The minimal model program (MMP) in Algebraic Geometry is a (almost complete) method for finding in each birational class a good representative. Singularities naturally crop up even if one starts from a smooth variety and computes its minimal model. We'll briefly describe the study of such singularities with respect to the (log) MMP to provide some background / motivations for examining the log canonical threshold (lct), a numerical invariant describing the severity of these singularities.

In Magma, I've been working on a package that will aid in the computation of lct globally on Fano G-varieties X, where G is a finite subgroup of Aut(X). The computation involves examining the splitting of the G-action on the Riemann-Roch space of a (pluri-)anti-canonical divisor to find the invariant parts of the (pluri-)anti-canonical linear system and then resolving the singularities of the worst of these parts to calculate their lct, all of which I'll illustrate with some examples. If there's time at the end we'll look at the correspondences between the global lct on Fano G-varieties and Kähler geometry, quotient singularities and with studying conjugacy classes in higher rank Cremona groups.