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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 π to 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 π 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 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 π*(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.

Plane Cremona maps, exceptional curves and roots