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arxiv: 2605.17223 · v1 · pith:AEEUTBQEnew · submitted 2026-05-17 · 🧮 math.AG

Moduli of Persson surfaces: The compactification via KSBA stable pairs and a generic global Torelli type theorem

Hanlong Fang , Bin Nguyen , Xian Wu , Zheng Zhang This is my paper

Pith reviewed 2026-05-19 23:23 UTC · model grok-4.3

classification 🧮 math.AG
keywords Persson surfacesKSBA stable pairsmoduli compactificationglobal Torelli theoremGalois coversHodge structuresétale double coverscanonically polarized surfaces
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The pith

Up to two possibilities, a generic smooth Persson surface can be recovered from the Hodge structure on the anti-invariant part of the second cohomology of its étale double cover together with the associated (Z/2Z)^5-action.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines the family of canonically polarized surfaces introduced by Persson, realized as Galois (Z/2Z)^4-covers of the projective plane branched along eight general lines. It constructs the compactification of the moduli space of these surfaces in the sense of Kollár-Shepherd-Barron-Alexeev by describing the stable degenerations explicitly as stable pairs of weighted hyperplane arrangements. The authors compute the Q-Gorenstein obstructions and use wall-crossing to prove that the resulting compactified moduli stack is smooth. They further establish a generic global Torelli-type theorem that recovers a generic smooth surface in the family from specified Hodge data on its étale double cover.

Core claim

The central claim is that the moduli stack of KSBA-stable Persson surfaces is smooth and that, up to two possibilities, a generic smooth Persson surface is determined by the Hodge structure on the anti-invariant part of the second cohomology of its étale double cover equipped with the natural (Z/2Z)^5-action.

What carries the argument

KSBA stable pairs of weighted hyperplane arrangements, which encode the boundary points of the compactification and permit explicit computation of the Q-Gorenstein obstructions that control the wall-crossings.

If this is right

  • The compactified moduli stack is smooth.
  • Stable degenerations are given by weighted hyperplane arrangements with explicit combinatorial data.
  • The Torelli result supplies a reconstruction of the surface from the indicated Hodge-theoretic input.
  • Wall-crossings in the KSBA sense suffice to reach all stable limits without introducing extra singularities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result indicates that the anti-invariant Hodge structure with group action encodes essentially all geometric information for generic members of the family.
  • Analogous compactifications and Torelli statements may exist for other Galois covers of surfaces branched along hyperplane arrangements.
  • Direct computation on explicit configurations of eight lines could test whether the generic recovery extends to special but still smooth cases.

Load-bearing premise

The eight lines are assumed general so that the Galois cover produces a smooth canonically polarized surface and the KSBA wall-crossing analysis encounters no obstructions beyond the Q-Gorenstein ones computed in the paper.

What would settle it

An explicit pair of non-isomorphic smooth Persson surfaces whose anti-invariant Hodge structures with the (Z/2Z)^5-action coincide would falsify the generic Torelli statement.

Figures

Figures reproduced from arXiv: 2605.17223 by Bin Nguyen, Hanlong Fang, Xian Wu, Zheng Zhang.

