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KSBA compactification of the moduli space of K3 surfaces with a purely non-symplectic automorphism of order four

Part of: Families, fibrations Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  12 April 2021

Han-Bom Moon  and
Luca Schaffler
Han-Bom Moon
Affiliation:
Department of Mathematics, Fordham University, New York, NY 10023, USA (hmoon8@fordham.edu)
Luca Schaffler
Affiliation:
Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden (lucsch@math.kth.se)
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Abstract

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with a purely non-symplectic automorphism of order four and $U(2)\oplus D_4^{\oplus 2}$ lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of $\mathbb {P}^{1}\times \mathbb {P}^{1}$ branched along a specific $(4,\,4)$ curve. We show that, up to a finite group action, this stable pairs compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient $(\mathbb {P}^{1})^{8}{/\!/}\mathrm {SL}_2$ with the symmetric linearization.

Keywords

K3 surfacestable pairmoduli spacecompactification

MSC classification

Primary: 14J10: Families, moduli, classification
Secondary: 14J28: $K3$ surfaces and Enriques surfaces 14D06: Fibrations, degenerations

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 1 , February 2021 , pp. 99 - 127
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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