Compactifications of moduli space of (quasi-)trielliptic K3 surfaces

2022/12/30 by Chen, Yitao, Wu, Haoyu, Yao, Hanyu
#14J15 #14J28 #14J70 #14L24 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2212.14635

Abstract

We study the moduli space F_T1 of quasi-trielliptic K3 surfaces of type I, whose general member is a smooth bidegree (2,3)-hypersurface of ℙ¹× ℙ². Such moduli space plays an important role in the study of the Hassett-Keel-Looijenga program of the moduli space of degree 8 quasi-polarized K3 surfaces. In this paper, we consider several natural compactifications of F_T1, such as the GIT compactification and arithmetic compactifications. We give a complete analysis of GIT stability of (2,3)-hypersurfaces and provide a concrete description of the boundary of the GIT compactification. For the Baily--Borel compactification of the quasi-trielliptic K3 surfaces, we also compute the configurations of the boundary by classifying certain lattice embeddings. As an application, we show that (ℙ¹× ℙ²,εS) with small ε is K-stable if S is a K3 surface with at worst ADE singularities. This gives a concrete description of the boundary of the K-stability compactification via the identification of the GIT stability and the K-stability. We also discuss the connection between the GIT, Baily--Borel compactification, and Looijenga's compactifications by studying the projective models of quasi-trielliptic K3 surfaces.

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• Compact moduli of elliptic K3 surfaces
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