The KSBA compactification for the moduli space of degree two \(K3\) pairs
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 \(\overline{\mathcal P}_2\) the author constructs first a different compactification \(\mathcal P_2 \subset \widehat{\mathcal P}_2\) via GIT, then shows that \(\overline{\mathcal P}_2\) is obtained from \(\widehat{\mathcal P}_2\) by a flip along the semistable locus.NEWLINENEWLINENEWLINE Reviewer: Andreas Höring (Nice)NEWLINENEWLINENEWLINENEWLINEThis is a contribution to the long-standing problem of finding a geometrically meaningful compactification of moduli spaces of \(K3\) surfaces. In general the moduli spaces \({\mathcal F}_d\) of polarised \(K3\) surfaces have various compactifications, for instance Baily-Borel and toroidal compactifications, but it is not clear how to ascribe geometric meaning to the boundary points. For degree~\(2\) the GIT compactification of Shah parametrises Gorenstein degenerations with slc singularities, but this is not separated.NEWLINENEWLINENEWLINENEWLINEAnother approach is to look at pairs \((X,H)\) consisting of a variety \(X\) and an ample divisor \(H\), rather than \({\mathcal O}_X(H)\). This idea has been around for some time, as the author points out: examples include Alexeev's approach to the Voronoi compactification of the moduli of abelian varieties, and work of \textit{J. Kollár} and \textit{N. I. Shepherd-Barron} [Invent. Math. 91, No. 2, 299--338 (1988; Zbl 0642.14008)] on moduli of log general type surfaces. The KSBA compactification of the title is the compactification described by Kollár and Shepherd-Barron, and in a different context by Alexeev: particular cases can be traced back to the 1970s.NEWLINENEWLINENEWLINENEWLINEThe author studies this in Shah's case, obtaining a detailed description of the compactification \(\overline{\mathcal P}_2\) of the moduli space \({\mathcal P}_2\) of pairs \((X,H)\) with \(X\) a \(K3\) surface and \(H\) an ample divisor of degree~\(2\). This has a forgetful map to the Baily-Borel compactification \({\mathcal F}_2^*\) whose general fibre is \({\mathbb P}^2\).NEWLINENEWLINENEWLINENEWLINEThe situation is still not ideal. The procedure produces a Deligne-Mumford stack whose coarse moduli space has \(\overline{\mathcal P}_2\) as one of its components, but does not give a functorial description of \(\overline{\mathcal P}_2\) alone. However, what does hold is enough to supply some geometric meaning for the boundary components of \(\overline{\mathcal P}_2\), and hence of compactifications of \({\mathcal F}_2\).NEWLINENEWLINENEWLINENEWLINEThe main result is that the compactification \(\overline{\mathcal P}_2\) and the map to \({\mathcal F}^*\) exist, and that there are six irreducible boundary components in \({\mathcal P}_2\). These boundary components are quite different from one another: they have different dimensions (3, 4, 10, 12, 13 and 19) and correspond to geometrically different degenerations, which are described in detail in each case. For example, in the case of the 10-dimensional component, the degenerate fibre has a component whose normalisation is a degree~\(2\) del Pezzo surface, polarised by a double curve \(E\). It is associated with the root lattice \(E_7+D_{10}\), which is essentially \(H^2\) of the central fibre: this identifies the component of the Baily-Borel boundary over which this component lies. At the cusps of the modular curve in the Baily-Borel boundary, where instead of a type~II degeneration of \(K3\) surfaces one has a type~III degeneration, the curve \(E\) acquires a node.NEWLINENEWLINENEWLINENEWLINEThe author notes a slight overlap with, and makes some use of, the results of \textit{A. Thompson} [Trans. Am. Math. Soc. 366, No. 1, 219--243 (2014; Zbl 1303.14023)]: the two papers are, however, for the most part complementary, as Thompson is concerned with possible degenerations from the point of view of minimal model theory and does not need to keep track of the polarisation, or to attempt to construct global moduli as is done here.NEWLINENEWLINENEWLINENEWLINEIn fact, because this case is so well studied, the paper includes a large amount of detail and many illuminating remarks about the connections between different approaches (GIT, KSBA, Bail-Borel\dots) to constructing and compactifying moduli, which are likely to prove useful even in less well developed cases.NEWLINENEWLINENEWLINE Reviewer: G. K. Sankaran (Bath)
scientific article; zbMATH DE number 6549391