{"id":"3b1c78d4-b927-49d6-938c-80a1d6f1affb","arxiv_id":"1908.09330","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every surface isogenous to a product of mixed type with p_g=0 is a Mori dream surface, and its effective, nef, and semiample cones all coincide.","lead":"Surfaces made by taking a product of two identical curves and quotienting by a finite group that swaps the factors, with vanishing holomorphic forms, are shown to be Mori dream surfaces. This completes the proof that all reducible fake quadrics, surfaces with the same Hodge numbers as a smooth quadric, are Mori dream surfaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Families 2-5 rest on unarchived MAGMA output, and the stated application of Lemma 1.9 to (D2,D9,D7,D14) contradicts the listed pairings; the theorem is not independently verifiable as written.","rationale":"The reader identified the unverified MAGMA computations for families 2-5 as the weakest assumption; I agree this is the central load-bearing point. The theorem's strategy is to reduce to Lemma 1.9 and verify its hypotheses by computation, and for four of the five families that verification is neither printed nor archived. My stress-test adds a concrete internal inconsistency: the sentence applying Lemma 1.9 to (D2,D9,D7,D14) fails the lemma's hypotheses literally, though a reordering (D2,D7,D14,D9) would satisfy them. This makes the missing computation more salient rather than changing the underlying mathematical claim. Because the concern is about missing verification rather than a demonstrated falsehood, the verdict should remain conditional: the paper should provide the full MAGMA output (or an archived script) for families 2-5 and correct the Lemma 1.9 ordering before full acceptance.","tokens_in":10174,"tokens_out":12740,"duration_ms":121214,"concrete_test":"Reconstruct and run a MAGMA computation for families 2-5: for each family take the group G0 and generating vector of type [0;4^3] listed in Table 1, form the index-6 supergroup H = SmallGroup(768,1085341) with a generating vector [0;2,3,8] inducing the given G0-vector, then run the Appendix A orbit-divisor loop with H in place of G0 and print the full intersection matrix. Check that the matrix reproduces the five numerical equivalences in Theorem 3.5 for Case 2 and the analogous pattern for Cases 3-5, and verify that Lemma 1.9's zero-pairing hypothesis holds for the ordered quadruple (D2,D7,D14,D9). If any entry differs, the claimed cone equality for that family is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the main theorem for the four one-parameter families in Table 1 depends on a MAGMA computation that is only summarized as 'in the other 3 cases we have an analogous output' and backed by a personal webpage rather than an archived script. The printed Appendix A script covers only Family 1. The full intersection matrix for the 15 orbit divisors in families 2-5 is never shown, so the existence of four divisors satisfying Lemma 1.9 - the entire basis for EFF(S)=NEF(S)=SAMP(S) - is unverifiable from the manuscript. This is load-bearing: one wrong intersection number would destroy the four-divisor configuration. A concrete sign that the numerical claims need checking is that the instruction 'Applying Lemma 1.9 to D2,D9,D7,D14' cannot be read literally: the stated equivalences give D2.D14 = D2.D7 = 16 and D9.D7 = D2.D7 = 16, whereas Lemma 1.9 requires D1.D4 = 0 and D2.D3 = 0 for the ordered quadruple. A valid ordering would be (D2,D7,D14,D9). The theorem may still be true, but as printed the proof for four of the five families rests on unarchived, partially asserted computation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies surfaces isogenous to a product of mixed type with p_g = p_q = 0, which form five irreducible families (Table 1). The authors construct G-invariant divisors on C × C by taking orbits of graphs of automorphisms, compute their intersection numbers via fixed-point counts from Riemann Existence (Lemmas 3.1–3.4), and then use a numerical criterion (Lemma 1.9) to conclude that for every such surface S one has Eff(S) = Nef(S) = SAmp(S). Consequently S is a Mori dream surface, and in particular every reducible fake quadric is a Mori dream surface. The proof is case-by-case: Family 1 is supported by a printed MAGMA script, while Families 2–5 rely on MAGMA computations summarized in the text and accessible from a personal webpage.","tokens_in":10402,"tokens_out":3805,"duration_ms":36622,"significance":"If the result is correct, it is a substantial contribution: it gives the first uniform statement that all reducible fake quadrics are Mori dream spaces, and it provides an explicit description of their effective, nef, and semiample cones. The method is elegant: it avoids Cox-ring computations by using orbit divisors and the elementary Lemma 1.9, and the fixed-point counting in Lemma 3.4 is a clean application of the Riemann Existence Theorem. The paper also ships a machine-checkable script for Family 1, which is a strength. The main weakness is