{"id":"938ec7c8-1b04-4349-9c0d-084539599fb9","arxiv_id":"2512.03213","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Explicit bicanonical equations are produced for the fake projective plane (C18,p=3,{2I}), adding the 12th pair of fake projective planes to the list of those with known equations.","lead":"This paper writes down explicit polynomial equations for a new fake projective plane, an exotic algebraic surface that shares the invariants of the ordinary projective plane - only the 12th of 50 conjugate pairs to be captured by equations. The authors reach it by starting from a sibling surface and climbing up and down a carefully engineered tower of covers and quotients.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Final equations are never shown to carry any invariant distinguishing (C18,p=3,{2I}) from other fake projective planes; the Cartwright-Steger label rests entirely on the authors' GAP computation and construction path.","rationale":"The paper is a serious computational construction with candid disclaimers and a genuine exact verification of the fake-projective-plane property in Step 7. I do not regard the finite-field smoothness certificate as a decisive obstacle: for an integral projective model, the rank of the Jacobian matrix can only drop under specialization, so smoothness of the reduction over F_4363 already rules out singularities over the number field once the equations have coefficients in Z[sqrt(-2)]; the missing flatness argument is standard, though it should be written down. The label identification is different: it is the one load-bearing assertion that has no independent witness. Unlike the FPP certificate, which is a property of the final equations, the label (C18,p=3,{2I}) is asserted to follow from the authors' own GAP computation and the construction path, and the paper explicitly leaves a choice unresolved in Step 5 (Remark 3.3). An output that satisfied Step 7 would still be a fake projective plane even if it corresponded to a different C18-class pair, so the headline claim 'new pair ... labeled by (C18,p=3,{2I})' would fail. The proposed automorphism-group check is concrete and feasible because Aut(X) embeds into PGL(H^0(2K)) for a fake projective plane in its bicanonical embedding. Hence the reader's CONDITIONAL verdict is appropriate; no verdict change is needed.","tokens_in":15784,"tokens_out":21273,"duration_ms":208798,"concrete_test":"Compute the automorphism group of the final ideal: find all T in PGL_10 over Q(sqrt(-2)) such that T preserves the 84-dimensional space of cubics in BetterNewFPPrr. Compare the resulting finite group (order and conjugacy classes) with the automorphism group of (C18,p=3,{2I}) in the Cartwright-Steger classification / GAPdataAll. If they do not match, the label is wrong. If the candidate C18-class pairs share the same automorphism group, additionally compute the nonreduced linear cuts (linear forms L whose restriction to the surface is twice a curve); the set of such cuts encodes the Picard 2-torsion data in the label and separates the four pairs.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is the label identification, not the finite-field smoothness certificate. Step 7 establishes that the 84 cubics define some fake projective plane, but nothing in that verification computes a numerical, homological, or group-theoretic invariant of the final surface that separates (C18,p=3,{2I}) from the other fake projective planes—in particular from the three remaining commensurable C18-class pairs (C18,p=3,{2},D3), (C18,p=3,{2},(dD)_3), and (C18,p=3,{2},(d^2D)_3). The label is inherited from Proposition 1.2, whose proof is 'a result of the GAP computation in GAPdataAll', and from the construction path through 72.P^2_fake. This is not a purely formal deduction from the output: in Step 5 the authors solve 'one of the systems' (Remark 3.3 explicitly says they do not know which irreducible subrepresentations the solution corresponds to) and then choose lifts of generators of C3×SL(2,Z/3Z). A different valid choice could in principle lead to a different quotient, and Step 7 would not detect it because all fake projective planes share the same Hodge numbers, K^2=9, and bicanonical Hilbert polynomial. Thus the paper's central 'new pair' claim depends on an unaudited identification; a distinguishing invariant of the final equations is needed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper gives an explicit set of 84 cubic equations in 10 variables (auxiliary file BetterNewFPPrr) which, it claims, cut out the bicanonical embedding in CP^9 of the fake projective plane labeled (C18,p=3,{2I}) in the Cartwright–Steger classification. The construction starts from the known equations of the commensurable fake projective plane (C18,p=3,∅,d_3D_3), works up a chain of cyclic covers through a common Galois cover 72.P^2_fake with automorphism group of order 648, and then