Karol Palka

Title: Almost minimal log surfaces and the logMMP with half-integral coefficients.
 
Abstract: 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