Karol Palka

If we want to study a quasiprojective complex surface Y, it is natural to take its completion (X,D), where X is projective, X\D=Y and D is a reduced snc-divisor, the curve 'at infinity', and to study the log minimal model program run for the pair (X,D). However, the theorems of the program have to be supported by a detailed study of the effect on the open part X\D. This theory of 'open surfaces' has been developed by Miyanishi, Fujita and others for more than 40 years and turned out to be very successful. Now the program itself, although without an obvious geometric interpretation, works equally well when D is a Q-divisor. Recently we were able to adapt it to obtain progress in the classical
subject of classification of planar rational cuspidal curves.