Plane Cremona maps, exceptional curves and roots |
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Authors: | Maria Alberich-Carramiñ ana |
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Institution: | Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal, 647, 08028-Barcelona, Spain |
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Abstract: | We address three different questions concerning exceptional and root divisors (of arithmetic genus zero and of self-intersection and , respectively) on a smooth complex projective surface which admits a birational morphism to . The first one is to find criteria for the properness of these divisors, that is, to characterize when the class of is in the -orbit of the class of the total transform of some point blown up by if is exceptional, or in the -orbit of a simple root if is root, where is the Weyl group acting on ; 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 is a necessary and sufficient condition in order that 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. |
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Keywords: | |
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