takes invariants to descend to the target plane. Steps 1–6 are numerical and use p-adic lifting, high-precision point sampling, and algebraic number recognition; Step 7 is an exact verification in Magma: it checks the Hilbert polynomial over the number field, smoothness via three Jacobian minors over F_4363, vanishing of h^1 of the structure sheaf and cotangent bundle, and h^2(2K(-1))=0. The authors conclude that the equations define a fake projective plane and \"confidently\" identify it as (C18,p=3,{2I}).","tokens_in":16036,"tokens_out":16769,"duration_ms":180148,"significance":"If the label and smoothness claims are fully justified, the paper adds a twelfth conjugate pair to the short list of fake projective planes with explicit bicanonical equations. The high-degree common-cover strategy is technically nontrivial and may open a path to the three remaining pairs in the same commensurability class. The exact verification files are a genuine strength: they turn the numerics into machine-checked certificates for the Hilbert polynomial, smoothness, and the relevant cohomology vanishings. The main unresolved point is not whether the equations define some fake projective plane—the Step 7 invariants all point that way—but whether the specific Cartwright–Steger label is actually established by the data presented.","major_comments":[{"comment":"The title and abstract claim the equations define the specific pair (C18,p=3,{2I}), but Step 7 verifies only properties common to every fake projective plane: correct Hilbert polynomial, smoothness, h^1(O)=h^2(O)=0, h^1(cotangent)=0, K^2=9, and bicanonical embedding. These invariants do not separate (C18,p=3,{2I}) from the other pairs in the same C18 commensurability class, in particular (C18,p=3,{2},D_3), (C18,p=3,{2},(dD)_3), and (C18,p=3,{2},(d^2D)_3). The label is inherited from Proposition 1.2, whose proof is a GAP computation in GAPdataAll, and from the construction path. However, Remark 3.3 explicitly states that the authors do not know which irreducible subrepresentations the solved system corresponds to, and Step 5 says \"we solved one of the systems\" and \"picking lifts of the generators.\" Different valid choices in Step 5 could lead to a different quotient, and Step 7 would not","section":"§3.5, §3.7, Proposition 1.2"},{"comment":"The smoothness certificate needs a stated justification. The text says that adding three Jacobian minors to the equations gives zero Hilbert polynomial over F_4363, and concludes smoothness of the surface over the number field. This implication is true under a standard closedness argument—the singular locus of the generic fiber is a closed subscheme of the projective scheme over a localization of Z[√-2], and if nonempty it would specialize to a singular point of the F_4363 fiber—but the manuscript does not provide that argument. Since smoothness is load-bearing for the fake projective plane verification, the authors should either state this spreading-out/closedness argument explicitly or supply an alternative flatness/degeneration argument. This is a fixable omission, not an observed failure.","section":"§3.7"}],"minor_comments":[{"comment":"The title contains typographical artifacts: \"F AKE PROJECTIVE PLANE\" and the author name \"W ANG\" should be typeset normally.","section":"Title/Abstract"},{"comment":"\"Mathemaica\" is a typo for \"Mathematica\". Also the phrase \"For better or for worse\" in §3.7 is informal for a journal article.","section":"§1.2"},{"comment":"The commutative diagram labels such as \"9. [P2_fake\" and \"8.P2_fake/C3\" are hard to parse because of missing spaces and the unusual use of a bracket for the complex-conjugate pair. Please introduce a consistent notation for conjugate surfaces and label all arrows.","section":"§1.1 diagram"},{"comment":"The torsion subgroup notation is ambiguous: \"C2_2 × C13\" and similar entries should be typeset as C_2^2 × C_13, C_8^2 × C_2^2, etc., or explained in a sentence.","section":"Table (2.1)"},{"comment":"The words \"we speculated\" and \"it seems plausible\" are appropriate for heuristics, but since Step 7 later verifies the final output, it would be clearer to mark these as provisional assumptions that are subsequently certified, rather than as evidence for the final claim.","section":"§3.6"}],"recommendation":"major_revision","confidential_remarks":"The computational achievement is substantial and the exact verification scripts are a strength. My main concern is the label: the paper's headline claim names a specific Cartwright–Steger pair, but the final verification does not separate that pair from the other three C18-class pairs, and the construction path itself contains explicit branch ambiguities. This is fixable if the authors can compute a distinguishing invariant of the final equations or give a rigorous tracking of the branch choices. The finite-field smoothness certificate also deserves one short paragraph explaining why it implies smoothness over the number field. I would be comfortable with the paper after those additions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one if you care about explicit equations of fake projective planes. Borisov and Wang deliver a 12th conjugate pair, (C18,p=3,{2I}), and they do it the honest way: heavy numerics up front, then a genuinely exact check in Step 7. The Hilbert polynomial, smoothness certificate, h^1 vanishing, and bicanonical condition are all computed over an exact finite field or number field. So the existence of a fake projective plane cut out by their 84 cubics is on solid ground.