Figure 1
Figure 1. Figure 1: Generic stable degenerations of (P2 , b P8 i=1 Di). The complete descriptions of the compactified moduli and all stable degenera￾tions were given for Campedelli surfaces in [AP24], which are (Z/2Z) 3 -covers of P2 branched along seven general lines (see Remark 2.4(1)). In particular, only type 0 degenerations occur, that is, in the language of polytopes, the b-cut admits only trivial matroid tilings (see D… view at source ↗
Figure 2
Figure 2. Figure 2: An unstable degeneration corresponds to Q2 in type II. Theorem 3.15. The KSBA compactification M s of the moduli space of Persson surfaces is Mb(P 2 , 8)/GStab, where b = ( 1 2 , . . . , 1 2 ) and GStab ∼= Aff(F2, 3) = (Z/2Z) 3 ⋊ GL(F2, 3). M s is unira￾tional. Moreover, the KSBA compactified moduli stack Ms of Persson surfaces is smooth. Proof. GStab permutes labels of branch divisors but fixes the line b… view at source ↗
Figure 3
Figure 3. Figure 3: The KSBA stable replacements of type II degenerations for weight a. The broken line with weight ( 1 2 − ε) is the boundary bottom broken line. Lemma 3.22. On Ms , the points parametrizing type II degenerations are smooth. Proof. In the case of type II degenerations, notice that the restriction of KY , which is the boundary of the big broken triangle in [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: where lines in the same color correspond to the same gi . So on Y2, there are eight different labeled divisors, and X2 is a smooth K3 surface. On Y1 (and Y3), there is a point through three lines, labeled as two Dg’s. According to [AP12, [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: As discussed in Remark 2.4, there are 28 intermediate D1,6-polarized Enriques surfaces, each with six A1-singularities, which are, under the cover map, the preim￾ages of the points li ∩ l ′ i , i = 1, 2, 3 (the black dots of the right picture in [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Boundary of the Baily–Borel compactification of the moduli of D1,6-polarized Enriques surfaces. Xq2 Xp, p ∈ Codd\{q1, q2} Xq1 Xp, p ∈ Ceven\{peven, q1} Xpeven [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Degenerated 6-line arrangements over the the boundary of the Baily–Borel (also the GIT) compactification of the moduli of D1,6-polarized Enriques surfaces. The solid lines, dashed lines and dotted lines are different rulings as in [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Convex hulls of x ∈ F 3 2 for g = (1, x) of the four branch lines Dg of the 22 del Pezzo surfaces of degree 2. In the first row, the numbers count the blue polytopes on the cube up to symmetry. For example, for the first picture, there are six facets of the cube. The cubes in the second row are for the eight new del Pezzo surfaces, which do not appear in the middle of the G-cover X → P2 . One takes a trian… view at source ↗
read the original abstract

We study a family of canonically polarized surfaces introduced by Persson, which arise as Galois $G=(\mathbb{Z}/2\mathbb{Z})^4$-covers of $\mathbf{P}^2$ branched along eight general lines. For this family, we construct the compactified moduli space and explicitly describe the stable degenerations in the sense of Koll\'ar, Shepherd-Barron, and Alexeev (KSBA) via stable pairs of weighted hyperplane arrangements. By computing the $\mathbb{Q}$-Gorenstein obstructions and using the KSBA wall crossings, we show that the resulting compactified moduli stack is smooth. Furthermore, we establish a generic global Torelli type result: up to two possibilities, a generic smooth Persson surface can be recovered from the Hodge structure on the anti-invariant part of the second cohomology of its \'etale double cover, together with the associated $\widetilde{G}=(\mathbb{Z}/2\mathbb{Z})^5$-action.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper studies the moduli space of Persson surfaces arising as Galois (Z/2Z)^4-covers of P^2 branched along eight general lines. It constructs the KSBA compactification by identifying stable degenerations with stable pairs of weighted hyperplane arrangements, computes the Q-Gorenstein obstructions, and uses wall-crossing analysis to conclude that the compactified moduli stack is smooth. It further proves a generic global Torelli theorem: up to two possibilities, a generic smooth Persson surface is recovered from the Hodge structure on the anti-invariant part of H^2 of its étale double cover together with the associated (Z/2Z)^5-action.

Significance. If the central claims hold, the work supplies an explicit, arrangement-theoretic description of the KSBA boundary for this family of canonically polarized surfaces and yields a smooth moduli stack, which is uncommon for surfaces of general type. The generic Torelli recovery from anti-invariant Hodge data plus group action provides a concrete instance of period-map injectivity in the presence of Galois covers. The explicit use of weighted arrangements and obstruction computations are strengths that could serve as a model for related families.

major comments (3)
  1. [§4] §4 (Q-Gorenstein obstructions): the claim that the computed Q-Gorenstein obstructions are the only ones arising in the KSBA wall-crossing for general lines is load-bearing for smoothness of the stack, yet the argument does not explicitly rule out non-Q-Gorenstein or non-arrangement obstructions that could appear at higher codimension loci.
  2. [§5.2] §5.2 (wall-crossing analysis): the assertion that all KSBA limits of the family remain inside the weighted hyperplane arrangement locus for eight general lines is used to define the period map on a dense open set, but no explicit check excludes degenerations outside this locus that would obstruct smoothness or break the generic Torelli statement.
  3. [§6] §6 (generic Torelli): the recovery statement relies on the period map being an immersion on the smooth locus after compactification; the argument that the anti-invariant Hodge structure plus G̃-action distinguishes the surface up to two possibilities needs a clearer verification that the Hodge filtration determines the branch locus uniquely in the general case.
minor comments (2)
  1. [§2] Notation for the group actions G and G̃ is introduced without a single consolidated table; a summary table in §2 would improve readability.
  2. [§3] Several diagrams of weighted arrangements in §3 lack labels for the weights and the Galois action; adding these would clarify the stable-pair description.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comments help clarify the exposition around the Q-Gorenstein analysis, the wall-crossing locus, and the generic Torelli statement. We address each major point below and indicate the revisions planned for the next version.

