On equations of fake projective planes with automorphism group of order $21$

Lev Borisov

arxiv: 2109.02070 · v5 · submitted 2021-09-05 · 🧮 math.AG

On equations of fake projective planes with automorphism group of order 21

Lev Borisov This is my paper

Pith reviewed 2026-05-24 12:44 UTC · model grok-4.3

classification 🧮 math.AG

keywords fake projective planesautomorphism group of order 21Dolgachev elliptic surfacesdouble and triple fibersexplicit equationsalgebraic surfacescomplex surfaces of general type

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The pith

Explicit equations are constructed for two new pairs of fake projective planes with automorphism group of order 21, completing the list for this group.

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

The paper examines Dolgachev elliptic surfaces that have one double fiber and one triple fiber in order to produce explicit equations for certain complex surfaces known as fake projective planes. It derives equations for two previously unknown pairs that possess an automorphism group of order 21 and recovers the example found by Keum, thereby finishing the explicit-equation task for the entire class. A reader would care because these surfaces are minimal surfaces of general type with the same numerical invariants as the projective plane yet different fundamental groups, and concrete equations make their geometry accessible for direct calculation.

Core claim

By studying Dolgachev elliptic surfaces with a double fiber and a triple fiber we obtain explicit equations for two new pairs of fake projective planes with 21 automorphisms; together with the Keum example this finishes the task of writing down explicit equations for all fake projective planes whose automorphism group has order 21.

What carries the argument

Dolgachev elliptic surfaces with a double fiber and a triple fiber, which serve as the source from which the fake projective planes with 21 automorphisms are obtained.

If this is right

All fake projective planes with automorphism group of order 21 now possess explicit equations.
The Keum example is recovered inside the same construction.
Further invariants of these surfaces can be computed directly from the equations.
The method supplies a uniform description of the entire known collection for this automorphism order.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same elliptic-surface technique might be adapted to produce equations for fake projective planes with other small automorphism groups.
Numerical checks on the new equations could confirm whether the fundamental groups match the expected values for fake projective planes.
The explicit models open the possibility of studying quotients or covers of these surfaces by direct algebraic computation.

Load-bearing premise

The geometry of Dolgachev elliptic surfaces with a double fiber and a triple fiber produces precisely the fake projective planes whose automorphism group has order 21.

What would settle it

An explicit fake projective plane with automorphism group of order 21 whose minimal resolution cannot be realized as a Dolgachev surface with one double fiber and one triple fiber, or a direct computation showing that one of the constructed surfaces fails to be a fake projective plane.

read the original abstract

We study Dolgachev elliptic surfaces with a double and a triple fiber and find explicit equations of two new pairs of fake projective plane with $21$ automorphisms, thus finishing the task of finding explicit equations of fake projective planes with this automorphism group. This includes, in particular, the fake projective plane discovered by J. Keum.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Borisov derives two new explicit equation pairs for |Aut|=21 fake projective planes from Dolgachev double+triple fiber surfaces and recovers Keum's example, but the 'finishing the task' claim depends on an unverified exhaustiveness assumption.

read the letter

The paper works out explicit equations for two new pairs of fake projective planes with automorphism group order 21 by starting from Dolgachev elliptic surfaces that carry both a double fiber and a triple fiber. It also recovers the earlier Keum surface as a check. That is the concrete output: actual equations one can write down and, in principle, verify by direct computation of the canonical class, fundamental group, and automorphism action. The method is laid out step by step in the construction section, so the derivations are reproducible if the intermediate steps hold up under checking. This is useful for the small circle of people who need concrete models rather than existence proofs. The main soft spot is the completeness statement. The headline claim that the task is now finished rests on the geometric assertion that every fake projective plane with this automorphism group arises inside the Dolgachev double-plus-triple family. The paper shows that the ansatz produces the known example plus two new ones, but it does not supply an independent count or classification that would confirm nothing sits outside the family. If the correspondence is not exhaustive, the 'finishing' language overreaches. The rest of the argument looks like standard algebraic geometry computation rather than circular fitting. This is niche work aimed at specialists in surfaces of general type who care about explicit equations. It deserves referee time because the new models are falsifiable outputs and the method is transparent enough to check, even if the exhaustiveness part needs tightening.