\n\nThe paper is also unusually candid. Section 1.2 says the intermediate cover computations are not fully rigorous, and the main body flags where they didn't verify that a map is an embedding. That matches the actual practice: the final verification is what carries the claim.\n\nNow the soft spots, in decreasing order of importance. The label (C18,p=3,{2I}) is not derived from any invariant of the final equations. Step 7 proves the surface is a fake projective plane, but all FPPs have the same Hodge numbers, K^2, and bicanonical Hilbert polynomial. The identification rests on Proposition 1.2, whose proof is \"a result of the GAP computation in GAPdataAll.\" The GAP data is in the auxiliary files, so a determined reader could audit it, but the paper itself gives no way to check that the quotient they built is the Cartwright-Steger object with that label rather than one of the three other C18-class pairs. This is not a fatal flaw—the construction path is a legitimate identification if the GAP computation is correct—but the authors should make the audit trail clearer, or compute a distinguishing invariant on the output (e.g., torsion in Pic or fundamental group).\n\nThe finite-field smoothness check (three Jacobian minors over F_4363) is actually sufficient for smoothness over the number field, because the non-smooth locus is closed and proper over the base: empty special fiber kills the generic fiber. The paper doesn't say this, so it reads as a gap; it's a missing sentence, not a missing argument.\n\nMinor: the final equations live in an auxiliary file with no checksum, and the paper doesn't give the output quantities from Step 7 (the Hilbert polynomial, cohomology dimensions) in the text. Both are fixable in revision.\n\nThe paper is for the small community of people computing bicanonical equations of fake projective planes and arithmetic ball quotients. It deserves a serious referee and, after the label discussion and the smoothness sentence are added, it should be accepted. I'd take it to the reading group to talk about how much auditability is required of a computational classification claim.","headline":"A solid, honest computational paper that almost certainly produces the equations of a new fake projective plane; the only real soft spot is that the Cartwright-Steger label is inherited from a GAP computation rather than from an invariant of the final equations.","tokens_in":16663,"tokens_out":7336,"would_cite":true,"duration_ms":72194,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J29","14Q10","14E20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Explicit 84 cubic equations in ten variables cut out the bicanonical embedding of the fake projective plane (C18,p=3,{2I}) in CP^9, produced by descending from a common Galois cover of degree 648.","keywords":["fake projective plane","explicit equations","bicanonical embedding","algebraic surfaces of general type","cyclic covers","quotients","common Galois cover","K^2=9"],"falsifier":"Run the smoothness test for the 84 cubics over ℚ(√−2) directly—for example, check that the ideal plus three Jacobian minors has empty variety at several primes or by a Gröbner basis computation—or compute the fundamental group of the resulting surface; if the group or the Picard torsion differs from the classified pair, the claim fails.","tokens_in":15518,"feed_emoji":"📐","tokens_out":6705,"duration_ms":61491,"temperature":0.7,"pith_summary":"This paper aims to give explicit equations for a fake projective plane—a smooth complex surface with the same Hodge numbers as CP^2 but not isomorphic to it. It claims that 84 cubic equations in 10 variables, with coefficients in Z[√−2], define the bicanonical embedding of the fake projective plane labeled (C18,p=3,{2I}), a new conjugate pair in the standard classification. The route is a computational descent: start from a known commensurable fake projective plane, climb to a common Galois cover by cyclic covers and quotients, then take invariants to reach the target. If correct, this adds a 12th conjugate pair to the list of fake projective planes with explicit polynomial equations and demonstrates a transfer method that may finish the remaining three pairs.","feed_headline":"84 cubic equations define a