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 sncdivisor, 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 Qdivisor. Recently we were able to
adapt it to obtain progress in the classical subject of classification of planar
rational cuspidal curves.
