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Compactifications of moduli of elliptic K3 surfaces: Stable pair and toroidal

Valery Alexeev, Adrian Brunyate and Philip Engel

Geometry & Topology 26 (2022) 3525–3588
DOI: 10.2140/gt.2022.26.3525
Abstract

We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal compactifications of the moduli space, one for the ramification divisor and another for the rational curve divisor.

In the course of the proof, we further develop the theory of integral–affine spheres with 24 singularities. We also construct moduli of rational (generalized) elliptic stable slc surfaces of types An, Cn and En.

Keywords
K3 surfaces, elliptic surfaces, moduli, KSBA compactification, stable pairs
Mathematical Subject Classification
Primary: 14J10
References
Publication
Received: 30 October 2020
Revised: 8 August 2021
Accepted: 6 September 2021
Published: 16 March 2023
Proposed: Dan Abramovich
Seconded: Gang Tian, Anna Wienhard
Authors
Valery Alexeev
Department of Mathematics
University of Georgia
Athens, GA
United States
Adrian Brunyate
Baltimore, MD
United States
Philip Engel
Department of Mathematics
University of Georgia
Athens, GA
United States