New techniques in resolution of singularitiesMPIM
by
MPIM, Vivatsgasse, 7 - Lecture Hall
Max Planck Institute for Mathematics
MPI-Oberseminar
Since Hironaka's famous resolution of singularities in characteristics zero in 1964, it took about 40 years of intensive work of many mathematicians to simplify the method, describe it using conceptual tools and establish its functoriality. However, one point remained quite mysterious: despite different descriptions of the basic resolution algorithm, it was essentially unique. Was it a necessity or a drawback of the fact that all subsequent methods relied on Hironaka's ideas essentially?
The situation changed in the last decade, when a logarithmic, a weighted and a foliated analogues and generalizations were discovered in works of Abramovich-Temkin-Wlodarzcyk, McQuillan, Quek, Abramovich-Temkin-Wlodarzcyk-Belotto and others. At this stage we can already try to figure out general ideas and principles shared by all these methods and the picture is quite surprising -- it seems that each method is quite determined by its basic setting consisting of the class of geometric objects and basic blowings up one works with. In particular, the classical method is probably the only natural resolution (via principalization) method obtained by blowing up smooth centers in the ambient manifold.
In my talk I'll describe the settings and the methods on a very general level. If time permits I will add some details about the simplest dream (or weighted) method, which has no memory and improves the singularity invariant by each weighted blowing up. Thus, the algorithm becomes simplest possible and the (modest) price one has to pay consists of extending the setting of varieties (or schemes) and blowings up of smooth centers to the setting of orbifolds and blowings up weighted centers.