Computational Algebra Seminar: Wilson -- Computing invariant global log canonical thresholds in Magma

Speaker: Andrew Wilson
Title: Computing invariant global log canonical thresholds in Magma
Time & Place: 3-4pm, Thursday 10 February, Carslaw 535

Abstract:
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.