Plane Cremona maps, exceptional curves and roots

We address three different questions concerning exceptional and root divisors (of arithmetic genus zero and of self-intersection $-1$ and $-2$, respectively) on a smooth complex projective surface $S$ which admits a birational morphism $\pi$ to $\mathbb{P} ^{2}$. The first one is to find criteria for the properness of these divisors, that is, to characterize when the class of $C$ is in the $W$-orbit of the class of the total transform of some point blown up by $\pi$ if $C$ is exceptional, or in the $W$-orbit of a simple root if $C$ is root, where $W$ is the Weyl group acting on $\operatorname{Pic}S$; we give an arithmetical criterion, which adapts an analogous criterion suggested by Hudson for homaloidal divisors, and a geometrical one. Secondly, we prove that the irreducibility of the exceptional or root divisor $C$ is a necessary and sufficient condition in order that $\pi_{\ast} (C)$ could be transformed into a line by some plane Cremona map, and in most cases for its contractibility. Finally, we provide irreducibility criteria for proper homaloidal, exceptional and effective root divisors.