Abstract:

Building on the work of Brunella, McQuillan has recently obtained some strong results on the Minimal Model Program for algebraic foliations on algebraic surfaces, generalising most of the important results of the Castelnuovo-Enrique-Severi school to the study of foliations. In the case of foliations, the Abundance Conjecture fails already in dimension two because of the existence of a foliation with negative Kodaira dimension but positive numerical dimension.On the other hand, it is possible to characterise those foliations for which Abundance fails.

 

I will survey some recent results on the Minimal Model Program for foliations on algebraic surfaces.