Featured Speaker: Philip Engel (UGA)

Title: Compactification of K3 moduli

Abstract: By the Torelli theorem, the moduli space [M] of lattice-polarized K3 surfaces is the quotient of a Hermitian symmetric domain by an arithmetic group. In this capacity, it has compactifications such as the Baily-Borel  [\overline{M}^{\rm BB}] and toroidal compactifications  [\overline{M}^F]  which depend on some choice of fan [F]. On the other hand, choosing canonically an ampledivisor on every such K3, one can build a compactification [\overline{M}^{\rm slc}] of so-called stable pairs. I will discuss joint work with V. Alexeev on how one proves that the normalization of [\overline{M}^{\rm slc}] is [\overline{M}^F]  for some choice of fan [F]. We will focus on the example of elliptic K3s, polarized either by either the trisection [R^{\rm ram}] of nontrivial -torsion or by [R^{\rm rc}]: The section plus the sum of the singular fibers.