Compactifications of the moduli space of plane quartics and two lines
type: Item
instance of: scholarly article
title: Compactifications of the moduli space of plane quartics and two lines
zbMATH Open document ID: 1427.14097
author: Patricio Gallardo
author: Jesus Martinez-Garcia
author: Zheng Zhang
published in: European Journal of Mathematics
publication date: 2019-06-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1708.08420
review text: The aim of the paper under review is to study the moduli space of \(K3\) surfaces of degree \(2\) that are double covering of the projective space branched at the union of a quartic curve and two lines. The construction here is given in two ways that are finally turned to coincide. The first part is devoted to study the space of semi-/stable conditions for a tuple of a hypersurface and \(k\) hyperplanes in the projective space as a generalisation of the result obtained by the first two authors. Applying this, they describe the moduli space as the GIT quotient for triples of a quartic curve and two lines. As a result, they obtain an isomorphic period map constructed in Section 3.6. In the second part, they construct the moduli space as that of polarised \(K3\) surfaces by the lattice \(M\) presented in Notation 3.3. The strategy is to first describe the period map for generic intersections, and then, by using the method of the normalised-lattice polarisation for degree-5 pairs due to \textit{R. Laza} [Trans. Am. Math. Soc. 361, No. 4, 2109--2161 (2009; Zbl 1174.14004)] they construct a period mapping to be a morphism, specifically to describe the extension at non-generic intersections.
reviewed by: Makiko Mase
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 32G20
Mathematics Subject Classification ID: 14Q10
zbMATH DE Number: 7073897
zbMATH Keywords: \(K3\) surfaces
zbMATH Keywords: variations of GIT quotients
zbMATH Keywords: period map
zbMATH Keywords: quartic curves
describes a project that uses: vgit
MaRDI profile type: Publication
OpenAlex ID: W3101907511
arXiv ID: 1708.08420
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Wikidata QID: Q129947078
DOI: 10.1007/S40879-018-0248-7
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description: scientific article