K-stability and birational models of moduli of quartic K3 surfaces

Kenneth Ascher; Kristin DeVleming; Yuchen Liu

doi:10.1007/s00222-022-01170-5

K-stability and birational models of moduli of quartic K3 surfaces

Kenneth Ascher^*, Kristin DeVleming, Yuchen Liu

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

We show that the K-moduli spaces of log Fano pairs (P³, cS) where S is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily–Borel compactification of moduli of quartic K3 surfaces as c varies in the interval (0, 1). We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza–O’Grady’s prediction on the Hassett–Keel–Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of P³.

Original language	English (US)
Pages (from-to)	471-552
Number of pages	82
Journal	Inventiones Mathematicae
Volume	232
Issue number	2
DOIs	https://doi.org/10.1007/s00222-022-01170-5
State	Published - May 2023

Funding

We would like to thank Dori Bejleri, Justin Lacini, Zhiyuan Li, Andrea Petracci, David Stapleton, Xiaowei Wang, and Chenyang Xu for helpful discussions, and Yuji Odaka for useful comments. We thank the referee for their helpful comments and suggestions. The authors were supported in part by the American Insitute of Mathematics as part of the AIM SQuaREs program. Research of KA was supported in part by the NSF Grant DMS-2140781 (formerly DMS-2001408). Research of YL was supported in part by the NSF Grant DMS-2148266 (formerly DMS-2001317).

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General Mathematics

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abstract = "We show that the K-moduli spaces of log Fano pairs (P3, cS) where S is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily–Borel compactification of moduli of quartic K3 surfaces as c varies in the interval (0, 1). We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza–O{\textquoteright}Grady{\textquoteright}s prediction on the Hassett–Keel–Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of P3.",

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AU - DeVleming, Kristin

AU - Liu, Yuchen

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N2 - We show that the K-moduli spaces of log Fano pairs (P3, cS) where S is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily–Borel compactification of moduli of quartic K3 surfaces as c varies in the interval (0, 1). We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza–O’Grady’s prediction on the Hassett–Keel–Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of P3.

AB - We show that the K-moduli spaces of log Fano pairs (P3, cS) where S is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily–Borel compactification of moduli of quartic K3 surfaces as c varies in the interval (0, 1). We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza–O’Grady’s prediction on the Hassett–Keel–Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of P3.

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K-stability and birational models of moduli of quartic K3 surfaces

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