Geometric interpretation of toroidal compactifications of moduli of points in the line and cubic surfaces

Abstract

It is known that some GIT compactifications associated to moduli spaces of either points in the projective line or cubic surfaces are isomorphic to Baily–Borel compactifications of appropriate ball quotients. In this paper, we show that their respective toroidal compactifications are isomorphic to moduli spaces of stable pairs as defined in the context of the MMP. Moreover, we give a precise mixed-Hodge-theoretic interpretation of this isomorphism for the case of eight labeled points in the projective line.

Introduction

Understanding the interplay between geometric and Hodge-theoretic compactifications of a given moduli space is one of the main challenges in algebraic geometry. This type of problem was investigated for moduli spaces of different types of algebraic varieties: for instance, abelian varieties [2], cubic threefolds [10], and very recently degree 2 K3 surfaces [4]. In the current paper we describe in detail such interplay for certain moduli spaces of points in ${\mathbb{P}}^1$ studied by Deligne and Mostow, and the moduli space of cubic surfaces. More specifically, we expand Deligne–Mostow's results by showing that the toroidal compactifications of their ball quotients are isomorphic to appropriate Hassett's moduli spaces of weighted stable rational curves. We pursue this isomorphism in detail for the case of eight labeled points in ${\mathbb{P}}^1$ by explicitly describing the Hodge theoretic boundary components. In another direction, as a consequence of our results, it follows that the KSBA compactification in [52] of a certain family of K3 surfaces arising from eight points in the projective line is isomorphic to a toroidal compactification up to a finite group action. For cubic surfaces, we prove that Naruki's compactification is toroidal, and that it has a modular interpretation in terms of Kollár–Shepherd-Barron–Alexeev stable pairs. Some of these results were expected (see [6, §1], [25, Remark 1.3 (4)], and [11, Appendix C]). However, their proofs were not pursued so far.

In [14], Deligne and Mostow described certain moduli spaces of n labeled points in ${\mathbb{P}}^1$ such that the GIT compactification ${({\mathbb{P}}^1)}^n/{/}_{\mathbf{w}}{\mathrm{SL}}_2$ with respect to a specific linearization $\mathbf{w}=(w_1,\dots ,w_n)$ is isomorphic, after possibly quotienting by the action on the labels of an appropriate symmetric group $S_m$, to the Baily–Borel compactification ${\overline{{\mathrm{\Gamma }}_{\mathbf{w}}\backslash {\mathbb{B}}_{n-3}}}^{\mathrm{bb}}$ of a ball quotient, where ${\mathrm{\Gamma }}_{\mathbf{w}}$ is arithmetic. The general theory developed in [7] provides us with an alternative compactification of the moduli space of n points in ${\mathbb{P}}^1$: the unique toroidal compactification ${\overline{{\mathrm{\Gamma }}_{\mathbf{w}}\backslash {\mathbb{B}}_{n-3}}}^{\mathrm{tor}}$, which is a blow up of the Baily–Borel compactification ${\overline{{\mathrm{\Gamma }}_{\mathbf{w}}\backslash {\mathbb{B}}_{n-3}}}^{\mathrm{bb}}$ at the cusps. This toroidal compactification has divisorial boundary, with each boundary component being the quotient of a CM-abelian variety by a (possibly trivial) finite group. In general, toroidal compactifications have milder singularities, but they lack a geometrically modular interpretation.

On the other hand, a geometric compactification of the moduli space of n points in ${\mathbb{P}}^1$ mapping birationally onto ${({\mathbb{P}}^1)}^n/{/}_{\mathbf{w}}{\mathrm{SL}}_2$ is provided by the Hassett weighted moduli space ${\bar{\mathrm{M}}}_{0,\mathbf{w}+ϵ}$ of stable n-pointed rational curves with weights $\mathbf{w}+ϵ:=(w_1+ϵ,\dots ,w_n+ϵ)$ [29]. It is very natural to ask how ${\bar{\mathrm{M}}}_{0,\mathbf{w}+ϵ}$ and ${\overline{{\mathrm{\Gamma }}_{\mathbf{w}}\backslash {\mathbb{B}}_{n-3}}}^{\mathrm{tor}}$ are related, which leads to our main result.

