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

. We study Dolgachev elliptic surfaces with a double and a triple ﬁber and ﬁnd explicit equations for two new pairs of fake projective planes with 21 automorphisms. This includes, in particular, the fake projective plane discovered by J. Keum.


Introduction
The theory of fake projective planes originated with the famous example of D. Mumford [Mum79] of a surface of general type with the same Hodge numbers as the usual projective plane CP 2 .By its nature, the construction did not yield any explicit equations of it.Over the subsequent decades, work by multiple authors (see for instance [AK17, Ish88, Keu06, Keu08, KK02, Kli03, PY07, PY10]) produced additional examples and general results and culminated in the classification of all fake projective planes by D. Cartwright and T. Steger [CS11,Car14].These surfaces are classified as free quotients of the complex 2-dimensional ball B 2 = {(z 1 , z 2 ), |z 1 | 2 + |z 2 | 2 < 1} by certain discrete arithmetic subgroups.There are exactly 50 conjugate pairs of such surfaces, separated into 28 classes.This extremely useful classification does not lead to any polynomial equations either since there are no known methods for constructing explicit automorphic forms for these groups.
Over the last several years, the author of this paper has been involved in multiple collaborations with the goal of discovering explicit polynomial equations that define fake projective planes and related surfaces; see [BK20,BY20,BF20,BBF22].This paper is a continuation of such efforts, originally aimed at finding the equations of Mumford's fake projective plane.While this goal is still elusive, we find equations of the fake projective plane constructed by J. Keum in [Keu06], which is commensurable to the Mumford's surface.We also find another interesting fake projective plane in the process.As always, new approaches had to be developed for the case at hand.
It is not surprising that most of the currently computed fake projective planes have nontrivial automorphism groups, as this provides some avenues for exploration, and this paper continues the trend.According to the classification of Cartwright and Steger, the maximum order of the automorphism group of a fake projective plane is 21.There are three conjugate pairs of fake projective planes with automorphism group of this size, and in all three cases the group is the semi-direct product of a normal subgroup C 7 and a non-normal subgroup C 3 .Specifically, here are the planes, with their names in the CS classification and brief comments: • (a = 7, p = 2, ∅, D 3 2 7 ) is the first example of the fake projective plane for which explicit equations were found; see [BK20].There are two other conjugate pairs of fake projective planes in its class.• (a = 7, p = 2, {7}, D 3 2 7 ) is the surface constructed by Keum in [Keu06].It has three more pairs of fake projective planes in its class, including Mumford's fake projective plane.
• (C20, p = 2, ∅, D 3 2 7 ) is the last of the three surfaces; it does not seem to be implicated in any other construction.There are no other pairs in its class.
In all three cases, the quotient of P 2 fake by the subgroup C 7 of its automorphism group has a minimal resolution Y with rather peculiar geometry; see [Keu06,Keu08].The quotient has three singular points of type 1 3 (1, 7), which are permuted by the residual C 3 -action of the automorphism group of P 2 fake .The minimum resolution Y of P 2 fake /C 7 has three disjoint chains of three lines with self-intersections −3, −2, −2 which we denote by Here − indicates a transversal intersection point.In addition, Y is fibered over CP 1 , with generic fibers of genus 1, two multiple fibers, three nodal fibers and one fiber of type I 9 (a ring of nine CP 1 with self-intersection (−2) each) with components The residual automorphism group C 3 preserves the fibration structure and acts by A → A 1 → A 2 → A and similarly for the S, B and C curves.
The multiplicities of the multiple fibers in the case of (a = 7, p = 2, ∅, D 3 2 7 ) are 2 and 4, and they are 2 and 3 in the other two cases, which are the focus of this paper.In particular, in the cases of interest, the curves S, S 1 and S 2 are 6-sections of the fibration.The two special fibers 3F 2 and 2F 3 have multiplicity 3 and 2, respectively.The reductions F 2 and F 3 are linearly equivalent to 2F and 3F with F = K Y , and the generic fiber is equivalent to 6F.