that the computational verification for Families 2–5 is not fully shipped or archived, and one application of Lemma 1.9 is stated with an incorrect ordering. These issues are local and fixable, but they are load-bearing because the main theorem depends on them.","major_comments":[{"comment":"The proof for Families 2–5 depends entirely on MAGMA computations that are not included in the manuscript. The text states that in the other three cases one obtains an analogous output, and the supporting material is hosted on a personal webpage that is not a stable archive. The full intersection matrix for the 15 orbit divisors in each family is never printed, so the existence of four divisors satisfying the hypotheses of Lemma 1.9 cannot be verified from the manuscript alone. Since a single incorrect intersection number would invalidate the cone equality, I ask that the complete computations (code, output tables, or a permanent DOI) be provided for all five families, not just Family 1.","section":"Section 3, Theorem 3.5, Families 2–5"},{"comment":"The sentence 'Applying Lemma 1.9 to D2, D9, D7, D14' is inconsistent with the numerical equivalences stated in items 1 and 2. With the ordered quadruple (D1,D2,D3,D4) = (D2,D9,D7,D14), Lemma 1.9 requires D1.D4 = 0, but the stated equivalences give D2.D14 = D2.D7 = 16, and similarly D9.D7 = D2.D7 = 16. The correct ordering is (D2,D7,D14,D9), for which D2.D9 = D2^2 = 0 and D7.D14 = D7^2 = 0. Please correct the ordering and verify the analogous ordering in Families 3–5 as well.","section":"Section 3, Theorem 3.5, Case 2"},{"comment":"The construction of the large automorphism group H = G(768, 1085341) and the claim that the covering c : C → P^1 is Galois are justified by a MAGMA script available only at an external URL. This is a separate, load-bearing computational step: without it, the existence of the orbit divisors for Families 2–5 is not established. The script and its output should be archived, or the relevant group-theoretic checks (e.g., the uniqueness of H and the lifting of generating vectors) should be reproduced in the appendix.","section":"Section 2.1 and Theorem 3.5"}],"minor_comments":[{"comment":"The printed script computes self-intersections and pairings but does not label the orbit divisors D_i in a way that lets the reader match the numerical output to the names used in Theorem 3.5. Adding a table that maps the orbit index to the divisor name would make the verification transparent.","section":"Appendix A"},{"comment":"There is a typo: 'contruct extremal rays' should be 'construct extremal rays'.","section":"Introduction"},{"comment":"The proof asserts that Jac(f_1^{-1} f_2)_{x0} ≠ 1 for a non-identity automorphism fixing x0. This is true for a finite-order automorphism of a germ, but the justification should be stated explicitly, since Cartan's Lemma alone gives the linearization, not the non-identity of the linear part.","section":"Lemma 3.3"},{"comment":"The notation KX is used for the canonical divisor, whereas elsewhere the surface is denoted S and the canonical divisor KS; please unify the notation.","section":"Theorem 3.5, Family 1"},{"comment":"The first data row of Table 1 appears garbled in the arXiv typeset version; please ensure the journal version renders the H1(S,Z) entry correctly.","section":"Table 1"}],"recommendation":"major_revision","confidential_remarks":"The paper's central idea is sound and the result is likely correct, but the current manuscript does not allow independent verification of the computations that carry four of the five families. I recommend requiring the authors to supply complete, archived computational artifacts for Families 2–5 and to fix the misordered application of Lemma 1.9. The external personal webpage is not acceptable as the sole repository for load-bearing computations; a permanent archive (e.g., a DOI or a journal-hosted supplement) should be required."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is that this is a genuine extension, not a paradigm shift. Frapporti and Lee prove that every mixed-type surface isogenous to a product with p_g=0 is a Mori dream surface with Eff=Nef=SAmp, completing the reducible fake quadrics story when combined with the unmixed case in [KL19]. The new idea is the orbit-divisor construction with a subgroup H of Aut(C) properly containing G0, which is needed exactly for the bimodule cases where G0 alone gives only one divisor class. Lemma 1.9 is a clean reduction: once you find four irreducible divisors with the right intersection pattern, the cone equality follows formally. The theoretical framework is sound. Lemma 3.2 and 3.4 are standard, the application of Riemann Existence is correct, and the reduction to the classification in [BCG08, Fra13] is legitimate. Family 1 is fully verified by the printed MAGMA script, which is reproducible in principle. The construction of H for families 2-5 is explicit and credible, and the numerical equivalences for Case 2 are consistent with the stated intersection numbers—modulo the typo below.