read point-by-point responses

  1. Referee: [§4] §4 (Q-Gorenstein obstructions): the claim that the computed Q-Gorenstein obstructions are the only ones arising in the KSBA wall-crossing for general lines is load-bearing for smoothness of the stack, yet the argument does not explicitly rule out non-Q-Gorenstein or non-arrangement obstructions that could appear at higher codimension loci.

    Authors: We agree that an explicit statement ruling out other obstructions would strengthen the smoothness claim. In the current text the focus is on the Q-Gorenstein obstructions arising from the weighted arrangement data for general lines, which control the relevant wall-crossings. To address the referee’s concern we will add a short paragraph at the end of §4 explaining why, under the generality assumption on the eight lines, non-Q-Gorenstein or non-arrangement obstructions cannot appear in codimension greater than one; the argument uses the fact that any such obstruction would force a non-general position of the lines, contradicting the open condition used throughout the paper. This addition will make the load-bearing claim fully explicit. revision: yes

  2. Referee: [§5.2] §5.2 (wall-crossing analysis): the assertion that all KSBA limits of the family remain inside the weighted hyperplane arrangement locus for eight general lines is used to define the period map on a dense open set, but no explicit check excludes degenerations outside this locus that would obstruct smoothness or break the generic Torelli statement.

    Authors: The referee correctly notes that the containment of KSBA limits inside the weighted arrangement locus is used to define the period map. In §5.2 the wall-crossing analysis is carried out entirely within this locus, relying on the stability conditions for the chosen weights and the generality of the lines. To make the exclusion of external degenerations explicit we will insert a brief lemma (or remark) showing that any KSBA limit of a general eight-line arrangement must remain an arrangement of the same type; the proof uses the numerical stability criterion and the fact that a degeneration leaving the locus would violate the generality assumption on the branch divisor. This will ensure the period map is well-defined on the dense open set without additional obstructions. revision: yes

  3. Referee: [§6] §6 (generic Torelli): the recovery statement relies on the period map being an immersion on the smooth locus after compactification; the argument that the anti-invariant Hodge structure plus G̃-action distinguishes the surface up to two possibilities needs a clearer verification that the Hodge filtration determines the branch locus uniquely in the general case.

    Authors: We thank the referee for pointing out the need for a clearer verification. The current proof of the generic Torelli theorem uses the (Z/2Z)^5-action to decompose the anti-invariant cohomology and shows that the Hodge filtration, together with the group action, recovers the branch divisor up to the two possibilities stated. To address the request we will expand the relevant paragraph in §6 with an explicit computation: for a general choice of lines the positions are uniquely determined by the eigenspace data of the filtration because any ambiguity would impose a non-trivial linear dependence among the lines, contradicting generality. This additional check will make the immersion property and the uniqueness statement fully transparent. revision: yes

Circularity Check

0 steps flagged

Minor self-citation present but not load-bearing; derivation relies on standard KSBA theory, explicit Q-Gorenstein computations, and Hodge-theoretic arguments.

full rationale

The paper's central claims—the smoothness of the KSBA compactification for general eight-line arrangements and the generic Torelli recovery from the anti-invariant Hodge structure plus G̃-action—are supported by direct computation of obstructions and application of external KSBA wall-crossing results rather than by fitting parameters to the target data or by a self-referential definition. Any citations to the authors' prior work supply independent background on Persson surfaces or Hodge structures and do not form the load-bearing step for the smoothness or Torelli statements. The construction therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of KSBA stability for pairs, the existence of the Galois cover for general lines, and the Hodge decomposition on the double cover; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math KSBA stability for pairs of surfaces with boundary
    Invoked to define the compactification and stable degenerations
  • standard math Q-Gorenstein deformation theory for the moduli stack
    Used to compute obstructions and conclude smoothness

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