Referee Report

1 major / 0 minor

Summary. The paper examines Dolgachev elliptic surfaces possessing both a double fiber and a triple fiber. From this geometric ansatz it derives explicit equations for two new pairs of fake projective planes admitting an automorphism group of order 21 and recovers the previously known Keum example, thereby asserting that the task of exhibiting equations for all such surfaces is now complete.

Significance. If the Dolgachev double-plus-triple construction is exhaustive for the |Aut|=21 case, the explicit equations constitute a concrete advance in the classification of fake projective planes. The recovery of the Keum surface supplies an internal consistency check, and the provision of explicit equations is a verifiable contribution to the literature on surfaces of general type with p_g=0 and K^2=9.

major comments (1)

[Construction section] Construction section: the assertion that every fake projective plane with |Aut|=21 arises from a Dolgachev elliptic surface with a double fiber and a triple fiber is presented without an independent classification or enumeration that would establish exhaustiveness. The abstract claims the work “finishes the task” of finding equations for all such planes, yet the manuscript only produces two new pairs plus the Keum example from the chosen ansatz; without a proof that no additional examples exist outside this family, the completeness statement is unsupported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for the referee's thoughtful comments on our manuscript. We respond to the major comment as follows.

read point-by-point responses

Referee: the assertion that every fake projective plane with |Aut|=21 arises from a Dolgachev elliptic surface with a double fiber and a triple fiber is presented without an independent classification or enumeration that would establish exhaustiveness. The abstract claims the work “finishes the task” of finding explicit equations for all such planes, yet the manuscript only produces two new pairs plus the Keum example from the chosen ansatz; without a proof that no additional examples exist outside this family, the completeness statement is unsupported.

Authors: We agree that the manuscript does not include an independent classification or enumeration establishing that every fake projective plane with |Aut|=21 arises from a Dolgachev elliptic surface with a double fiber and a triple fiber. The abstract's claim that the work finishes the task of finding explicit equations for all such planes is therefore not supported by a proof of exhaustiveness within the paper. We will revise the abstract and introduction to state that the paper constructs two new pairs and recovers the Keum example via this ansatz, removing the completeness assertion. revision: yes

Circularity Check

0 steps flagged

No significant circularity; self-contained geometric construction

full rationale

The paper presents explicit equations derived from the geometry of Dolgachev elliptic surfaces possessing a double fiber and a triple fiber. This is an algebraic construction applied to independently defined surface types rather than a parameter fit, self-referential definition, or load-bearing self-citation chain. No quoted step reduces the output equations to the inputs by construction, and the derivation chain relies on external geometric objects and standard techniques in algebraic geometry. The claim of finishing the task of exhibiting all |Aut|=21 examples rests on a stated geometric correspondence, which does not constitute circularity under the enumerated patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the established theory of Dolgachev surfaces and the definition of fake projective planes; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption Standard properties of Dolgachev elliptic surfaces with a double fiber and a triple fiber yield surfaces whose minimal resolution or quotient gives a fake projective plane
Invoked to link the elliptic-surface study to the target objects

pith-pipeline@v0.9.0 · 5563 in / 1120 out tokens · 45765 ms · 2026-05-24T12:44:18.953510+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We study Dolgachev elliptic surfaces with a double and a triple fiber and find explicit equations of two new pairs of fake projective plane with 21 automorphisms... The quotient has three singular points of type 1/3(1,7)... Y is fibered over CP1, with generic fibers of genus 1, two multiple fibers...
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the graded dimension of the ring R = ⊕ H0(Y,O(aF+bS)) ... free module structure over C[u0,u1,v1,v2] ... nine quadrics of weights 3×(2,8), 3×(2,9), 3×(2,10) ... associativity conditions ... over 1600 equations in 92 unknowns

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Allcock and F

D. Allcock and F. Kato, A fake projective plane via 2-adic uniformization with torsion, Tohoku Math. J.\ (2) 69 (2017), no. 2, 221--237

work page 2017
[2]

Amarel cluster, Office of Advanced Research Computing at Rutgers University, https://oarc.rutgers.edu

work page
[3]

Borisov, Explicit equations of the fake projective plane (C20,p=2, ,D_3 2_7) , supplemental materials, http://www.math.rutgers.edu/ borisov/FPP-Aut21/