Theorem 1.1

Let w be a set of n rational weights and m a nonnegative integer for which we have the Deligne–Mostow isomorphism

$${\overline{{\mathit{\Gamma }}_{\mathbf{w}}\backslash {\mathbb{B}}_{n-3}}}^{\mathrm{bb}}\cong {({\mathbb{P}}^1)}^n/{/}_{\mathbf{w}}{\mathrm{SL}}_2\times S_m.$$

(For a complete list of these cases see Table 2, Table 3; all satisfy $\sum w_i=2$.) Then the toroidal compactification of ${\overline{{\mathit{\Gamma }}_{\mathbf{w}}\backslash {\mathbb{B}}_{n-3}}}^{\mathrm{tor}}$ is isomorphic to the quotient by $S_m$ of the Hassett moduli space of n-pointed rational curves with weights $\mathbf{w}+ϵ$. In particular, we have the following commutative diagram:

where the horizontal arrows are isomorphisms.

The main idea for the proof of Theorem 1.1, see Section 2, is to first prove the result for two special cases called the ancestral ones. Then we extend our result to the remaining Deligne–Mostow cases using work of Doran [20], which guarantees that the remaining ball quotients can be viewed as sub-ball quotients of the two ancestral cases. Note that the common denominator d of w is 3, 4, or 6 in all cases, and when $d=4$ we always have $m=1$ (that is, $S_m$ is trivial).

The isomorphisms in Theorem 1.1 compactify period maps ${\mathrm{\Phi }}_{\mathbf{w}}:{\mathrm{M}}_{0,n}/S_m\hookrightarrow {\mathrm{\Gamma }}_{\mathbf{w}}\backslash {\mathbb{B}}_{n-3}$ associated with certain weight-1 variations of Hodge structure ${\tilde{\mathcal{V}}}_{\mathbf{w}}$ on ${\mathrm{M}}_{0,n}/S_m$ with monodromy group ${\mathrm{\Gamma }}_{\mathbf{w}}$ and Hodge numbers $(n-2,n-2)$. Namely, the fiber of ${\tilde{\mathcal{V}}}_{\mathbf{w}}$ over (the $S_m$-orbit of) $\mathbf{x}=(x_1,\dots ,x_n)\in {\mathrm{M}}_{0,n}$ is the subspace in $H^1$ of the curve $C_{\mathbf{w},\mathbf{x}}:=\{Y^d={\mathrm{\Pi }}_{j=1}^n{(X-x_{j}Z)}^{d{w}_j}\}\subset \mathbb{WP}[1:1:2]$ on which the automorphism ${\rho }^{⁎}$ defined by $Y\mapsto e^{\frac{2\pi i}{d}}Y$ acts through $e^{\pm \frac{2\pi i}{d}}$, cf. [19, Thm. 8.4ff]. The injectivity of ${\mathrm{\Phi }}_{\mathbf{w}}$ is thus transformed into a global Torelli theorem, and one wonders to what extent ${\bar{\mathrm{\Phi }}}_{\mathbf{w}}$ underlies an extended global Torelli theorem matching geometric and Hodge-theoretic moduli.

In Section 3, we work this out for the case $\mathbf{w}=\frac{\mathbf{1}}{\mathbf{4}}=(\frac{1}{4},\dots ,\frac{1}{4})$ ($n=8$, $m=1$), where the VHS has fibers ${\tilde{\mathcal{V}}}_{\frac{\mathbf{1}}{\mathbf{4}},\mathbf{x}}=H^1{(C_{\mathbf{w},\mathbf{x}})}^{-{\rho }^2}$. To wit, we provide a mixed-Hodge-theoretic interpretation of the restriction of $\bar{\mathrm{\Phi }}={\bar{\mathrm{\Phi }}}_{\frac{\mathbf{1}}{\mathbf{4}}}$ to the exceptional divisor of φ, along which (generically) $C_{\frac{\mathbf{1}}{\mathbf{4}},\mathbf{x}}$ degenerates to a pair of genus-3 curves. That is, each component of this divisor has a Zariski open $\mathcal{S}$ parameterizing curves ${\{D_{\nu }=D_{\nu }^{(1)}\cup D_{\nu }^{(2)}\}}_{\nu \in \mathcal{S}}$ on which ρ acts, with four fixed points on each $D_{\nu }^{(j)}$, and $D_{\nu }^{(1)}\cap D_{\nu }^{(2)}={\{q_{\ell }={\rho }^{\ell }(q_0)\}}_{\ell =0}^3$. Write N for the monodromy logarithm of ${\tilde{\mathcal{V}}}_{\frac{\mathbf{1}}{\mathbf{4}}}$ along $\mathcal{S}$, and $\mathcal{R}$ for the (finite) monodromy group of $H^1{(D_{\nu }^{(j)},\mathbb{Z})}^{-{\rho }^2}$ on $\mathcal{S}$. We show in Proposition 3.4, Proposition 3.6 that $\bar{\mathrm{\Phi }}{|}_{\mathcal{S}}$ records the limiting mixed Hodge structure of $\tilde{\mathcal{V}}$ at ν in the Hodge-theoretic boundary component ${\mathrm{\Gamma }}_N\backslash B(N)\cong {\times }_{j=1}^2\{\mathcal{R}\backslash {({\mathrm{\Omega }}^1{(D_{\nu }^{(j)})}^{-{\rho }^2})}^{\vee }/(1-\rho )H_1(D_{\nu }^{(j)},\mathbb{Z})\}\cong {\mathbb{P}}^2\times {\mathbb{P}}^2$ (which is independent of $\nu \in \mathcal{S}$ despite appearances). This leads to the following result, proved in Section 3.5:

Theorem 1.2

Given $\nu \in \mathcal{S}$, let ${\sigma }_{+}^{(j)}$ be any path on $D_{\nu }^{(j)}$ ($j=1,2$) from a fixed point of ρ to a point of $D_{\nu }^{(1)}\cap D_{\nu }^{(2)}$, and ${\sigma }^{(j)}:={\sigma }_{+}^{(j)}-{\rho }^2{\sigma }_{+}^{(j)}$. Then the functional $({\int }_{{\sigma }^{(1)}},{\int }_{{\sigma }^{(2)}})\in {\oplus }_{j=1}^2{({\mathit{\Omega }}^1{(D_{\nu }^{(j)})}^{-{\rho }^2})}^{\vee }$ becomes well-defined in ${\mathit{\Gamma }}_N\backslash B(N)$ and computes $\bar{\mathit{\Phi }}(\nu )$.

The above ball quotients can also be identified with specific moduli spaces of surfaces (see [19], [21], [18], [44], [53]). For instance, if $\mathbf{w}=\frac{\mathbf{1}}{\mathbf{4}}$ as above, an appropriate finite quotient of an open subset of the ball quotient ${\mathrm{\Gamma }}_{\frac{\mathbf{1}}{\mathbf{4}}}\backslash {\mathbb{B}}_5$ parametrizes the K3 surfaces with order-four, purely non-symplectic automorphism and $U(2)\oplus D_4^{\oplus 2}$ lattice polarization. These surfaces were studied in [44] from the point of view of automorphic forms, and they arise as the minimal resolution of the double cover $X\to {\mathbb{P}}^1\times {\mathbb{P}}^1$ branched along a specific curve of class $(4,4)$ which depends on the choice of eight points in ${\mathbb{P}}^1$. More precisely, if $[{\lambda }_1:1],\dots ,[{\lambda }_8:1]$ are the eight distinct points, then the equation of the branch curve is in the following form:

$$y_0{y}_1(y_0^2\prod _{i=1}^4(x_0-{\lambda }_i{x}_1)+y_1^2\prod _{i=5}^8(x_0-{\lambda }_i{x}_1))=0.$$

The involution of ${\mathbb{P}}^1\times {\mathbb{P}}^1$ given in an affine patch by $(x,y)\mapsto (x,-y)$ lifts to the K3 surface giving the order 4 purely non-symplectic automorphism. A compactification $\bar{\mathbf{K}}$ of such a family by KSBA stable pairs (see [43], [1], [42]) was studied in [52], where it is shown $\bar{\mathbf{K}}$ is isomorphic to the quotient of ${\bar{\mathrm{M}}}_{0,\frac{\mathbf{1}}{\mathbf{4}}+ϵ}$ by a finite group. Therefore, as an immediate consequence of Theorem 1.1, we have the following result.

Corollary 1.3

Let $\bar{\mathbf{K}}$ be the KSBA compactification of the moduli space of K3 surfaces with a purely non-symplectic automorphism of order four and $U(2)\oplus D_4^{\oplus 2}$ lattice polarization. Then, $\bar{\mathbf{K}}$ is isomorphic to the quotient of the toroidal compactification ${\stackrel{‾}{{\mathit{\Gamma }}_{\frac{\mathbf{1}}{\mathbf{4}}}\backslash {\mathbb{B}}_5}}^{\mathrm{tor}}$ by $(S_4\times S_4)⋊S_2$.

