GIT versus Baily-Borel compactification for \(K3\)'s which are double covers of \(\mathbb{P}^1 \times \mathbb{P}^1\) (Q2020394)

The moduli space of degree 2 polarised \(K3\) surfaces has been compactified in various ways, including the Baily-Borel compactification and the compactification defined by Shah arising from the GIT moduli of plane sextics: Shah's work and subsequent work of Looijenga, Laza and Voisin explains very well how these two are connected. Already for degree 4 surfaces the picture is more complicated. The main difficulty is that Looijenga's compactifications depend on a hyperplane arrangement in the period domain and when these arrangements become more complicated, for instance with many intersections among the hyperplanes, the compactifications also become subtler. In their earlier work (e.g. [Compos. Math. 155, No. 9, 1655--1710 (2019; Zbl 1429.14009)]), the authors showed how to extend Looijenga's work and conjectured that there should be wall-crossing phenomena associated with a chain of birational transformations linking the (original) GIT compactification (essentially in this case, the moduli \(\mathfrak{M}\) of stable \((4,4)\)-curves in \(\mathbb{P}^1\times\mathbb{P}^1\)) to the Baily-Borel compactification \(\mathscr{F}^*\). This general idea is called by the authors the Hassett-Keel-Looijenga program, as it is similar in outline to the Hassett-Keel program in the context of moduli of curves. In this paper, they prove their conjectures in this case, providing a full example of the Hassett-Keel-Looijenga program in a wider context than previously. The main result says that if \(\mathscr{F}\) is the moduli space of polarised \(K3\) surfaces of degree 4, with \(\lambda\) the Hodge line bundle and \(\Delta\) the boundary, then for \(\beta\in [0,1]\cap\mathbb{Q}\) the graded ring \[ R(\mathscr{F},\lambda+\beta\Delta)=\bigoplus_k H^0(\mathscr{F},k(\lambda+\beta \Delta)) \] is finitely generated, so one may set \(\mathscr{F}(\beta)=\operatorname{Proj}R(\mathscr{F},\lambda+\beta\Delta)\) and then \(\mathscr{F}(0)=\mathscr{F}^*\) and \(\mathscr{F}(0)=\mathfrak{M}\). Then as \(\beta\) varies in \([0,1]\cap \mathbb{Q}\) there is a chamber decomposition in which the variety \(\mathfrak{M}(\beta)\) jumps at the critical values \(0\), \(\frac{1}{8}\), \(\frac{1}{6}\), \(\frac{1}{5}\), \(\frac{1}{4}\), \(\frac{1}{3}\), \(\frac{1}{2}\) and \(1\) only. By construction there is a natural rational map (the period map) from \(\mathscr{F}(0)\) to \(\mathscr{F}(1)\) and this factorises as follows. For each critical value \(\beta_k\) there are birational morphisms \(\varphi_{k,\pm}\colon \mathscr{F}(\beta_k\pm\epsilon)\to \mathscr{F}(\beta_k)\) (this is not quite the notation in the paper, which requires more detail) and the period map is the composition of the \(\varphi_{k,+}^{-1}\circ\varphi_{k,-}\), with the obvious adjustments for \(\beta=0\) and \(\beta=1\). In this list, however, \(\frac{1}{4}\) occurs twice, that is \(\beta_3=\beta_4=\frac{1}{4}\). The exceptional loci are described precisely in terms of special subvarieties of \(\mathscr{F}\) and \(\mathfrak{M}\). All of this is in agreement with the earlier conjectures. An extra result is that \(\mathscr{F}(\frac{1}{8},-)=\mathscr{F}(0,+)\), i.e. the model closest to \(\mathscr{F}\), is the moduli space of double covers of quadrics in \(\mathbb{P}^3\) having at worst slc singularities. The authors point out an intriguing parallel between this and the results of \textit{J. Shah} [Ann. Math. (2) 112, 485--510 (1980; Zbl 0412.14016)], and a relation with KSBA compactifications. The proofs are inevitably quite technical but there is a clear strategy. Much of that comes from the authors' earlier papers, and most of the rest comes from a VGIT argument similar in style to the one presented in, inter alia, [\textit{S. Casalaina-Martin} et al., J. Algebr. Geom. 23, No. 4, 727--764 (2014; Zbl 1327.14207)]. A feature of the analysis is the use of the notion of basin of attraction, which has occurred in VGIT arguments in the context of the Hassett-Keel program but not in exactly this form. Very roughly, in trying to analyse the flips at critical values one is led to considering orbit closures, and these can be reduced to 1-PS orbit closures: the basin of attraction of a semi-stable point \(x^*\) is then the set of points \(x\) that converge to \(x^*\) under the action of some 1-PS, and these sets turn out to be computable in a fairly direct way. It is remarkable that the entire program can be carried out so explicitly, since in the context of curves and the Hassett-Keel program it has generally been found very difficult even to understand the first flip.