The first idea of this paper is to consider the ring It has a double grading, and we can derive a formula for the graded dimension which is suggestive of a free module structure over the subring generated by the variables u 0 , u 1 , v 1 , v 2 with weights (0, 2), (0, 3), (1, 0), (1, 3), respectively.The appropriate GIT quotient is a birational model Y 0 of Y that collapses all of the curves that intersect F and S trivially.These are the curves C, B 1 , C 1 , B 2 , C 2 , and the image of the special fiber in Y 0 becomes so that the intersection point of (the images of) B and A 1 is a simple node and the intersections of A 2 with both A and A 1 are 1 3 (1, 2) singularities.The construction of the new fake projective planes then proceeds as follows.
The idea is to postulate the above free module structure and the weights of the quadratic relations and to use the associativity conditions of the ring to solve for the coefficients of the quadrics.It entails solving a system of over 1600 equations with 92 unknowns, which is done by an ad hoc method utilizing the Mathematica software system.

L. Borisov 4
L. Borisov Step 2. We construct seven-, five-and two-parameter subfamilies with additional conditions on the special fiber.Respectively, we require the special fiber to contain a line, two disjoint lines, two disjoint lines with two nodes on one and one node on the other.In particular, a generic element of the two-parameter family has special fiber in the sense that the intersection points of B with A 1 and A 2 with A and A 1 are nodes.
Step 3. We find a finite-field reduction of the surface Y 0 by looking through the parameter choices over a finite field and checking whether the resulting surfaces have worse-than-nodal singularities at two special points on the curve A 2 .The smallest prime for which we were able to find such surfaces was 79.
Step 4. We proceed by successively solving the (hard to write) conditions on being more singular at the intersection points of A-curves for parameters modulo powers of 79.We then recognize the parameters as algebraic numbers and construct Y 0 over a number field of degree 12.We make a coordinate change to realize Y 0 over the number field Q( √ −7).
Step 5. We study Y 0 to find its geometric features, such as the curves S 1 and S 2 and the birational action of C 3 .We find the degree 7 extension of the field of rational functions of Y 0 that gives P 2 fake and calculate the bicanonical linear system of the latter.We then realize P 2 fake as an intersection of 84 cubic equations in CP 9 , following the blueprint of [BK20].
Step 6.We identify the fake projective plane as (C20, p = 2, ∅, D 3 2 7 ) by exhibiting too many torsion line bundles for it to be (a = 7, p = 2, {7}, D 3 2 7 ).The method is to find non-reduced C 3 -invariant elements of |2K P 2 fake | modulo a prime (this time it is 29) by an exhaustive search and then lift them to powers of the said prime and finally the algebraic numbers.We also use one of the torsion line bundles to pick a more natural basis of H 0 (P 2 fake , 2K) so that its equations have smaller coefficients.Finally, we verify that it is indeed a fake projective plane, as in [BK20].
Step 7. In Step 2, one actually finds two different five-parameter families of Dolgachev surfaces which contain two disjoint lines in the special fiber.Unfortunately, for the second family we were unable to reduce the number of parameters further by considering the condition of having nodes.However, we are still able to go through Steps 3-5 in this case, by a brute force approach to the finite-field search.By a process of elimination, we see that the new pair of fake projective planes is the one constructed in [Keu06].
The paper is organized as follows.Section 2 contains the first two steps of the construction.Section 3 contains Steps 3 and 4. Section 4 describes Steps 5 and 6.In Section 5 we discuss the last step.Finally, in Section 6 we talk about the open problems associated with our construction.We also have an appendix in which we put some equations that are too lengthy for the main body of the paper.However, many of the key formulas are far too large to even be included in the appendix.They are collected in [Bor22+] instead.

Families of (2,3) Dolgachev surfaces
We start by studying smooth projective surfaces Y with a genus 1 fibration Y → CP 1 with the following properties: • The class of the general fiber is 6F, where F is the canonical class of Y .In particular, F 2 = 0.
• There exist a double fiber 2F 3 and a triple fiber 3F 2 .The classes of F 2 and F 3 are 2F and 3F, respectively.
• There is a rational 6-section S with SF = 1 and S 2 = −3.
• We have p g (Y ) = q(Y ) = 0. Our motivation is that the minimal resolutions of the fake projective planes we are interested in satisfy the above; see [Keu08].
As implied in the introduction, we first compute (1) the graded dimension of the ring in a series of lemmas.