\n\nThe soft spots are real but not fatal. First, the verification for families 2-5 depends on MAGMA output that is not shipped. The appendix prints only the Family 1 script; for the rest the authors point to a personal webpage without a stable archive and summarize the output as 'analogous'. That is a reproducibility gap, and it is load-bearing for the main theorem. A referee cannot check that four divisors satisfying Lemma 1.9 actually exist without re-running an unarchived computation. Second, the specific application of Lemma 1.9 in Theorem 3.5 is mistyped: the ordered quadruple (D2,D9,D7,D14) does not satisfy the lemma's hypothesis, because D2.D14 and D9.D7 equal D2.D7=16, not 0. The stress-test note is right about this. The fix is trivial—use (D2,D7,D14,D9) instead—but as printed the proof contains a false assertion. Third, the phrase 'in the other 3 cases we have an analogous output' is too compressed for a serious referee.\n\nNone of this undermines the central theorem; the argument is well-structured and the errors are easily corrected. The paper deserves a serious referee. I would send it to peer review and ask for: an archive of all five MAGMA scripts (or a stable repository), the full intersection matrices for families 2-5, and a corrected Lemma 1.9 ordering. The reader's soundness score of 6 is proportionate.","headline":"Solid extension proving the Mori dream property for mixed-type surfaces isogenous to a product with p_g=0, but families 2-5 rest on unarchived MAGMA output and one Lemma 1.9 application is mistyped.","tokens_in":10955,"tokens_out":2292,"would_cite":true,"duration_ms":23357,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J29","14E30","14J50","14H37","14L30","14Q10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every surface isogenous to a product of mixed type with p_g=0 is a Mori dream surface, and its three divisor cones coincide.","keywords":["Mori dream space","fake quadrics","surfaces isogenous to a product","mixed type","effective cone","nef cone","semiample cone","orbit divisors"],"falsifier":"Recompute the intersection numbers among the orbit divisors for families 2–5 and check whether the asserted numerical pattern holds: $D_2^2=0$, $D_7^2=0$, $D_2\\cdot D_7=16$, $D_1\\sim (D_2+D_7)/2$, $D_5\\sim D_2+D_7$, and $D_3\\sim 2(D_2+D_7)$. A single failure would remove the four divisors required by Lemma 1.9, and the cone equality would no longer follow.","tokens_in":9955,"feed_emoji":"📐","tokens_out":13232,"duration_ms":114500,"temperature":0.7,"pith_summary":"The paper proves a uniformity statement about a class of surfaces of general type: those obtained as free quotients of a product of a curve with itself by a mixed group action, with geometric genus zero. For each such surface $S$ the effective, nef, and semiample cones in the real space $N^1(S)$ are not merely nested but equal, and they are generated by the classes of two orbit divisors. The interest is that these surfaces include all reducible fake quadrics—smooth minimal surfaces with the same Hodge diamond as a quadric, $p_g=q=0$ and $K^2=8$—whose divisor theory was largely open. A Mori dream surface is one whose birational geometry is governed by finitely many polyhedral cones and a finitely generated Cox ring.","feed_headline":"All reducible fake quadrics are Mori dream surfaces","feed_subtitle":"The paper proves Eff=Nef=SAmp for mixed-type surfaces with p_g=0, settling the cone geometry.","key_machinery":"The load-bearing construction is the orbit divisor induced by a subgroup $H$ of $\\mathrm{Aut}(C)$. For $f\\in H$, the graph $\\Delta_f=\\{(x,f(x))\\}\\subset C\\times C$ is moved by the ambient group $G$; the reduced sum of the $G$-orbit of $\\Delta_f$ descends to an irreducible effective divisor on $S$. The paper computes intersection numbers of these divisors by counting fixed points of automorphisms through the Riemann Existence Theorem, which relates fixed points to branch data, and then applies a numerical lemma: four distinct effective irreducible divisors $D_1,\\dots,D_4$ with $D_i^2=0$, $D_1\\cdot D_4=D_2\\cdot D_3=0$, and the four remaining mixed intersections positive and equal, force $\\mathrm{Eff}(S)=\\mathrm{Nef}(S)=\\mathrm{SAmp}(S)=\\mathbb{R}_{\\geq 0}\\langle D_1,D_2\\rangle$. In the five-point branch case the subgroup $H$ equal to the diagonal part $G_0$ suffices; in the three-point branch cases the symmetries of the three branch points are lifted to a larger automorphism group of $C$.","core_discovery":"Let $S=(C\\times C)/G$ be a surface isogenous to a product of mixed type with $p_g=q=0$, where $G$ is a finite group acting freely and exchanging the two factors. The paper establishes that $\\mathrm{Eff}(S)=\\mathrm{Nef}(S)=\\mathrm{SAmp}(S)$ and that this common cone is $\\mathbb{R}_{\\geq 0}\\langle D_1,D_2\\rangle$ for two explicit orbit divisors. Hence $S$ is a Mori dream surface and carries no negative curves. Since the unmixed case was already settled, every reducible fake quadric is a Mori dream surface. The proof is case-by-case over the five