The KSBA compactification for the moduli space of degree two \(K3\) pairs
type: Item
instance of: scholarly article
title: The KSBA compactification for the moduli space of degree two \(K3\) pairs
zbMATH Open document ID: 1375.14126
DOI: 10.4171/JEMS/589
author: Radu Laza
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2016-03-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1205.3144
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 6549391
zbMATH Keywords: \(K3\) surface
zbMATH Keywords: moduli space of \(K3\) surface
zbMATH Keywords: compactifications of moduli spaces
zbMATH Keywords: KSBA compactification
MaRDI profile type: Publication
arXiv ID: 1205.3144
OpenAlex ID: W2963545755
review text: Editorial remark: Accidentally, this article has been issued twice for reviewing. Therefore, we display both reviews, by adding the second one to the originally published review:NEWLINENEWLINENEWLINENEWLINEThe moduli space \(\mathcal F_2\) of complex \(K3\) surfaces of degree two is a locally symmetric variety \(\mathcal D/\Gamma_2\). There are several known compactifications of \(\mathcal F_2\), e.g. the Baily-Borel compactification \((\mathcal D/\Gamma_2)^*\), but in general the geometric meaning of the boundary components is not clear. In this paper the author follows a different approach: instead of considering couples \((X, \mathcal O_X(H))\) with \(X\) a \(K3\) surface and \(\mathcal O_X(H)\) a polarisation of degree two, one considers pairs \((X, H)\) where \(H \subset X\) is an effective ample divisor of degree two. The corresponding moduli space \(\mathcal P_2\) has a natural forgetful map \(\mathcal P_2 \rightarrow \mathcal F_2\) which is a \(\mathbb P^2\)-bundle. The main theorem of this paper is that there exists a geometric compactification \(\mathcal P_2 \subset \overline{\mathcal P}_2\) admitting an extended natural map \(\overline{\mathcal P}_2 \rightarrow (\mathcal D/\Gamma_2)^*\) to the Baily-Borel compactification. The compactification is geometric/modular in the sense that for each irreducible boundary component there is a description of the polarised pairs corresponding to its general points. The space \(\overline{\mathcal P}_2\) arises via the general framework of compactifications of moduli spaces of pairs of log general type introduced by Kollár, Shepherd-Barron and Alexeev (KSBA) [\textit{J. Kollár} and \textit{N. I. Shepherd-Barron}, Invent. Math. 91, No. 2, 299--338 (1988; Zbl 0642.14008); \textit{V. Alexeev}, in: Higher dimensional complex varieties. Proceedings of the international conference, Trento, Italy, June 15--24, 1994. Berlin: Walter de Gruyter. 1--22 (1996; Zbl 0896.14014)]. For the geometric description of \(\overline{\mathcal …
description: scientific article; zbMATH DE number 6549391