{"id":"46b1de2c-7f3e-4d1b-b8f6-7ca12a7a2790","arxiv_id":"2509.08198","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new explicit fake quadric is constructed from a Z/2-Godeaux surface with two A1 and two A3 singularities, and shown to be the first example not arising as a quotient of a product of curves.","lead":"This paper uses computer searches over finite fields to find rare algebraic surfaces with many singularities, then builds a new 'fake quadric', a smooth surface with the same numerical invariants as a standard quadric. The result is the first explicit example of a fake quadric that cannot be written as a quotient of a product of curves.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"§7's cover is justified by numerical equivalences only; Pardini's building data require linear equivalence, and the Z/2-Godeaux torsion makes the gap non-formal.","rationale":"The reader's weakest assumption already targets the Magma verifications, and my concern sharpens one specific unverified step: the divisor congruences are apparently checked only numerically, while Pardini's abelian-cover theorem needs linear equivalence. If Pic(X') is not torsion-free—which is plausible for a Z/2-Godeaux surface—the cover may fail even if every intersection number computed in §§5–6 is correct. This is more load-bearing than the compressed Section 9 classification appeal, because it threatens the existence of the fake quadric itself, not just the non-product-quotient novelty claim. I am not asserting the construction is wrong: the paper ships Magma code, and the equations may in fact be verified by explicit divisors or rational functions. The defect is that the text does not establish the stronger equivalence, and the surrounding language ('numerical divisibility relations') suggests the weaker check. A targeted inspection of the code, or a torsion-sensitive re-verification, would settle the point. If the linear relations hold, the construction is credible and the reader's conditional acceptance should stand.","tokens_in":9717,"tokens_out":24571,"duration_ms":308515,"concrete_test":"Inspect the ancillary Magma verification of equations (3): determine whether it checks linear equivalence (e.g., by constructing the rational function whose divisor is the difference, or by calling a function like IsLinearlyEquivalent on the actual divisors on X') or only equality of intersection numbers. If only the latter, compute the torsion subgroup of Pic(X') (for instance via H_2(X',Z) or by testing the line bundle L_i ⊗ O_X'(-...) for triviality) and rerun the Pardini building-data check after any torsion twist, including the seven h^0(X',K+L_i) vanishings. If the linear relations hold exactly, the concern is settled; if not, the fake quadric claim is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The existence of the fake quadric rests on the G-cover in §7, which is defined using congruences (3). But the relations (1) that feed into (3) are established in §5–§6 only as numerical equivalences: §5 describes 'testing for numerical dependencies' via nullspaces of intersection matrices, and §6 explicitly speaks of 'numerical divisibility relations' and determines the a_i,b_i by intersecting with the N_i. Pardini's theorem [Par91] requires linear equivalence of the building data, not just numerical equivalence. This distinction is not automatically harmless here. X is a Z/2-Godeaux surface, so its fundamental group is Z/2 and the same 2-torsion is expected in H_1 of the resolution X' (the exceptional curves of the rational double points are rational trees); hence Pic(X') can have a nontrivial 2-torsion class. A divisor class can be numerically trivial while being linearly nontrivial, so the displayed equalities 2L2 ≡ N1+N2+N6+N8 and 4L5 ≡ ... could hold in the intersection lattice and yet fail in Pic(X'). In that case the line bundles L_i in §7–§8 do not satisfy the exact relations required for a G-cover, and the cover S'—and with it the fake quadric—may not exist. The h^0(X',K+L_i) vanishings are computed for the stated classes; a torsion twist would require recomputing them. The ancillary Magma files may well verify linear equivalence via explicit rational functions, but the printed argument does not say so.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10066,"tokens_out":15046,"duration_ms":178152,"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":[{"comment":"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.","section":"§5–§7, Eqs. (1)–(4)"},{"comment":"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.","section":"§9"},{"comment":"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.","section":"Appendix, Theorem 4"}],"minor_comments":[{"comment":"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.","section":"§4, Step 3"},{"comment":"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.","section":"§6"},{"comment":"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.","section":"§8"},{"comment":"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.","section":"Appendix and §1"}],"recommendation":"major_revision","confidential_remarks":"The central construction is promising and the computational scaffolding is a real strength, but the gap between numerical and linear equivalence in §5–§7 is exactly the kind of non-formal step that can invalidate the cover. I would therefore require the authors to supply the missing linear-equivalence verification or a convincing torsion argument before reconsidering. The final product-quotient exclusion should also be made explicit. If these points are addressed, the paper would be a strong candidate for acceptance; in its current form it is not yet a fully rigorous proof of the main claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nI 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.\n\nThis 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.","headline":"A likely genuine new fake quadric outside the product-quotient class, but the written proof skips the linear-equivalence check Pardini's theory demands.","tokens_in":10531,"tokens_out":4920,"would_cite":false,"duration_ms":56357,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J29"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["fake quadrics","Godeaux surfaces","surfaces of general type","abelian covers","surface singularities","product-quotient surfaces","finite-field interpolation","lifting to characteristic zero"],"falsifier":"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.","tokens_in":9624,"feed_emoji":"🧮","tokens_out":10031,"duration_ms":102811,"temperature":0.7,"pith_summary":"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.","feed_headline":"First explicit fake quadric outside curve-quotient classes","feed_subtitle":"A Godeaux surface with singularities 2A1+2A3 yields a smooth K^2=8, p_g=0 surface with explicit equations.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the abelian-cover machinery: reduced building data, and the formulas for p_g and chi used to compute the invariants of the Z/2 × Z/4 cover.","marker":"[Par91]"},{"why":"Identifies 2A1+2A3 as the singular set of a quotient of a quaternionic fake quadric by Z/2 × Z/4, motivating the search for this configuration.","marker":"[DR14]"},{"why":"Part of the product-quotient classification whose lists are consulted to prove that no Z/2-Godeaux surface with singular set 2A1+2A3 is a quotient of a product of curves.","marker":"[BP12]"},{"why":"Extends the product-quotient classification to mixed quasi-etale quotients with arbitrary singularities, completing the exclusion used in Section 9.","marker":"[FP15]"},{"why":"The computer algebra system in which the interpolation, finite-field search, and all certified verifications are performed.","marker":"[BCP97]"},{"why":"Supplies the parameterized moduli family of Z/2-Godeaux surfaces, including the subfamily from which the singular examples are extracted.","marker":"[DR20]"},{"why":"Classifies surfaces with p_g=q=0 isogenous to a product of curves, giving the framework for the claim that the fake quadric is not such a quotient.","marker":"[BCG08]"},{"why":"In the appendix, establishes the minimal realization and the lifting theorem that every automorphism of a variety isogenous to a product lifts to the product, the key input excluding product-quotient origin.","marker":"[2]"}],"fun_headline_variants":["First explicit fake quadric not from curve products","Godeaux surface gives first explicit fake quadric","New fake quadric via abelian cover of Godeaux","Explicit fake quadric from singular Godeaux surface","Concrete fake quadric beyond quotient of curves"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["First explicit fake quadric not from curve products","Godeaux surface gives first explicit fake quadric","New fake quadric via abelian cover of Godeaux","Explicit fake quadric from singular Godeaux surface","Concrete fake quadric beyond quotient of curves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000549,"raw_usage":{"total_tokens":2469,"prompt_tokens":767,"completion_tokens":1702,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":511,"completion_tokens_details":{"reasoning_tokens":1625}},"tokens_in":511,"tokens_out":1702,"duration_ms":13558,"temperature":1.0,"reasoning_tokens":1625,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T21:03:38.769577+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}