Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric (With an Appendix by Christian Gleissner and Noah Ruhland)

Carlos Rito; Christian Gleissner; Noah Ruhland

REVIEW 3 major objections 4 minor 12 references

Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric (With an Appendix by Christian Gleissner and Noah Ruhland)

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper constructs a Godeaux surface with singular set 2A1+2A3, shows its Z/2 × Z/4 abelian cover is a fake quadric, and proves this fake quadric is not a quotient of a product of curves.

desk verdict A likely genuine new fake quadric outside the product-quotient class, but the written proof skips the linear-equivalence check Pardini's theory demands. read the letter →

arxiv 2509.08198 v2 pith:VN3RLNYB submitted 2025-09-10 math.AG

Carlos Rito , Christian Gleissner , Noah Ruhland This is my paper

classification math.AG MSC 14J29

keywords fakequadricsGodeauxsurfacesofgeneraltypeabeliancoverssurfacesingularitiesproduct-quotientfinite-fieldinterpolationliftingtocharacteristiczero

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims to construct a numerical Godeaux surface whose singular locus is exactly two A1 and two A3 points, and then to show that a certain Z/2 × Z/4 abelian cover of its minimal resolution is a smooth minimal surface of general type with invariants K^2=8 and p_g=q=0—a fake quadric. The construction is explicit, with equations over the rational numbers, and the paper proves the resulting surface is not a quotient of a product of curves. This matters because all previously constructed fake quadrics arose as such quotients, and identifying the universal cover of a fake quadric is a central open problem in the geography of surfaces. The paper also develops a computational method, based on interpolation over finite fields followed by Chinese-remainder lifting, for detecting highly singular members inside families of varieties, and applies it to locate the required Godeaux surface.

What carries the argument

The central object is the pair (X, X'): a Godeaux surface with singular set 2A1+2A3 and its minimal resolution, whose eight exceptional (−2)-curves are recorded in an intersection matrix. The central identity is the pair of numerical divisibility relations (1), which encode the 2- and 4-divisibility needed to define a Z/2 × Z/4 abelian cover via the building-data formalism of [Par91]. The computational engine is finite-field interpolation followed by Chinese-remainder reconstruction and rational reconstruction, used to lift the singular examples and the auxiliary curves C and D to characteristic zero.

What would settle it

Independently recompute the entire construction from the published ancillary Magma files, preferably in a second computer algebra system: verify that the lifted singularity set is 2A1+2A3, that the nullspace computation reproduces relations (1), and that h^0(X', K+L_i)=0 for i=2,...,8. A single nonzero h^0 would give p_g(S)>0 and disprove the fake quadric claim.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a new explicit object: a Godeaux surface X defined over Q with singular locus 2A1+2A3, and a Z/2 × Z/4 abelian cover of its minimal resolution X' that is a smooth minimal surface of general type with K^2=8 and p_g=0, hence a fake quadric. The cover is constructed through explicit divisor relations on X', including 8K_{X'} ≡ 4C' + 2N1 + N3 + 2N4 + 3N5 + 3N6 + 2N7 + N8 and 4K_{X'} ≡ 2D' + N1 + N2 + N6 + 2N7 + N8, which provide the building data for the abelian-cover theory in [Par91]; the required vanishings h^0(X', K+L_i)=0 are certified by computer. An appendix theorem—every automorphism of a minimal realization of a variety isogenous to a product l

Load-bearing premise

The load-bearing premise is that the reported computer verifications are correct: the lifted surface really has the claimed two-plus-two singularity type, the divisor relations (1) really hold, and the needed spaces of sections all have dimension zero. If any of these checks is wrong, the fake quadric construction collapses.