Proof.The ring R 0 is freely generated by the elements u 0 ∈ H 0 (Y , O(2F)) and u 1 ∈ H 0 (Y , O(3F)) whose divisors are F 2 and F 3 , respectively.Indeed, these generate a subring of R 0 , and we will show that there are no other forms.The subring for a = 0 mod 6 is isomorphic to the homogeneous ring of the CP 1 of the base of the fibration, so it is a polynomial ring in u 3 0 and u 2 1 .If a is odd, then O(aF) restricts to a nontrivial bundle to F 3 (because the normal bundle of the double fiber is nontrivial), so all its global sections vanish on F 3 , and the corresponding elements in R 0 are divisible by u 1 .Similarly, if a is not divisible by 3, then all global sections of O(aF) vanish on F 2 , and the elements are divisible by u 0 .Together, these observations imply the result.□ Lemma 2.2.The dimension of H 0 (Y , O(aF + S)) is 1 for a = 0 and is a − 1 for a > 0.
Proof.For 0 ≤ a ≤ 2 we have (aF + S)S < 0, so any section of O(aF + S) must vanish on S, and the statement follows from Lemma 2.1.For a ≥ 3 we have χ(aF + S) = 1 2 (aF + S)((a − 1)F + S) + 1 = a − 1, so it suffices to show that the invertible sheaf O(aF +S) has no higher cohomology.The vanishing of H 2 (Y , O(aF +S)) for all integers a is clear from Serre duality and the fact that every effective divisor on Y must have a non-negative intersection with F. To see the vanishing of H 1 (Y , O(aF + S)), we run induction on a. Specifically, consider the long exact sequence.
Since (aF + S)F 2 = 2, the last term is 0, so it suffices to prove that H 1 (Y , O(aF + S)) = 0 for a ∈ 1, 2, which follows from our computation of the global sections of these divisors.
Proof.We will prove it by induction on b with the base case provided by Lemma 2.2.As before, dim H 2 (Y , O(aF For the induction step, the short exact sequence ).The terms on the left and on the right are zero by the induction hypothesis and (aF + bS)S = a − 3b ≥ 0. □ (1) Technically, we could just conjecture everything, with the justification for it being the final outcome, but it is worth proving what we can.
The above lemmas allow us to compute the graded dimension of R.
Proof.We write dim H 0 (Y , O(aF + bS)) = c a,b to simplify notation.Since (aF + bS)S < 0 implies that S is a fixed component of |aF + bS|, we see that for a < 3b there holds We then use Lemmas 2.1 and 2.3 to write the above as which is then easily computed by Mathematica.
(2) □ As a consequence of Proposition 2.4, we conjecture that R has the structure of a rank 6 graded free module over the ring generated by the sections u 0 , u 1 , v 1 and v 2 of O(2F), O(3F), O(S) and O(3F + S), respectively.Note that while u 0 , u 1 , v 1 are defined uniquely up to scaling, the section v 2 can also be changed by adding a scalar multiple of u 1 v 1 .We denote the generators of the module at weights (1, 4) and (1, 5) by v 3 , v 4 , v 5 , v 6 .We conjecture that the generator at degree (9, 2) is equal to v 3 v 5 .It can likely be proved that the ring R is Gorenstein, which would then imply that such a choice is possible, but we just take it as a sensible guess.
We will consider polynomial relations on u 0 , . . ., v 6 .By looking at the graded dimension of R, we observe that these variables must satisfy three linearly independent relations of degree (2, 8), three linearly independent relations of degree (2, 9) and three relations of degree (2, 10), which are linearly independent together with u 0 times the (2, 8) relations.We will refer to these relations as quadrics (in v i ).In general, genus 1 curves of degree 6 in P 5 can be written as an intersection of nine degree 2 polynomials, and we expect that for most values of u 0 and u 1 , these quadrics describe the corresponding fiber of Y → CP 1 .
