Next Meeting

Abstracts

Siao Chi Mok (Cambridge) - Logarithmic Fulton–MacPherson configuration spaces

The Fulton–MacPherson (FM) configuration space is a well-known compactification of the ordered configuration space of a projective variety. Given a semistable degeneration of X, we construct a log smooth degeneration of the FM space of X. This degeneration satisfies a “degeneration formula” — each irreducible component of its special fibre can be described as a proper birational modification of a product of log FM configuration spaces. These log FM spaces parametrise point configurations on certain target degenerations arising from both tropical/logarithmic geometry and the original Fulton–MacPherson construction.

Pierrick Bousseau (Oxford) - The KSBA moduli space of stable log Calabi-Yau surfaces

The KSBA moduli space of stable pairs (X,B), introduced by Kollár--Shepherd-Barron, and Alexeev, is a natural generalization of the moduli space of stable curves for higher dimensional varieties. This moduli space is described concretely only in a handful of situations. For instance, if X is a toric variety and B=D+\epsilon C, where D is the toric boundary divisor and C is an ample divisor, it was shown by Alexeev that the KSBA moduli space is a toric variety. More generally, for stable pairs of the form (X,D+\epsilon C) with (X,D) a log Calabi-Yau variety and C an ample divisor, it was conjectured by Hacking--Keel--Yu that the KSBA moduli space is still toric (up to passing to a finite cover). In joint work with Alexeev and Arguz, we prove this conjecture for all log Calabi-Yau surfaces. This uses tools from the minimal model program, log smooth deformation theory and mirror symmetry.

Sam Molcho (La Sapienza, Rome) - Intersection theory of degenerating abelian schemes

When studying complete families of abelian varieties, one naturally encounters their degenerations, which are certain toroidal compactifications of the family's associated semi-abelian locus. The intersection theory of such degenerating families is not well understood. In this talk, I will report on recent progress in the intersection theory of two central examples of such families: the universal compactified Jacobians over the moduli space of curves and the universal family over Mumford’s canonical partial compactification of Ag. I will describe the main tools underlying our approach -- degenerating versions of the Fourier transform and Beauville’s weight decomposition -- and present the resulting structural theorems. If time permits, I will discuss open problems and obstructions to extending these methods to more general families. This is based on joint work with Bae–Pixton and Bae–Feusi–Iribar López.