Editorial extensions

If this is right

If the construction is correct, the paper produces the first explicit fake quadric that is not a quotient of a product of curves, giving a concrete test object for the geography of surfaces of general type.
The defining equations are defined over Q and are complex-conjugation invariant, so the surface can be interrogated computationally; whether its universal cover is the bidisk remains open.
The singular Godeaux surface with 2A1+2A3 realizes one of the two quotient configurations predicted in [DR14] to arise from automorphisms of quaternionic fake quadrics, although the paper does not establish that it comes from such a quaternionic example.
The finite-field interpolation and lifting procedure is presented as a general recipe for finding highly singular members in parameterized families, and here it uncovered an unexpected 4-dimensional locus of four-nodal surfaces and a 2-dimensional family of six-nodal surfaces.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the same finite-field interpolation plus CRT-lifting recipe should adapt to other singularity types, such as A2 or D4, since the paper notes the method is not intrinsically limited to nodes; a natural test is to search other Godeaux families for configurations with those singularities.
Editorial inference: the need to repeat computations over more than 600 primes suggests that rational coefficient blow-up is the main bottleneck, so p-adic lifting or lattice-basis reduction could make the method practical in higher dimensions where interpolation sets grow quickly.
Editorial inference: because the fake quadric is defined over Q and invariant under complex conjugation, if it were later shown to be uniformized by the bidisk it would be a quaternionic fake quadric, connecting this explicit example to the hypothetical lattice lists discussed in [LSV19]; the paper stops short of that connection.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Carlos, here's my take on the Rito–Gleissner–Ruhland paper. The headline is real: they construct a Godeaux surface with singular set 2A1+2A3 and then a Z/2×Z/4 abelian cover that looks like a fake quadric, K²=8, pg=0. If correct, it's the first explicit fake quadric not isogenous to a product of curves, and the computational pipeline for finding singular members is a useful contribution in its own right. The appendix theorem on lifting automorphisms of product-quotient varieties is also new and cleanly proved.

The trouble is in the gap between numerical equivalence and linear equivalence. Sections 5 and 6 explicitly describe "numerical divisibility relations" computed via nullspaces of intersection matrices. Those give relations in the numerical class group, not in Pic(X'). Pardini's abelian cover theorem requires the building data to satisfy genuine linear equivalences. The resolution X' has q=0, so Pic^0 is trivial, but because X' is a Z/2-Godeaux resolution, its Picard group has a 2-torsion class. So a numerically trivial divisor need not be linearly trivial. If the relations (3) hold only up to that torsion, the G-cover may not exist, and the h^0(K+L_i) vanishings would have been computed for the wrong line bundles. The ancillary Magma files might verify the needed linear equivalences via explicit rational functions, but the printed paper never says so. This is a load-bearing omission, not a cosmetic one.

The final Section 9 exclusion is also under-argued: it relies on a brief look at Bauer–Pignatelli and Frapporti–Pignatelli. That's probably correct, but a referee will want the actual check spelled out.

I want to give credit where due: the construction is concrete, the code is supplied, and the authors clearly know the Pardini framework. Nothing in the text suggests cooked results or circular reasoning. The gap is fixable—if the Magma files do contain explicit linear equivalence checks, the paper becomes a solid major result. If they don't, the central assertion is unproven.

This deserves peer review, not a desk reject, because the result is important enough that the referee should spend the time inspecting the ancillary files. My recommendation: send it out, with an explicit request that the referee verify the linear equivalence claims and that the authors state clearly in the text how those were checked. I would not cite the fake quadric as a fact until that's resolved.

Referee Report

3 major / 4 minor

Summary. The paper develops a computational interpolation-and-lifting method for detecting highly singular members in families of algebraic varieties. It applies this method to a family of Z/2-Godeaux surfaces, producing a surface X whose singular locus is 2A1 + 2A3. On the minimal resolution X' the authors search for divisor relations that allow a Z/2 × Z/4 abelian cover via Pardini's theory. They define such a cover, compute its invariants, and conclude that its minimal model is a fake quadric with K^2 = 8 and p_g = q = 0. An appendix by Gleissner and Ruhland proves a lifting theorem for automorphisms of varieties isogenous to a product; this is used to argue that the new fake quadric is not a quotient of a product of curves, giving the first explicit example of this kind.

Significance. If the construction is correct, this is a substantial advance: it provides the first explicit fake quadric that is not a product-quotient, addressing a long-standing question in the geography of surfaces of general type. The computational method for detecting singular members in families is of independent interest. The paper is also commendable for shipping Magma ancillary files that are intended to certify the main numerical claims, and the appendix contains a useful general statement about automorphism groups of product varieties. However, the printed argument contains a serious gap between numerical and linear equivalence that is load-bearing for the existence of the cover, and the final product-quotient exclusion is only sketched via citations.

major comments (3)