We then set up the possible quadrics with undetermined coefficients, taking care to undo multiple symmetries of the construction.For example, we make sure that the coefficients of the (2, 8) quadrics in v 2 3 , v 3 v 4 , v 2 4 are the standard basis vectors, and similarly for the (2, 9) and (2, 10) quadrics.We assume that (v 3 , v 4 ) and (v 5 , v 6 ) are dual to each other in the socle pairing of R/⟨u 0 , u 1 , v 1 , v 2 ⟩.We can also make additional assumptions on the coefficients in view of the possible changes of v i such as v 2 → αv 2 + βu 1 v 1 .The only such assumptions which appear crucial to the success of the method are that the coefficients of the first (2, 8) quadric at v 3 v 1 u 2 0 an v 4 v 1 u 2 0 are 1 and 0, the coefficients of the second quadric at these monomials are both 0, and the coefficients of the third quadric are 0 and 1, respectively.It can be shown that a generic collection of the quadrics can be manipulated into this form by appropriate linear changes of v 3 , v 4 together with adding multiples of u 2 0 v 1 to them.However, there are six ways of doing so, which means that while we expect Y 0 to be defined over a quadratic imaginary field, we cannot expect the coefficients of these quadrics to be this simple.
(2) It is, of course, computable by hand, but this seems to be a fool's errand given the subsequent use of various software.
Once the quadrics are written down, one can compute the multiplication table for the generators of R as a free module over C[u 0 , u 1 , v 1 , v 2 ] and then set up the associativity relations as equations on the coefficients of the quadrics; see [Bor22+, DolgachevSurfaces.nb].The associativity relations led to over 1600 equations in 92 variables.Fortunately, some of these equations were quite simple, but still the task appeared daunting.We were able to use the Mathematica "Solve" command to eventually reduce to nine free parameters.The basic idea was to try to solve the easier equations first, with either byte count or the number of terms used as measure of complexity.The drawback of this technique is that one can at best hope to recover rational parameterizations by a subset of the set of variables, but we were fortunate in this case.The resulting equations are presented in Section A.1 in the appendix.The variables d i are the parameters, and the variables u 0 , u 1 , v 1 , . . ., v 6 are the coordinates.
Having found the equations of a nine-parameter family, we then tried to impose additional geometric conditions on it.We postulated without loss of generality that the I 9 fiber of Y occurs at u 3 0 = u 2 1 (or simply u 0 = u 1 = 1 if one wants to dehomogenize).We know from the work of Keum [Keu08] that possible intersections of S with (A, A 1 , A 2 ) fall into cases (2, 2, 1) and (1, 1, 3), so in either setting, the image of the I 9 fiber in Y 0 should have two disjoint lines.We first solved for the condition that the equations in Section A.1 vanish on one line, by parameterizing the said line.This gave a seven-parameter family, and we used its explicit description to put in a condition of having two such lines, taking care not to get the degenerate cases where the lines intersect.Much trial and error was involved, and even in the best case, the computations took a long time.Due to an unfortunate mistake on the author's part, some of the intermediate steps were lost; however, the final five-parameter answer survived and is explicitly written in [Bor22+, DolgachevSurfaces.nb].
The general member of this five-parameter family has a double and a triple fiber at u 1 = 0 and u 0 = 0, respectively, and a special fiber at u 3 0 = u 2 1 which has the following configuration of curves A, B, A 1 , A 2 and their intersection points p 1 , . . ., p 4 : special fiber Here, A 2 and B are the two disjoint lines in the special fiber, and A and A 1 are both of degree 2. Our next step was to impose the conditions that Y 0 is singular at p 1 , p 3 and p 4 , as would be expected by the geometry of the surface.One immediate technical difficulty was that the points p 1 and p 2 were not defined over the parameter space.However, we were able to make a change of variables, similar to a rational parameterizations of a plane conic, to resolve this difficulty.We also made a simple coordinate change to simplify the formulas somewhat; see [Bor22+, DolgachevSurfaces.nb] for details.
The conditions of being singular at each p i were computed as follows.We looked at the Jacobian matrix of the quadrics at p i and computed its minors.We then computed the greatest common factor of these minors, which left us with large, but manageable expressions in the parameters t 1 , . . ., t 5 .An ad hoc manipulation of the equations and the parameters allowed us to replace t i with s 1 , . . ., s 5 and then solve for s 1 and s 5 .The polynomial equation on the remaining parameters s 2 , s 3 and s 4 has several factors, with all but one of them leading to undesirable degeneracies.We were left with a polynomial with integer coefficients of degree 3, 13 and 12 in s 2 , s 3 and s 4 , respectively.It is about 130 Kb long in its expanded form and thus is not worth trying to write down explicitly in this paper; see [Bor22+,DolgachevSurfaces.nb].It appears that the resulting 2-dimensional parameter space is not rational, but it was simple enough for our purposes.

