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An important tool in studying open algebraic surfaces is the
logarithmic Minimal Model Program. Although in dimension two its general
mechanism is well understood, in concrete problems we need an analysis
which goes far beyond the general framework. For log surfaces with reduced
boundaries the usage of almost minimal models (due to Miyanishi, Fujita
and others), which allows to avoid singularities, turned out to be very
successful. Recently we generalized this approach to boundaries with
coefficient 1/2 and used it in particular in the proof of the
Coolidge-Nagata conjecture (2015, coauthored with M. Koras) concerning
cuspidal planar curves. We will discuss a more general form of this
approach showing how to apply it to other problems concerning surfaces of
log general type. References: [1] The Coolidge-Nagata conjecture, part I, http://www.sciencedirect.com/science/article/pii/S0001870814002783 [2] The Coolidge-Nagata conjecture, http://arxiv.org/abs/1502.07149 [3] Cuspidal curves, minimal models and Zaidenberg's finiteness conjecture, http://arxiv.org/abs/1405.5346 |