Geometric interpretation of toroidal compactifications of moduli of points in the line and cubic surfaces
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 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 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 such that the GIT compactification with respect to a specific linearization is isomorphic, after possibly quotienting by the action on the labels of an appropriate symmetric group , to the Baily–Borel compactification of a ball quotient, where is arithmetic. The general theory developed in [7] provides us with an alternative compactification of the moduli space of n points in : the unique toroidal compactification , which is a blow up of the Baily–Borel compactification 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 mapping birationally onto is provided by the Hassett weighted moduli space of stable n-pointed rational curves with weights [29]. It is very natural to ask how and 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 (For a complete list of these cases see Table 2, Table 3; all satisfy .) Then the toroidal compactification of is isomorphic to the quotient by of the Hassett moduli space of n-pointed rational curves with weights . 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 we always have (that is, is trivial).
The isomorphisms in Theorem 1.1 compactify period maps associated with certain weight-1 variations of Hodge structure on with monodromy group and Hodge numbers . Namely, the fiber of over (the -orbit of) is the subspace in of the curve on which the automorphism defined by acts through , cf. [19, Thm. 8.4ff]. The injectivity of is thus transformed into a global Torelli theorem, and one wonders to what extent underlies an extended global Torelli theorem matching geometric and Hodge-theoretic moduli.
In Section 3, we work this out for the case (, ), where the VHS has fibers . To wit, we provide a mixed-Hodge-theoretic interpretation of the restriction of to the exceptional divisor of φ, along which (generically) degenerates to a pair of genus-3 curves. That is, each component of this divisor has a Zariski open parameterizing curves on which ρ acts, with four fixed points on each , and . Write N for the monodromy logarithm of along , and for the (finite) monodromy group of on . We show in Proposition 3.4, Proposition 3.6 that records the limiting mixed Hodge structure of at ν in the Hodge-theoretic boundary component (which is independent of despite appearances). This leads to the following result, proved in Section 3.5: Theorem 1.2 Given , let be any path on () from a fixed point of ρ to a point of , and . Then the functional becomes well-defined in and computes .
The above ball quotients can also be identified with specific moduli spaces of surfaces (see [19], [21], [18], [44], [53]). For instance, if as above, an appropriate finite quotient of an open subset of the ball quotient parametrizes the K3 surfaces with order-four, purely non-symplectic automorphism and 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 branched along a specific curve of class which depends on the choice of eight points in . More precisely, if are the eight distinct points, then the equation of the branch curve is in the following form: The involution of given in an affine patch by lifts to the K3 surface giving the order 4 purely non-symplectic automorphism. A compactification of such a family by KSBA stable pairs (see [43], [1], [42]) was studied in [52], where it is shown is isomorphic to the quotient of by a finite group. Therefore, as an immediate consequence of Theorem 1.1, we have the following result.
Corollary 1.3 Let be the KSBA compactification of the moduli space of K3 surfaces with a purely non-symplectic automorphism of order four and lattice polarization. Then, is isomorphic to the quotient of the toroidal compactification by .
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 was constructed by Naruki [57] using cross-ratios of tritangents to the cubic surfaces. Another perspective comes from GIT: an appropriate cover of the GIT quotient of cubic surfaces (for details see Definition 4.1.2) provides a compactification , which is related to Naruki's compactification by a birational morphism . By [6, Theorem 3.17] we also know that there is an isomorphism between and the Baily–Borel compactification 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 of the moduli space Y of marked cubic surfaces is isomorphic to the toroidal compactification 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 -curves, and we denote by B the divisor on S given by their sum (note that 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 . The authors also showed that Naruki's compactification is isomorphic to the log canonical model of Y, and they believed that should parametrize stable pairs and their degenerations, where , . In the next theorem we confirm this belief. As a result, we give a modular interpretation of the toroidal compactification .
Theorem 1.5 The Naruki compactification is isomorphic to the normalization of the KSBA compactification of the moduli space parameterizing marked smooth cubic surfaces with divisor given by the sum of the 27 lines with weight and their degenerations. The stable pairs parametrized by the boundary of are in the form , where is a singular cubic surface and is the sum of the lines in , which we further describe as follows (see the corresponding picture in Table 1): has exactly one singularity and has six double lines passing through this singularity. The remaining 15 lines have multiplicity one. has exactly two singularities. has one quadruple line passing through the two singular points, four double lines passing through one singularity, and other four double lines passing through the other singularity. The remaining seven lines have multiplicity one. has exactly three singularities. has three quadruple lines, each one containing a pair of 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. has exactly four singularities (this is known as the Cayley cubic surface). has six quadruple lines, each one containing a pair of singularities. The remaining three lines have multiplicity one.
In the remaining cases, is equal to the normal crossing union of three planes in , each one containing exactly nine of the lines of , 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 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 , which we check is finite. The result then follows from Zariski's Main Theorem.
Section snippets
Geometric interpretation of the toroidal compactifications of Deligne–Mostow ball quotients
In §2.1 and §2.2 we recall the necessary background for the Deligne–Mostow ball quotients and Hassett's moduli spaces of weighted stable rational curves, which are necessary for the proof of Theorem 1.1. The first step of the proof is carried out in §2.3, where we show that our theorem holds for the so called ancestral cases associated to 8 and 12 points in . Finally, using work of Doran ([22, Theorem 4]), in §2.4 we show that these two ancestral cases imply all the others.
Hodge theoretic interpretation for eight points in the line
The purpose of the following section is to give a precise mixed-Hodge-theoretic description of the extension of the period map associated to 8 labeled points in . That is, we show how to reinterpret the isomorphism as an extended global Torelli result, matching the geometrically-modular description of the source with the description of the target in terms of equivalence-classes of mixed Hodge structures.
Throughout, we shall denote field extensions by subscripts, viz.
Moduli of cubic surfaces: Naruki's compactification is toroidal
The goal of this section is to show that the Naruki compactification of the moduli space of smooth marked cubic surfaces is isomorphic to the toroidal compactification of an appropriate ball quotient (see [6], [18]). We start by briefly recalling the necessary background.
Naruki's compactification is a moduli space of KSBA stable pairs
In this section, our goal is to show that Naruki's compactification has a modular interpretation in terms of KSBA stable pairs. We start by briefly recalling the necessary background.
Extension criteria
The following lemma allows us to compare the toroidal compactification of a ball quotient with another compactification which in our applications has geometric origin. Lemma 6.1 Let be a ball quotient of dimension larger than or equal to 2 and let be a countable union of hyperplanes on such that Γ acts freely on . Let be a smooth normal crossing compactification of a finite cover of admitting a generically finite morphism . Then there exists a morphism inducing
Acknowledgments
We would like to thank Jeff Achter, Sebastian Casalaina-Martin, Eduardo Cattani, and Paul Hacking for insightful discussions, explanations, and for helping us correct imprecise statements in a preliminary draft of the paper. We thank Han-Bom Moon for his feedback on the first draft of the paper. We also thank the anonymous referees for carefully reading our paper and for their valuable suggestions. We are grateful for the working environments at the Department of Mathematics in Washington
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