Finding the special Dolgachev surface
This section describes how we found the surface Y 0 .We expected that Y 0 has 1 3 (1, 2) singularities at p 3 and p 4 , and it would be reasonable to try to encode these in terms of s 2 , s 3 and s 4 subject to the large polynomial equation.Unfortunately, this direct approach was not computationally feasible, as equations ballooned to hundreds of Mb in length.So we used the alternative method of finite-field reduction.Specifically, we wanted to find a reduction of Y 0 modulo a prime p and then lift it first to p-adics and then to a number field.
We looked at various primes p and parameters (s 2 , s 3 , s 4 ) ∈ F 3 p , with p chosen so that √ −7 exists in F p as it appeared likely that we would need this in the field of definition.For each triple of parameters we first checked whether it fits the polynomial relation that defines the parameter space.Then we used Magma to compute the structure of singularities at p 3 and p 4 (we have lengthy but explicit formulas for p i in terms of the parameters).If both singularities looked right, then we took note of the values of the parameters.For one reason or another, we had to go to p = 79 before any suitable examples were found!However, at p = 79 there were six possible solutions, see [Bor22+, Search79]: (s 2 , s 3 , s 4 ) ∈ {(14, 47, 52), (15, 65, 27), (19, 32, 14), (44, 14, 32), (58, 27, 65), (72, 52, 47)}.
It was then an interesting challenge to lift these solutions to powers of 79.By solving for v 5 and v 6 , one can reduce the problem to codimension 2. Then it is possible to encode the condition of having worse-than-nodal singularity as a certain polynomial in terms of first and second derivatives of the defining equations.Then we endeavored to have these polynomials produce values that are zero modulo higher and higher powers of 79, as we are adjusting our parameters s i to a more accurate p-adic approximation.Of course, we also have to keep the defining polynomial of the family zero to the appropriate power of 79.The resulting code is messy but not particularly slow, and we were able to compute the values of s i up to 79 101 in a reasonable amount of time.
Once a p-adic approximation was found, it was then routine to find "simple" algebraic numbers that give these parameters s i .As in [BF20, BBF22], we used a lattice reduction algorithm to find a small linear combination of powers of s i modulo 79 101 and 79 101 itself.This suggested that these are algebraic numbers of degree 12.By looking at the standard polynomial, we indeed found 12 possible triples (s 2 , s 3 , s 4 ).Perhaps not surprisingly, our finite-field search only picked up the six cases that correspond to the simple roots of the reduction of the defining polynomial of s 2 1048576 + 9633792s 2 + 47179776s 2 2 + 156022272s Since we expected that Y 0 can be defined over a quadratic field, we wanted to find a coordinate change in v i to see it.Specifically, we aimed to get the points p i to have simple coordinates.As it is a lot faster in Mathematica, this was done numerically, and then the coefficients were approximated by algebraic numbers.The resulting equations are listed in Section A.2 of the appendix, where we use w i to denote new variables, with u 0 and u 1 not affected by the coordinate change.

Constructing and identifying the fake projective plane
This section describes how we found the fake projective plane (up to conjugation) and identified it as (C20, p = 2, ∅, D 3 2 7 ).The relevant calculations are contained in the Mathematica file SevenfoldCover.nband Magma files Torsion and CheckSmoothness, as well as Macaulay2 file CheckFPP in [Bor22+].
We know that S is cut out by w 1 = 0, but it takes a bit of effort to find equations of S 1 and S 2 .The first idea is that since (4F + S)S = 4 − 3 = 1, the sections w 3 and w 4 are linear on S. Thus, points on S can be parameterized by t = w 4 w 3 .This is true for any member of the 9-dimensional family of Dolgachev surfaces; see Section A.1.So we know both the equation and the parameterizations of S, and we would like to do the same for S 1 and S 2 .
