LitNote-Alexeev et al.-2019-Stable pair compactification of moduli of K3 surfaces of degree 2-AET19

- CiteKey: “AET19” - Type: report - Title: “Stable pair compactification of moduli of K3 surfaces of degree 2,” - Author: “Alexeev, Valery; Engel, Philip; Thompson, Alan;” - Publisher: “arXiv,” - Year: 2019 - Collections: “Affine Dynkin Project,”

Stable pair compactification of moduli of K3 surfaces of degree 2



We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable slc pairs \((X,\epsilon R)\) over the toroidal compactification associated to the Coxeter fan. One-parameter degenerations of K3 surfaces in this family are described by integral-affine structures on a sphere with 24 singularities.

Extracted Annotations

Annotations(5/31/2022, 1:20:45 AM)

File:AET19_VZ57UFDN.png (Alexeev et al., 2019, p. 2)

  • By Torelli it is a quasiprojective variety which is a global quotient. Discussion of the Baily-Borel compactification and toroidal compactifications in terms of an admissible fan. See effective divisor.

File:AET19_FM3GJTNA.png (Alexeev et al., 2019, p. 2)

  • Discussion of the slc compactification in terms of stable pairs – pairs with slc singularities and very ample divisor log canonical class.

File:AET19_9K8TJVPT.png (Alexeev et al., 2019, p. 2)

  • Motivating question:  the boundary of BB and toroidal compactifications are easy to describe but not modular, while the slc compactification is modular but not easy to describe. Are there comparison maps?

File:AET19_YX3IW9IN.png (Alexeev et al., 2019, p. 2)

File:AET19_PLRBXLE3.png (Alexeev et al., 2019, p. 2)

  • Main result, part 1. See K3 surface, toroidal compactification, admissible fan.

File:AET19_AMKHLRNT.png (Alexeev et al., 2019, p. 2)

File:AET19_EUPN7SL3.png (Alexeev et al., 2019, p. 3)

File:AET19_ETANADDW.png (Alexeev et al., 2019, p. 3)

File:AET19_SI987AAV.png (Alexeev et al., 2019, p. 4)