Next, we discuss our work on cubic surfaces. Let Y be the moduli space parameterizing marked smooth cubic surfaces in [25, §6.3] (we recall the definition of marking in §4.1). This moduli space can be compactified using different techniques. A smooth normal crossing compactification $\mathbf{Y}\subseteq \bar{\mathbf{N}}$ was constructed by Naruki [57] using cross-ratios of tritangents to the cubic surfaces. Another perspective comes from GIT: an appropriate $W(E_6)$ cover of the GIT quotient of cubic surfaces (for details see Definition 4.1.2) provides a compactification $\mathbf{Y}\subseteq {\bar{\mathbf{Y}}}_{\mathrm{GIT}}$, which is related to Naruki's compactification by a birational morphism $\bar{\mathbf{N}}\to {\bar{\mathbf{Y}}}_{\mathrm{GIT}}$. By [6, Theorem 3.17] we also know that there is an isomorphism between ${\bar{\mathbf{Y}}}_{\mathrm{GIT}}$ and the Baily–Borel compactification ${\overline{{\mathrm{\Gamma }}_c\backslash {\mathbb{B}}_4}}^{\mathrm{bb}}$ of an appropriate ball quotient (see also [18]). In [6], Allcock, Carlson, and Toledo assert the following result without pursuing its proof (see the introduction in [6]). To complete the picture, in Section 4 we give a proof of this expected isomorphism.

Theorem 1.4

The Naruki compactification $\bar{\mathbf{N}}$ of the moduli space Y of marked cubic surfaces is isomorphic to the toroidal compactification ${\overline{{\mathit{\Gamma }}_c\backslash {\mathbb{B}}_4}}^{\mathrm{tor}}$ of its ball quotient. In particular, we have the following commutative diagram:

Another compactification of Y can be constructed using KSBA stable pairs. The marking of a smooth cubic surface S induces a labeling of its $(-1)$-curves, and we denote by B the divisor on S given by their sum (note that $(S,B)$ is a stable pair if B has normal crossings). In [25], Hacking, Keel, and Tevelev studied the KSBA compactification of the moduli space parameterizing such stable pairs $(S,B)$. The authors also showed that Naruki's compactification $\bar{\mathbf{N}}$ is isomorphic to the log canonical model of Y, and they believed that $\bar{\mathbf{N}}$ should parametrize stable pairs $(S,(\frac{1}{9}+ϵ)B)$ and their degenerations, where $ϵ\in \mathbb{Q}$, $0<ϵ\ll 1$. In the next theorem we confirm this belief. As a result, we give a modular interpretation of the toroidal compactification ${\overline{{\mathrm{\Gamma }}_c\backslash {\mathbb{B}}_4}}^{\mathrm{tor}}$.

Theorem 1.5

The Naruki compactification $\bar{\mathbf{N}}$ is isomorphic to the normalization of the KSBA compactification ${\bar{\mathbf{Y}}}_{\frac{1}{9}+ϵ}$ of the moduli space parameterizing marked smooth cubic surfaces with divisor given by the sum of the 27 lines with weight $\frac{1}{9}+ϵ$ and their degenerations. The stable pairs parametrized by the boundary of $\bar{\mathbf{N}}$ are in the form $(S_0,(\frac{1}{9}+ϵ)B_0)$, where $S_0\subseteq {\mathbb{P}}^3$ is a singular cubic surface and $B_0$ is the sum of the lines in $S_0$, which we further describe as follows (see the corresponding picture in Table 1):

• $S_0$ has exactly one $A_1$ singularity and $B_0$ has six double lines passing through this singularity. The remaining 15 lines have multiplicity one.

• $S_0$ has exactly two $A_1$ singularities. $B_0$ has one quadruple line passing through the two singular points, four double lines passing through one $A_1$ singularity, and other four double lines passing through the other $A_1$ singularity. The remaining seven lines have multiplicity one.

• $S_0$ has exactly three $A_1$ singularities. $B_0$ has three quadruple lines, each one containing a pair of $A_1$ singularities, and three pairs of double lines, where each pair passes through one of the three singular points. The remaining three lines have multiplicity one.

• $S_0$ has exactly four $A_1$ singularities (this is known as the Cayley cubic surface). $B_0$ has six quadruple lines, each one containing a pair of $A_1$ singularities. The remaining three lines have multiplicity one.

In the remaining cases, $S_0$ is equal to the normal crossing union of three planes in ${\mathbb{P}}^3$, each one containing exactly nine of the lines of $B_0$, possibly with multiplicities. For the precise multiplicities and the incidences of the lines we refer to the pictures in the second column of Table 1.

The main idea for the proof is to use the family over $\bar{\mathbf{N}}$ constructed by Naruki and Sekiguchi in [58] and endow it with the divisor intersecting the fibers of the family giving the 27 lines and their degenerations. The main part of the argument is checking that the degenerations obtained this way are stable. This gives a morphism $\bar{\mathbf{N}}\to {\bar{\mathbf{Y}}}_{\frac{1}{9}+ϵ}$, which we check is finite. The result then follows from Zariski's Main Theorem.