Finding equations of the fake projective plane $(C18,p=3,\{2I\})$

Bojue Wang; Lev Borisov

REVIEW 2 major objections 5 minor 21 references

Finding equations of the fake projective plane $(C18,p=3,\{2I\})$

T0 review · 2 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2512.03213 v2 pith:PMJ7E2QP submitted 2025-12-02 math.AG

Lev Borisov , Bojue Wang This is my paper · ORCID

classification math.AG MSC 14J2914Q1014E20

keywords fakeprojectiveplaneexplicitequationsbicanonicalembeddingalgebraicsurfacesofgeneraltypecycliccoversquotientscommonGaloiscoverK^2=9

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 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.

What carries the argument

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

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

The 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.

Now 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).

The 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.

Minor: 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.

The 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.

Referee Report

2 major / 5 minor

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}).

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 (2)

[§3.5, §3.7, Proposition 1.2] 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
[§3.7] 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.

minor comments (5)

[Title/Abstract] The title contains typographical artifacts: "F AKE PROJECTIVE PLANE" and the author name "W ANG" should be typeset normally.
[§1.2] "Mathemaica" is a typo for "Mathematica". Also the phrase "For better or for worse" in §3.7 is informal for a journal article.
[§1.1 diagram] 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.
[Table (2.1)] 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.
[§3.6] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new equations are produced from known input equations via a cover/quotient chain, and Step 7 verifies the output is an FPP using invariants independent of the target label.

full rationale

The derivation chain starts from the published equations of P^2_fake (C18,p=3,∅,d3D3) and builds 2.P^2_fake, 4.P^2_fake, 8.P^2_fake, 72.P^2_fake, then 9.[P^2_fake and [P^2_fake by invariant-theoretic operations. The final equations are not fitted to a target set of data; they are produced by solving polynomial systems, lifting p-adically, and recognizing algebraic numbers. Step 7's verification (quoted: 'we first observed that the surface S in question has the correct Hilbert polynomial... showed that it is smooth... computed the dimension of the cohomology spaces... This allowed us to conclude that the surface is an FPP') checks the output against FPP-defining invariants that do not depend on the construction path or on the Cartwright-Steger label. No equation is used as both input and output, and no fitted parameter is relabeled as a prediction. The main caveat is that the specific label (C18,p=3,{2I}) is inherited from Proposition 1.2, whose proof cites the authors' GAP data ('This is a result of the GAP computation in GAPdataAll, see [9]'), and Step 7 does not compute a separate invariant distinguishing this FPP from the three remaining commensurable C18-class pairs. That is a verification/auditability gap about the label, not a circular reduction: the FPP existence claim is checked independently, and the label is a conditional consequence of the construction path rather than a definition of the equations. Under the stated rules, this does not rise to circularity.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No physics-style fitted constants are present; the numbers in the cut (3.3) are outputs of an exhaustive finite-field search plus p-adic algebraic number recognition, not tunable parameters. The only genuine hand-chosen items are algorithmic (branch choices, basis normalizations, scaling equations, lattice reduction), which the paper argues do not affect the resulting surface. The intermediate surfaces (2.P2, 4.P2, 8.P2, 72.P2, 9.[P2) are existing objects from the arithmetic-ball-quotient classification, not invented entities.

free parameters (1)

Choice of solution branch (a,b) in (Z/3)^2 and irreducible subrepresentation in Step 5 = unknown (9 candidate pairs; one system solved; subrep choice left ambiguous per Remark 3.3)
The construction of the C3xC3 cover yielding 72.P2 is obtained by solving a polynomial system whose correct branch is not pinned down in the text; the authors argue the final result is independent of this (Remark 3.3), but the argument is not fully explicit. This is a hand-chosen branch, not a fitted numerical constant.

assumptions (6)

domain assumption Cartwright-Steger classification of the 100 fake projective planes as quotients of the complex 2-ball by arithmetic groups
Used throughout (Introduction, Section 1.1) for the FPP labels, the torsion table (Prop. 2.1), and the commensurability class; external benchmark [10].
domain assumption Correctness of the equations of P2_fake = (C18,p=3,empty,d3D3) from [3], including the C3xC3 automorphism action used in Steps 1-3
Input to the whole chain (Section 1.1, Steps 1-2); previously published and externally reviewed [3], not re-verified here.
domain assumption GAP-computed group structure: G648 = C3 x (SL(2,Z/3Z) semidirect C3xC3) with claimed G72 and bG72 subgroups (Prop. 1.2) and character decomposition (Prop. 2.3)
Computer-assisted, supplied by the authors in the auxiliary file GAPdataAll [9]; the label (C18,p=3,{2I}) of the quotient by bG72 enters here.
domain assumption Rigidity: covers and quotients of the arithmetic ball quotient 72.P2 are the classified FPP surfaces, making the constructed surface equal to the classified (C18,p=3,{2I})
Implicit in Step 7's 'we can then confidently identify it'; relies on the classification being exhaustive and on rigidity of ball quotients (Section 1.1, Section 3.7).
standard math Riemann-Roch, Kodaira vanishing and Holomorphic Lefschetz formula justify the dimension claims in Prop. 2.2 and Prop. 2.3
Standard results in the theory of ball quotients; used in Section 2 to compute h^0 of canonical and bicanonical sheaves.
ad hoc to paper Unstated: the finite-field smoothness certificate implies smoothness over the number field
Section 3.7: three Jacobian minors plus equations give zero Hilbert polynomial over F_4363; the text jumps to smoothness of the surface over the number field without giving the flatness/degeneration argument.

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Cite this review

Pith. "Pith review of Finding equations of the fake projective plane $(C18,p=3,\{2I\})$." pith.science (2026). https://pith.science/paper/PMJ7E2QP

@misc{pith2026251203213,
  author       = {Pith},
  title        = {Pith review of: Finding equations of the fake projective plane $(C18,p=3,\2I\)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PMJ7E2QP}},
  note         = {Machine review of arXiv:2512.03213}
}

abstract

We find explicit equations of a new pair of fake projective planes, labeled by $(C18,p=3,\{2I\})$ in the Cartwright-Steger classification. Our method involves starting with known equations of a commensurable fake projective plane $(C18,p=3,\emptyset,d_3 D_3)$ and working through a chain of cyclic covers and quotients to get to the new one.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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

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