---
- 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

## Meta

- **URL**: <http://arxiv.org/abs/1903.09742>
- **URI**: <http://zotero.org/users/1049732/items/GJIX3X8T>
- **Local File**: [Alexeev et al. - 2019 - Stable pair compactification of moduli of K3 surfa.pdf](file:///home/zack/Downloads/Zotero_Source/arXiv/2019/Alexeev%20et%20al.%20-%202019%20-%20Stable%20pair%20compactification%20of%20moduli%20of%20K3%20surfa.pdf); [arXiv.org Snapshot](file:///home/zack/Zotero/storage/AA8MUZGS/1903.html)
- **Open in Zotero**: [Zotero](zotero://select/library/items/GJIX3X8T)

## Abstract

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.

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## Extracted Annotations


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

-  Definition of moduli of polarized [[K3 surfaces]] in terms of [[ample]] line bundles.

![[AET19_VZ57UFDN.png]] [(Alexeev et al., 2019, p. 2)](zotero://open-pdf/library/items/74XP8EUZ?page=2&annotation=SAVEXAXR)

-  By [[Torelli]] it is a [quasiprojective](Unsorted/projective scheme.md) variety which is a global quotient. Discussion of the [[Baily-Borel]] compactification and [[Unsorted/toroidal compactifications]] in terms of an admissible [fan](Unsorted/toric varieties.md). See [[effective divisor]].

![[AET19_FM3GJTNA.png]] [(Alexeev et al., 2019, p. 2)](zotero://open-pdf/library/items/74XP8EUZ?page=2&annotation=556C5CPG)

-  Discussion of the [[slc compactification]] in terms of [[stable pairs]] -- pairs with [[slc]] singularities and [[ample]] [[log canonical class]].

![[AET19_9K8TJVPT.png]] [(Alexeev et al., 2019, p. 2)](zotero://open-pdf/library/items/74XP8EUZ?page=2&annotation=GZFRYYLE)

-  Motivating question:&nbsp; 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?

![[AET19_YX3IW9IN.png]] [(Alexeev et al., 2019, p. 2)](zotero://open-pdf/library/items/74XP8EUZ?page=2&annotation=GKJAERFT)

-  Motivation from [[PPAVs]], see [[Voronoi fan]] and [[theta divisor]].

![[AET19_PLRBXLE3.png]] [(Alexeev et al., 2019, p. 2)](zotero://open-pdf/library/items/74XP8EUZ?page=2&annotation=CM8KIDU2)

-  Main result, part 1. See [[K3 surface]], [[toroidal compactification]], [[admissible fan]].

![[AET19_AMKHLRNT.png]] [(Alexeev et al., 2019, p. 2)](zotero://open-pdf/library/items/74XP8EUZ?page=2&annotation=CHE5X8T8)

-  Main result, part 2. See [[Coxeter fan]], [[semitoric compactification]], [[stable pair compactification]], [[Stein factorization]], [[normalization]], and [[Dynkin diagrams]].

![[AET19_EUPN7SL3.png]] [(Alexeev et al., 2019, p. 3)](zotero://open-pdf/library/items/74XP8EUZ?page=3&annotation=ST7ECIM5)

![[AET19_ETANADDW.png]] [(Alexeev et al., 2019, p. 3)](zotero://open-pdf/library/items/74XP8EUZ?page=3&annotation=33YZA7TP)

![[AET19_SI987AAV.png]] [(Alexeev et al., 2019, p. 4)](zotero://open-pdf/library/items/74XP8EUZ?page=4&annotation=2NUPZPLC)