classified families, using orbit divisors—sums of graphs of automorphisms of $C$ pushed down to $S$—whose intersection products are computed from branch data and verified computationally.","pith_inferences":["The four-divisor criterion of Lemma 1.9 could serve as a cheap numerical test for the Mori dream property in other surfaces of general type with Picard number two: if four irreducible effective divisors with exactly that intersection pattern can be exhibited, the cone equality follows without further geometric input.","The number of orbit divisors (four in the first family, fifteen in the other four) is likely not essential; only the existence of two numerically equivalent pairs of divisors matters, so one could search for such witnesses in other constructions.","The success of lifting branch-point symmetries in the three-point cases suggests a general recipe: when the branch configuration of a cover has extra symmetries, those symmetries may enlarge the automorphism group and produce the orbit divisors needed to span the cones.","The authors raise whether every fake quadric, including irreducible ones, is a Mori dream surface; Lemma 1.9 gives a concrete way to attack that question by looking for four divisors with the same intersection table."],"forward_implications":["Every surface isogenous to a product of mixed type with p_g=0 is a Mori dream surface, so its Cox ring is finitely generated.","Combined with the previously known unmixed case, all reducible fake quadrics are Mori dream surfaces.","For each of the five families the divisor cone is the single chamber spanned by two semiample divisor classes; there are no negative curves and no nontrivial Mori chamber decomposition.","The cone equality means that inside this cone every effective divisor class is also semiample, so every effective divisor eventually has a base-point-free multiple.","The same numerical pattern—four orbit divisors satisfying the intersection conditions of Lemma 1.9—recurs across all five families, despite their different group-theoretic data."],"supporting_citations":[{"why":"Supplies the structure theorem for mixed actions on C×C, including the action formula and the invariants of the quotient surface.","marker":"[Cat00]"},{"why":"Classifies surfaces with p_g=q=0 isogenous to a product and provides the five families with their group data.","marker":"[BCG08]"},{"why":"Completes the classification of mixed surfaces with p_g=q=0, giving the group and generating-vector data used in the five families.","marker":"[Fra13]"},{"why":"Proves that unmixed-type surfaces with p_g=0 are Mori dream surfaces, the result this paper extends.","marker":"[KL19]"},{"why":"Defines Mori dream spaces, the property whose verification is the paper's goal.","marker":"[HK00]"},{"why":"Gives the criterion used to conclude a surface is a Mori dream space from a rational polyhedral effective cone and the equality Nef=SAmp.","marker":"[AHL10]"},{"why":"Provides the basic inclusions among semiample, movable, nef, and effective cones that structure the proof.","marker":"[AL11]"},{"why":"Supplies the Riemann Existence Theorem formulation used to count fixed points of automorphisms in Lemma 3.4.","marker":"[Mir95]"},{"why":"The computer algebra system used for the group-theoretic computations in the appendix script.","marker":"[BCP97]"}],"fun_headline_variants":["Reducible fake quadrics are all Mori dream surfaces","Eff=Nef=SAmp on mixed-type surfaces with p_g=0","Cone equality settles reducible fake quadrics as Mori dream","Mixed-type p_g=0 surfaces: Eff=Nef=SAmp","Every reducible fake quadric is Mori dream"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The printed intersection numbers among the orbit divisors for families 2–5 are correct, even though the paper shows the verifying script only for family 1 and states that the other four families give analogous output.","fun_headline_variants_meta":{"raw":{"variants":["Reducible fake quadrics are all Mori dream surfaces","Eff=Nef=SAmp on mixed-type surfaces with p_g=0","Cone equality settles reducible fake quadrics as Mori dream","Mixed-type p_g=0 surfaces: Eff=Nef=SAmp","Every reducible fake quadric is Mori dream"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001038,"raw_usage":{"total_tokens":4271,"prompt_tokens":748,"completion_tokens":3523,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":364,"completion_tokens_details":{"reasoning_tokens":3434}},"tokens_in":364,"tokens_out":3523,"duration_ms":24357,"temperature":1.0,"reasoning_tokens":3434,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:14:52.525606+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the intersection numbers among the orbit divisors for families 2–5 and check whether the asserted numerical pattern holds: $D_2^2=0$, $D_7^2=0$, $D_2\\cdot D_7=16$, $D_1\\sim (D_2+D_7)/2$, $D_5\\sim D_2+D_7$, and $D_3\\sim 2(D_2+D_7)$. A single failure would remove the four divisors required by Lemma 1.9, and the cone equality would no longer follow.","supporting_citations":[],"review_version":1}