new fake projective plane","feed_subtitle":"A 12th conjugate pair of these CP^2-lookalike surfaces now has explicit equations, with three pairs left to go.","key_machinery":"The central mechanism is a chain of cyclic covers and quotients centered on a common Galois cover 72.P^2_fake, whose automorphism group is the semidirect product C3 × (SL(2,Z/3Z) ⋉ (C3×C3)). The intermediate surface 8.P^2_fake is the quotient by the C3×C3 subgroup and is itself a Q8 cover of the starting plane. The critical step is an identity s1s4 = s2s3 among bicanonical sections on 8.P^2_fake; solving it, after p-adic lifting and numeric recognition, determines the C3×C3 cover from 8.P^2_fake to 72.P^2_fake and yields a basis of the 71-dimensional space of canonical sections of the common cover. Averaging over the appropriate subgroups then produces the invariants that become the equation","core_discovery":"The paper claims that the 84 cubic equations in ten variables listed in the auxiliary data, with coefficients in Z[√−2], define the bicanonical embedding in CP^9 of the fake projective plane labeled (C18,p=3,{2I}) in the classification of such surfaces. The construction starts from the known equations of a commensurable fake projective plane, builds a common Galois cover with automorphism group C3 × (SL(2,Z/3Z) ⋉ (C3×C3)) of order 648, and realizes the target plane as a quotient of that cover by a subgroup of order 72. The verification passes through the Hilbert polynomial, a smoothness check over a finite field, the dimensions of cohomology groups, and the condition h^2(2K(−1))=0 that ident","pith_inferences":["The label (C18,p=3,{2I}) is inherited from the group-theoretic computation that constructed the covers, not from an invariant computed directly on the final equations; a direct fundamental-group or torsion computation from the cubics would settle the identification independently.","The smoothness certificate at a single finite field is not accompanied by a flatness or degeneration argument; repeating the Jacobian-minor test at other primes or over the number field would be a cheap check that the certificate is not an artifact of that prime.","The coefficient size (20–30 digits over Z[√−2]) suggests the equations may admit an even more structured presentation, possibly connected to a different basis of the bicanonical space, which could be worth exploring for the remaining pairs."],"forward_implications":["If the equations are correct, the fake projective plane (C18,p=3,{2I}) becomes fully computational: one can evaluate divisors, compute automorphisms, and search for special curves on it directly from the polynomial system.","The cover-and-descend strategy provides a template for the three remaining fake projective planes commensurable with these two, since they all sit under the same kind of common cover.","The verification establishes that the surface has Hodge numbers of CP^2, self-intersection of the canonical class 9, and embeds bicanonically into CP^9, so it is a genuine fake projective plane rather than a singular or noncanonical model.","The explicit equations over Z[√−2] allow reduction modulo many primes, making the surface available for arithmetic tests such as point counts and zeta-function computations."],"fun_headline_variants":["84 cubics define a new fake projective plane","New fake projective plane solved with 84 equations","A 12th fake projective plane gets explicit form","Explicit equations for a new CP^2 lookalike"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The construction's load-bearing premise is that the high-precision numerical and p-adic computations converged to the true algebraic surface and that the finite-field smoothness test correctly certifies smoothness over the number field; if either fails, the identification as (C18,p=3,{2I}) collapses.","fun_headline_variants_meta":{"raw":{"variants":["84 cubics define a new fake projective plane","New fake projective plane solved with 84 equations","A 12th fake projective plane gets explicit form","Explicit equations for a new CP^2 lookalike"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000299,"raw_usage":{"total_tokens":1505,"prompt_tokens":621,"completion_tokens":884,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":365,"completion_tokens_details":{"reasoning_tokens":821}},"tokens_in":365,"tokens_out":884,"duration_ms":9092,"temperature":1.0,"reasoning_tokens":821,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:41:40.765148+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the smoothness test for the 84 cubics over ℚ(√−2) directly—for example, check that the ideal plus three Jacobian minors has empty variety at several primes or by a Gröbner basis computation—or compute the fundamental group of the resulting surface; if the group or the Picard torsion differs from the classified pair, the claim fails.","supporting_citations":[],"review_version":1}