It follows from intersection form considerations [Bor22+, IntersectionForms.nb] that S 2 is cut out by an equation of bidegree (1, 8) which also passes through A and A 2 with multiplicities 1 and 3, respectively.We pick multiple points on A and A 2 and encode the relevant conditions on the (1, 8) polynomial to find it to be (4.1) We also needed to compute the parametric equation of S 2 .After some of the more straightforward approaches failed to finish, we did the following.For a given value of the parameter t on S we can compute the corresponding point p(t) on S and six points of S 2 in the same fiber of the genus 1 fibration.One of these points is the image p 1 (t) of p(t) under the predicted order 3 birational automorphism of Y 0 .If t is an integer, then this point on S 2 has to be defined over Q(i √ 7), which allows us to distinguish p 1 (t) from the five other points.After the normalization w 1 = 1, the coordinates of p 1 (t) are polynomial in t of degree at most 5, and we use several values of t to find their coefficients.The resulting parameterization is given in Section A.3 in the appendix.
One could follow a similar approach to find S 1 , which has a degree (1, 10) equation vanish on it, but we chose to use the group law of the elliptic fibers instead.Indeed, we know that for a given value of t the points p(t), p 1 (t) and p 2 (t) satisfy p 1 (t) + p(t) = 2p 1 (t) under the group law of the fiber with any choice of origin.This allowed us to find p 1 (t) and then compute the parameterization; see Section A.4.We then found the degree (1, 10) polynomial (4.2) that vanishes on S 1 .
We then computed the order 3 birational automorphism by postulating its general form and using the parameterization of S, S 1 and S 2 to find the coefficients.It can be viewed as an order 3 automorphism of a genus 1 curve defined over C(u 0 , u 1 ).The formulas for the automorphism are in Section A.5.
We were then ready to compute the equations of the sevenfold cover of Y .The method was essentially the same as that of [BK20], but we briefly describe it here for the benefit of the reader.We know that the sevenfold Galois cover P 2 fake → Y is ramified at S, B, C, S 1 , B 1 , C 1 , S 2 , B 2 , C 2 .We consider the rational where f 1,8 and f 1,10 are given in (4.1) and (4.2), respectively.Its divisor is supported on the special curves, with all but the nine S, . . ., C 2 curves above having multiplicity divisible by 7.So the function field of P 2 fake is obtained from that of Y by adding the seventh root of the function (4.3).We then compute ten sections of the bicanonical linear system on P 2 fake by projecting them to Y and realizing them as sections of various line bundles on Y 0 .Care is taken to have the action of the automorphism group look nice in these coordinates.Specifically, we have ten variables P 0 , . . ., P 9 on which the order 7 element acts by g 7 (P 0 , . . ., P 9 ) = P 0 , ζ 7 P 1 , ζ and the order 3 element acts by g 3 (P 0 , . . ., P 9 ) = (P 0 , P 2 , P 3 , P 1 , P 5 , P 6 , P 4 , P 8 , P 9 , P 7 ).
Having constructed a fake projective plane with 21 automorphisms, we want to try to identify it in accordance with the Cartwright-Steger classification.Since the Dolgachev surface it is built from has a double and a triple fiber, there are two possibilities: Keum's surface (a = 7, p = 2, {7}, D 3 2 7 ) and (C20, p = 2, ∅, D 3 2 7 ).In the former case, the torsion in the Picard group is C 3 2 , with the automorphism group G acting transitively on seven nontrivial elements.Therefore, for every cyclic subgroup of order 3 in G there should be a unique G-invariant torsion element on P 2 fake .This naturally lead to us trying to construct such torsion classes.
If L is a torsion class in the Picard group of a fake projective plane, then H 0 (P 2 fake , K +L) is 1-dimensional; see for example [GKS23].If L is furthermore a 2-torsion element, then the square of the corresponding section is in H 0 (P 2 fake , 2K).In the other direction, 2-torsion elements of the Picard group can be constructed from sections of 2K which give non-reduced curves.The additional condition of C 3 invariance means that we should consider sections (4.4) a 0 P 0 + a 1 (P 1 + P 2 + P 3 ) + a 2 (P 4 + P 5 + P 6 ) + a 3 (P 7 + P 8 + P 9 ) up to scaling.We looked at the finite-field reduction of P 2 fake to look for such a i .Specifically, p = 29 was the smallest prime that both had √ −7 in it and had the expected Hilbert polynomial of the reduction of P 2 fake .We then ran through all possible a i with a 0 = 1 and used Magma to check if the resulting scheme is non-reduced; see [Bor22+,Torsion].We got three solutions, which was already a likely indicator that the P 2 fake is not (a = 7, p = 2, {7}, D 3 2 7 ) but rather is (C20, p = 2, ∅, D 3 2 7 ), which has a larger torsion subgroup C 6 2 in its Picard group.However, we needed to ascertain it by lifting to characteristic zero.