[§5–§7, Eqs. (1)–(4)] Pardini's theorem requires linear equivalence of the building data, but the verification described in §5 and §6 establishes only numerical equivalence: §5 computes nullspaces of intersection matrices, and §6 explicitly speaks of 'numerical divisibility relations'. Since X' has Picard 2-torsion (π1(X) = Z/2), a numerically trivial class need not be linearly trivial. The congruences 2L2 ≡ N1+N2+N6+N8 and 4L5 ≡ 2N1+N3+2N4+3N5+3N6+2N7+N8 could fail in Pic(X') while holding in the intersection lattice. Please prove these relations as linear equivalences, e.g. by displaying the rational functions or by a Magma verification that the relevant linear systems contain the required divisors, or explain explicitly why the 2-torsion is absorbed. The h^0 computations in §8 would also need to be repeated for the correct line bundles if a torsion twist is present.
[§9] The key newness claim—that the fake quadric is not a quotient of a product of curves—is dismissed with 'By looking to their results we see that this does not happen', citing [BP12] and [FP15]. This is load-bearing and not checkable as stated. Please give the precise classification statement (theorem or table row) in those papers that excludes a surface with singular set 2A1 + 2A3, and include a short verification that the present X cannot arise in their lists. Without this, the 'first explicit non-product-quotient fake quadric' assertion is not independently supported by the text.
[Appendix, Theorem 4] The proof of Theorem 4 is not self-contained: the step that an automorphism of X0 lifts to the product is quoted from the authors' preprint [2, Theorem 2.8] via Remark 3.2, with no proof in the appendix. Since the argument in §9 depends on this lifting theorem, either include a complete proof of the unmixed case or clearly state it as an external theorem and provide the preprint's precise statement. As written, the appendix only proves the reduction from the mixed case to the unmixed case.

minor comments (4)

[§4, Step 3] The lifting of the two roots a_i, b_i is described vaguely: 'By choosing the integer ones we show the existence ... (the rational ones give an isomorphic surface)'. Please clarify the arithmetic reconstruction and justify the isomorphism claim, or state that this is verified in the ancillary files.
[§6] The text says 'There exist positive integers a_i, b_i such that ...' and then uses a nullspace computation to determine them. It would help to state explicitly whether the nullspace computation verifies numerical equivalence only or also linear equivalence, and to point to the exact Magma function that performs the check.
[§8] The vanishing h^0(X', K+L_i)=0 is stated as a computational result. Please either include the relevant Magma script in the printed appendix or describe how the emptiness of the linear system is certified (e.g., via Gröbner bases), so that the reader can reproduce the check without rerunning large searches.
[Appendix and §1] There are several typos: 'chracteristic' (§8), 'biholomorpism' and 'the rfore' (Appendix proof), 'varietiy' (Theorem 4), 'diagonal subgoup' (Appendix). The notation M_2^1 in §3 is never defined explicitly.

Circularity Check

1 steps flagged · score 4.0 of 10

The G-cover construction is largely self-contained; the only load-bearing circularity candidate is the appendix's reliance on the authors' own preprint [2] to complete the automorphism-lifting theorem used for the non-product-quotient claim.

self citation load bearing [Appendix, Remark 3(2) and proof of Theorem 4]
"In [2, Theorem 2.8], the authors discuss the special case G=G0. They show that an automorphism f∈Aut(X) lifts to the product C1×...×Cn if all of the subgroups H_i are trivial. Moreover, there is always a unique realization of X with this property. It is called the minimal realization of X. ... It suffices to provide a lift of f to \hat f∈Aut(X0), since by minimality we already know that any automorphism of X0 has a lift to an automorphism of C1×...×Cn, cf. Remark 3."

The proof of the appendix's central theorem reduces the mixed case to the unmixed case and then completes the argument by invoking [2, Theorem 2.8], a preprint by the same authors (Fallucca–Gleissner–Ruhland). The cited special case is exactly the remaining step needed to conclude that an automorphism of the unmixed quotient X0 lifts to the product; without it, the proof does not reach Aut(C1×...×Cn). The paper does not reproduce or independently verify that cited theorem, and it is not machine-checked or code-reproduced here. This makes the cited self-result load-bearing for the conclusion that the new fake quadric is not a quotient of a product of curves. The main G-cover construction does not depend on this step, so the circularity is partial rather than total.

full rationale

Most of the derivation is not circular. The divisibility relations (1) are found by numerical nullspace search before C and D are constructed; the curves are then produced by independent geometric conditions (hyperplane sections and multiplicity conditions), and the integer coefficients in (2) are checked against the predicted values. The later definitions L2 ≡ 2K−D'−N7 and L5 ≡ 2K−C' are algebraic rearrangements of (1), not a re-use of the quantities being constructed. The fake-quadric invariants come from Pardini's external formulas together with the reported h^0 vanishings, so the core construction is not equivalent to its inputs. The one genuinely load-bearing self-citation is in the appendix: Theorem 4 reduces to [2, Thm 2.8] by the same authors, and this supports the 'not a product quotient' part of the main novelty claim. The numerical-versus-linear-equivalence objection is a correctness/completeness concern about whether the ancillary Magma verification establishes linear equivalence, not a circularity of the kind where an equation reduces to itself by definition. Similarly, the final exclusion cites the external classifications [BP12] and [FP15], which are not self-citations. Overall, the central cover construction is independent, but the non-product-quotient conclusion leans on an unverified-in-this-paper self-citation, giving a score of 4 rather than 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The construction relies on established theories (Pardini abelian covers, product-quotient classifications, Magma correctness) rather than on fitted parameters or new postulated objects. The guessed divisor relations are later verified, so they are not free parameters.

