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Birational geometry of the moduli space of quartic $K3$ surfaces

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

Published online by Cambridge University Press:  02 August 2019

Radu Laza  and
Kieran O’Grady
Radu Laza
Affiliation:
Stony Brook University, Stony Brook, NY 11794, USA email [email protected]
Kieran O’Grady
Affiliation:
‘Sapienza’ Universitá di Roma, Rome, Italy email [email protected]
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Abstract

By work of Looijenga and others, one understands the relationship between Geometric Invariant Theory (GIT) and Baily–Borel compactifications for the moduli spaces of degree-$2$ $K3$ surfaces, cubic fourfolds, and a few other related examples. The similar-looking cases of degree-$4$ $K3$ surfaces and double Eisenbud–Popescu–Walter (EPW) sextics turn out to be much more complicated for arithmetic reasons. In this paper, we refine work of Looijenga in order to handle these cases. Specifically, in analogy with the so-called Hassett–Keel program for the moduli space of curves, we study the variation of log canonical models for locally symmetric varieties of Type IV associated to $D$-lattices. In particular, for the $19$-dimensional case, we conjecturally obtain a continuous one-parameter interpolation between the GIT and Baily–Borel compactifications for the moduli of degree-$4$ $K3$ surfaces. The analogous $18$-dimensional case, which corresponds to hyperelliptic degree-$4$ $K3$ surfaces, can be verified by means of Variation of Geometric Invariant Theory (VGIT) quotients.

Keywords

moduliperiodsK3 surfaces

MSC classification

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles
Secondary: 14J28: $K3$ surfaces and Enriques surfaces
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 9 , September 2019 , pp. 1655 - 1710
Copyright
© The Authors 2019 

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Footnotes

Research of the first author is supported in part by NSF grants DMS-125481 and DMS-1361143. Research of the second author is supported in part by PRIN 2013.

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