The lifting procedure was pretty typical of such approach: we lifted the a i in (4.4) to modulo p k for large enough k and then guessed the algebraic numbers that could give these reductions.It is worth mentioning how exactly we did the lifting.We found via Magma several points on the non-reduced linear cut of the reduction of P 2 fake modulo p; then at each point we found two linearly independent tangent vectors in P 9 (F 29 ) which are orthogonal to the gradients of all 84 cubic polynomials and the linear polynomial (4.4).We lifted these vectors to small integers and then were successively adjusting them so that the aforementioned orthogonality held modulo higher powers of p.At each stage we modified the points, the tangent vectors and the polynomial (4.4) to the next power of p; this amounted to solving a system of linear equations modulo p, which was not time-consuming.Afterwards, we identified the corresponding algebraic numbers.
Last but not least, we used the description of the above non-reduced linear cut to find a more pleasant basis for H 0 (P 2 fake , 2K); see [Bor22+,SevenfoldCover.nb].Namely, we used a coordinate change from P i to Q i in which the aforementioned non-reduced cut (4.5)becomes This allowed us to find a model of it with smaller coefficients of the 84 equations, which we recorded in [Bor22+,EqsFPPwithrrQ].We then went through the verification procedure developed in [BK20] to make sure that the scheme cut out by the 84 cubic relations in 10 variables is indeed a fake projective plane in its bicanonical embedding; see [Bor22+, CheckFPP, CheckSmoothness].The smoothness calculation was performed by looking at three specific minors which are nonzero at the C 7 invariant points of P 2 fake .Each minor took a few hours to compute on our hardware.

Constructing the fake projective plane of Keum
While looking for Dolgachev surfaces with two disjoint lines, we have found a five-parameter subfamily of Dolgachev surfaces with two disjoint lines in the special fiber with the intersections of S with A, A 1 and A 2 given by (1, 1, 3).This makes it is provably different from the family in Step 2. This family was noticeably harder to work with.It was not even entirely straightforward to find the intersection points of the components of the special fiber.In particular, the conditions of being singular at three of these points were far too complicated to simplify.The details are in [Bor22+,SecondFamily.nb] To find a member of this family with additional singularities (1.1), we used a brute force approach.Namely, we ran a search over the parameter space F 5 p for small p, computed the intersection points of the components of the special fiber and checked to see if the tangent space at the three points of interest had the correct dimension.This was too time-consuming in Mathematica, so we used lower-level languages.We first tried PARI/GP and then eventually C; see [Bor22+, finitefieldnodes.c].Clearly, this is very easy to parallelize, and we ran multiple computations at a time on the Amarel cluster [Ama].This allowed us to reduce the set of possible parameters to roughly p 2 .For each of these we used Mathematica to check if there are worse-than-nodal singularities at the two expected points.
The first successful prime was p = 53.We proceeded to lift the solution to p-adics, as described in Section 3. We computed the sevenfold cover along the lines of Section 4; see [Bor22+,FindingKeumFPP.nb].We do not write down the parametric equations or the C 3 symmetry, but do list the equations in Section A.6 for the record; they are also available at [Bor22+,QuadricsWforKeum]. The field extension needed to (3) This verification was only done numerically rather than symbolically, but it is sufficiently convincing for our purposes.An interested reader is welcome to try their hand at verifying it symbolically.A.2. Equations of Y 0 in the case of (C20, p = 2, ∅, D 3 2 7 ) A.5. (Birational) automorphism of order 3 of Y 0 for (C20, p = 2, ∅, D 3 2 7 ) 2 7 P 2 , ζ 4 7 P 3 , ζ 3 P 4 , ζ 6 P 5 , ζ 5 P 6 , ζ 3 P 7 , ζ 6 P 8 , ζ 5 P 9