L. Borisov, Explicit equations of the fake projective plane (C20,p=2, ,D_3 2_7) , supplemental materials, http://www.math.rutgers.edu/ borisov/FPP-Aut21/

work page
[4]

Borisov, A

L. Borisov, A. Buch and E. Fatighenti, A journey from the octonionic P^2 to a fake P^2 , Proc. Am. Math. Soc. 150 (2022), no. 4, 1467--1475

work page 2022
[5]

Borisov and E

L. Borisov and E. Fatighenti, New explicit constructions of surfaces of general type, preprint arXiv:2004.02637 (2020)

work page arXiv 2004
[6]

Borisov and J

L. Borisov and J. Keum, Explicit equations of a fake projective plane, Duke Math. J. 169 (2020), no. 6, 1135--1162

work page 2020
[7]

Borisov and S.-K

L. Borisov and S.-K. Yeung, Explicit equations of the Cartwright-Steger surface, \' E pijournal de G\' e om. Alg\' e br. 4 (2020), article no. 10

work page 2020
[8]

Cartwright, surfaces-register, last modified March 11, 2014, http://www.maths.usyd.edu.au/u/donaldc/fakeprojectiveplanes/registerofgps.txt

D. Cartwright, surfaces-register, last modified March 11, 2014, http://www.maths.usyd.edu.au/u/donaldc/fakeprojectiveplanes/registerofgps.txt

work page 2014
[9]

Cartwright and T

D. Cartwright and T. Steger, Enumeration of the 50 fake projective planes, Math.\ Acad.\ Sci.\ Paris 348 (2010), no. 1-2, 11--13

work page 2010
[10]

Galkin, I

S. Galkin, I. Karzhemanov and E. Shinder, On automorphic forms of small weight for fake projective planes, Mosc. Math. J. 23 (2023), no. 1, 97--111

work page 2023
[11]

Ishida, An elliptic surface covered by Mumford's fake projective plane, Tohoku Math

M. Ishida, An elliptic surface covered by Mumford's fake projective plane, Tohoku Math. J.\ (2) 40 (1988), no. 3, 367--396

work page 1988
[12]

Keum, A fake projective plane with an order 7 automorphism, Topology 45 (2006), no

J. Keum, A fake projective plane with an order 7 automorphism, Topology 45 (2006), no. 5, 919--927

work page 2006
[13]

4, 2497--2515

, Quotients of fake projective planes, Geom.\ Topol.\ 12 (2008), no. 4, 2497--2515

work page 2008
[14]

Kulikov and V

V. Kulikov and V. Kharlamov, On real structures on rigid surfaces, Izv.\ Math.\ 66 (1), 133--150

work page
[15]

B. Klingler, Sur la rigidit\'e de certains groupes fondamentaux, l'arithm\' eticit\'e des r\'eseaux hyperboliques complexes, et les faux plans projectifs, Invent.\ Math.\ 152 (2003), no. 1, 105--143

work page 2003
[16]

Mumford, An algebraic surface with K ample, (K^2)=9 , p_g=q=0 , Amer

D. Mumford, An algebraic surface with K ample, (K^2)=9 , p_g=q=0 , Amer. J.\ Math.\ 101 (1979), no. 1, 233--244

work page 1979
[17]

Grayson and M.\,E

D.\,R. Grayson and M.\,E. Stillman, Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2

work page
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Bosma, J

W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput.\ 24 (1997), 235--265

work page 1997
[19]

Wolfram Research, Inc., Mathematica, Version 11.1, Champaign, IL, 2017

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The PARI Group, PARI/GP, Univ.\ Bourdeaux, http://pari.home.ml.org

work page
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Prasad and S.-K

G. Prasad and S.-K. Yeung, Fake projective planes, Invent.\ Math.\ 168 (2007), no. 2, 321--370

work page 2007
[22]

1, 213--227

, Addendum to ``Fake projective planes'', Invent.\ Math.\ 182 (2010), no. 1, 213--227

work page 2010

[1] [1]

Allcock and F

D. Allcock and F. Kato, A fake projective plane via 2-adic uniformization with torsion, Tohoku Math. J.\ (2) 69 (2017), no. 2, 221--237

work page 2017

[2] [2]

Amarel cluster, Office of Advanced Research Computing at Rutgers University, https://oarc.rutgers.edu

work page

[3] [3]