assumptions (4)

domain assumption Pardini's theory of abelian covers correctly describes the Z/2 x Z/4 cover and its invariants.
Invoked in Sections 7-8 to compute K_S^2, p_g(S), and chi(S) from the building data.
domain assumption The classifications of Bauer-Pignatelli [BP12] and Frapporti-Pignatelli [FP15] are complete and correctly exclude the singular set 2A1+2A3 for Z/2-Godeaux product-quotient surfaces.
Used in Section 9 to conclude S is not a quotient of a product of curves; the exclusion is not demonstrated in the paper.
domain assumption Theorem 4 in the Appendix (lifting automorphisms for minimal realizations) is valid; its proof relies on Fallucca-Gleissner-Ruhland [2, Theorem 2.8].
Key to the non-product-quotient conclusion; the paper cites its own preprint for the unmixed case.
domain assumption Magma's interpolation, polynomial arithmetic, and cohomology computations are correct.
All key checks (divisibility relations, h^0 vanishings, irreducibility of C and D) are delegated to Magma ancillary files.

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Pith. "Pith review of Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric (With an Appendix by Christian Gleissner and Noah Ruhland)." pith.science (2026). https://pith.science/paper/VN3RLNYB

@misc{pith2026250908198,
  author       = {Pith},
  title        = {Pith review of: Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric (With an Appendix by Christian Gleissner and Noah Ruhland)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VN3RLNYB}},
  note         = {Machine review of arXiv:2509.08198}
}

abstract

We present a computational method for detecting highly singular members in families of algebraic varieties. Applying this approach to a family of numerical Godeaux surfaces, we obtain explicit examples with many singularities. In particular, we construct a Godeaux surface whose singular locus consists of two $\mathsf A_1$ and two $\mathsf A_3$ singularities. We show that this surface admits a $\mathbb{Z}/2 \times \mathbb{Z}/4$ abelian cover which is a smooth minimal surface of general type with invariants $K^2=8$ and $p_g=0$, i.e. a fake quadric. Together with the result in the Appendix, this provides the first explicit construction of a fake quadric that does not arise as a quotient of a product of curves.

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Catanese, Fibred surfaces, varieties isogenous to a product and related moduli spaces, Amer

F. Catanese, Fibred surfaces, varieties isogenous to a product and related moduli spaces, Amer. J. Math., 122 (2000), no. 1, 1--44

work page 2000

[2] [2]

On Rigid Varieties Isogenous to a Product of Curves

F. Fallucca, C. Gleissner, and N. Ruhland, On rigid varieties isogenous to a product of curves, Preprint, arXiv:2504.11032 [math.AG], 2025

work page Pith review arXiv 2025

[3] [3]

Narasimhan, Several complex variables, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1995

R. Narasimhan, Several complex variables, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1995. (Reprint of the 1971 original)

work page 1995

[4] [4]

I. C. Bauer, F. Catanese, and F. Grunewald. The classification of surfaces with \(p_g=q=0\) isogenous to a product of curves. Pure Appl. Math. Q. , 4(2):547--586, 2008

work page 2008

[5] [5]

Bosma, J

W. Bosma, J. Cannon, and C. Playoust. The M agma algebra system. I . T he user language. J. Symbolic Comput. , 24(3-4):235--265, 1997. Computational algebra and number theory (London, 1993)

work page 1997

[6] [6]

Borisov and J

L. Borisov and J. Keum. Explicit equations of a fake projective plane. Duke Math. J. , 169(6):1135--1162, 2020

work page 2020

[7] [7]

Bauer and R

I. Bauer and R. Pignatelli. The classification of minimal product-quotient surfaces with \(p_ g =0\) . Math. Comput. , 81(280):2389--2418, 2012

work page 2012

[8] [8]

D z ambi \'c and X

A. D z ambi \'c and X. Roulleau. Automorphisms and quotients of quaternionic fake quadrics. Pac. J. Math. , 267(1):91--120, 2014

work page 2014