty} = \mathbb{P}^2 \setminus \mathbb{C}^2$

Thank you in advance.