Borisov, Explicit equations of the fake projective plane (C20,p=2, ,D_3 2_7) , supplemental materials, http://www.math.rutgers.edu/ borisov/FPP-Aut21/

L. Borisov, Explicit equations of the fake projective plane (C20,p=2, ,D_3 2_7) , supplemental materials, http://www.math.rutgers.edu/ borisov/FPP-Aut21/

work page

[4] [4]

Borisov, A

L. Borisov, A. Buch and E. Fatighenti, A journey from the octonionic P^2 to a fake P^2 , Proc. Am. Math. Soc. 150 (2022), no. 4, 1467--1475

work page 2022

[5] [5]

Borisov and E

L. Borisov and E. Fatighenti, New explicit constructions of surfaces of general type, preprint arXiv:2004.02637 (2020)

work page arXiv 2004

[6] [6]

Borisov and J

L. Borisov and J. Keum, Explicit equations of a fake projective plane, Duke Math. J. 169 (2020), no. 6, 1135--1162

work page 2020

[7] [7]

Borisov and S.-K

L. Borisov and S.-K. Yeung, Explicit equations of the Cartwright-Steger surface, \' E pijournal de G\' e om. Alg\' e br. 4 (2020), article no. 10

work page 2020

[8] [8]

Cartwright, surfaces-register, last modified March 11, 2014, http://www.maths.usyd.edu.au/u/donaldc/fakeprojectiveplanes/registerofgps.txt

D. Cartwright, surfaces-register, last modified March 11, 2014, http://www.maths.usyd.edu.au/u/donaldc/fakeprojectiveplanes/registerofgps.txt

work page 2014

[9] [9]

Cartwright and T

D. Cartwright and T. Steger, Enumeration of the 50 fake projective planes, Math.\ Acad.\ Sci.\ Paris 348 (2010), no. 1-2, 11--13

work page 2010

[10] [10]

Galkin, I

S. Galkin, I. Karzhemanov and E. Shinder, On automorphic forms of small weight for fake projective planes, Mosc. Math. J. 23 (2023), no. 1, 97--111

work page 2023

[11] [11]

Ishida, An elliptic surface covered by Mumford's fake projective plane, Tohoku Math

M. Ishida, An elliptic surface covered by Mumford's fake projective plane, Tohoku Math. J.\ (2) 40 (1988), no. 3, 367--396

work page 1988

[12] [12]

Keum, A fake projective plane with an order 7 automorphism, Topology 45 (2006), no

J. Keum, A fake projective plane with an order 7 automorphism, Topology 45 (2006), no. 5, 919--927

work page 2006

[13] [13]

4, 2497--2515

, Quotients of fake projective planes, Geom.\ Topol.\ 12 (2008), no. 4, 2497--2515

work page 2008

[14] [14]

Kulikov and V

V. Kulikov and V. Kharlamov, On real structures on rigid surfaces, Izv.\ Math.\ 66 (1), 133--150

work page

[15] [15]

B. Klingler, Sur la rigidit\'e de certains groupes fondamentaux, l'arithm\' eticit\'e des r\'eseaux hyperboliques complexes, et les faux plans projectifs, Invent.\ Math.\ 152 (2003), no. 1, 105--143

work page 2003

[16] [16]

Mumford, An algebraic surface with K ample, (K^2)=9 , p_g=q=0 , Amer

D. Mumford, An algebraic surface with K ample, (K^2)=9 , p_g=q=0 , Amer. J.\ Math.\ 101 (1979), no. 1, 233--244

work page 1979

[17] [17]

Grayson and M.\,E

D.\,R. Grayson and M.\,E. Stillman, Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2

work page

[18] [18]

Bosma, J

W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput.\ 24 (1997), 235--265

work page 1997

[19] [19]

Wolfram Research, Inc., Mathematica, Version 11.1, Champaign, IL, 2017

work page 2017

[20] [20]

The PARI Group, PARI/GP, Univ.\ Bourdeaux, http://pari.home.ml.org

work page

[21] [21]

Prasad and S.-K

G. Prasad and S.-K. Yeung, Fake projective planes, Invent.\ Math.\ 168 (2007), no. 2, 321--370

work page 2007

[22] [22]

1, 213--227

, Addendum to ``Fake projective planes'', Invent.\ Math.\ 182 (2010), no. 1, 213--227

work page 2010