Singularities of the MMP

To get in the mood for my MMP talk on Monday, I decided to take a more detailed look at the singularities that naturally occur as a result of the basic operations of the MMP: canonical and terminal singularities, and their log versions. The definitions are in terms of discrepancy, which I’ve conveniently just spent three posts describing in the case of cyclic quotient singularities.

In low dimensions…

• the surface canonical singularities are exactly the Du Val singularities, which have a Dynkin classification via the McKay correspondence,
• there are no terminal surface singularities (i.e. terminal implies smooth for surfaces)
• the 3-fold canonical and terminal singularities are classified, with some surprises. For instance, the only cyclic quotient singularities that are terminal are those of the form $\frac{1}{r}(1,a-a)$, the proof of which depends on some cryptic applications of $L$-series.