A journey from the octonionic $\mathbb{P}^{2}$ to a fake $\mathbb{P}^{2}$

As was mentioned in the Introduction, we set out to find a family of surfaces of general type with $K^{2}=3,\leavevmode\nobreak\ p=q=0$ and fundamental group $C_{13}$. Here is how this search led us to consider cuts of the octonionic projective plane $\mathbb{O}\mathbb{P}^{2}$.

Let $X$ be the universal cover of a surface in question, with a free action of an order $13$ automorphism $g$. Then $K_{X}^{2}=39$ and $\chi(K_{X})=13$. It is reasonable to expect that $h^{1}(X,K_{X})=0$ and $h^{0}(X,K_{X})={12}$. Then the pluricanonical ring $\bigoplus_{n\geq 0}H^{0}(X,nK_{X})$ of $X$ must have the graded dimension

$$\sum_{n\geq 0}\dim H^{0}(X,nK_{X})\,t^{n}=1+12t+52t^{2}+130t^{3}+\cdots=\frac{1+9t+19t^{2}+9t^{3}+t^{4}}{(1-t)^{3}}.$$

It is also reasonable to assume that the pluricanonical ring is generated in degree one, so $X$ is embedded into $\mathbb{P}^{11}$. It is also plausible that its image is cut out by $12(12+1)/2-52=26$ quadrics. By the Holomorphic Lefschetz formula, as in [H11, Theorem 2.1], the trace of the action of $g$ on $H^{0}(X,K_{X})$ is $(-1)$, which means that $H^{0}(X,K_{X})$ has a basis of eigenvectors of $g$ with eigenvalues $\zeta_{13}^{i}$ for $i=1,\ldots,12$. Similarly, the action of $g$ on the space of quadrics splits it into $13$ two-dimensional eigenspaces.

Inspired by [BF20], we undertook a rather exhaustive computer search for homogeneous varieties of degree $39$ and other relevant invariants, but were not successful. However, the octonionic projective plane $\mathbb{O}\mathbb{P}^{2}$ has degree $78$, and we observed the following remarkable coincidence: the homogeneous coordinate ring of $\mathbb{O}\mathbb{P}^{2}$ has graded dimension

$$\frac{(1+9t+19t^{2}+9t^{3}+t^{4})(1+t)}{(1-t)^{17}}.$$

More specifically, $\mathbb{O}\mathbb{P}^{2}$ is the dim $16$ singular locus of the $E_{6}$-invariant cubic in $\mathbb{P}^{26}$ cut out by the $27$ quadratic equations which are the partial derivatives of the cubic. So our idea was to take a linear cut of $\mathbb{O}\mathbb{P}^{2}$ by $15$ equations (so that we are in $\mathbb{P}^{11}$) which only drop the dimension by $14$. We also want one of the quadratic equations to reduce to zero on the linear cut. The best analogy would be cutting a quadric $xy=zw$ with Hilbert series $\frac{1+t}{(1-t)^{3}}$ by two linear equations $x=0$ and $z=0$ to get a line with the Hilbert series $\frac{1}{(1-t)^{2}}$ but, ultimately, it was a lucky guess.

There are several incarnations of the $E_{6}$-invariant cubic found in the literature. We used the one in Jacob Lurie’s undergraduate thesis [L11], namely

$$\begin{array}[]{l}-P_{10}P_{13}P_{16}-P_{11}P_{14}P_{17}-P_{12}P_{15}P_{18}-P_{16}P_{17}P_{18}+P_{1}P_{10}P_{19}-P_{1}P_{18}P_{2}+P_{11}P_{2}P_{20}-P_{1}P_{14}P_{21}\\ -P_{16}P_{20}P_{21}-P_{18}P_{22}P_{23}-P_{17}P_{19}P_{24}-P_{13}P_{2}P_{24}-P_{14}P_{15}P_{25}-P_{19}P_{22}P_{25}-P_{12}P_{13}P_{26}-P_{20}P_{23}P_{26}\\ -P_{10}P_{11}P_{27}-P_{21}P_{24}P_{27}-P_{25}P_{26}P_{27}+P_{12}P_{21}P_{3}-P_{10}P_{23}P_{3}-P_{2}P_{25}P_{3}-P_{15}P_{20}P_{4}+P_{13}P_{22}P_{4}\\ -P_{17}P_{3}P_{4}-P_{12}P_{19}P_{5}+P_{14}P_{23}P_{5}-P_{27}P_{4}P_{5}-P_{11}P_{22}P_{6}+P_{15}P_{24}P_{6}-P_{1}P_{26}P_{6}-P_{16}P_{5}P_{6}\\ -P_{11}P_{12}P_{7}-P_{23}P_{24}P_{7}+P_{16}P_{25}P_{7}-P_{1}P_{4}P_{7}-P_{10}P_{15}P_{8}-P_{21}P_{22}P_{8}+P_{17}P_{26}P_{8}-P_{2}P_{5}P_{8}\\ -P_{13}P_{14}P_{9}-P_{19}P_{20}P_{9}+P_{18}P_{27}P_{9}-P_{3}P_{6}P_{9}-P_{7}P_{8}P_{9}\end{array}$$

The $27$ variables $P_{1},\ldots,P_{27}$ are indexed by the lines on the Fermat cubic surface in $\mathbb{C}\mathbb{P}^{3}$ and the terms correspond to triples of coplanar lines. The sign prescription is more intricate, given in terms of the $C_{3}$ action on the cubic, see [L11]. The octonionic projective plane $\mathbb{O}\mathbb{P}^{2}$ is cut out by the $27$ partial derivatives of the above cubic.

There is a Cartan subgroup $(\mathbb{C}^{*})^{6}$ of $E_{6}$ that acts diagonally on the variables $P_{i}$. We picked an element $g$ of order $13$ of it which acts by $P_{i}\mapsto\zeta_{13}^{a_{i}}P_{i}$ with the weights $a_{i}$ given by

$$(6,7,7,10,3,6,10,3,0,5,8,8,4,9,5,4,9,0,2,11,11,12,1,2,12,1,0).$$

As the reader can see, the action of $g$ on the variables has a three-dimensional eigenspace of weight zero and $12$ two-dimensional eigenspace of other weights. For the three invariant variables $P_{9},P_{18},P_{27}$, the corresponding partial derivatives of the cubic

(1)

$$\begin{array}[]{l}-P_{13}P_{14}-P_{19}P_{20}+P_{18}P_{27}-P_{3}P_{6}-P_{7}P_{8},\\ -P_{12}P_{15}-P_{16}P_{17}-P_{1}P_{2}-P_{22}P_{23}+P_{27}P_{9},\\ -P_{10}P_{11}-P_{21}P_{24}-P_{25}P_{26}-P_{4}P_{5}+P_{18}P_{9}\end{array}$$

involve all of the variables $P_{i}$.

At this point, our expectations of the $g$-action on $H^{0}(X,K_{X})$ indicate that we need to take a linear cut by

	$\displaystyle(P_{9},P_{18},P_{27},P_{23}+d_{1}P_{26},P_{19}+d_{2}P_{24},P_{5}+d_{3}P_{8},P_{13}+d_{4}P_{16},P_{10}+d_{5}P_{15},P_{1}+d_{6}P_{6},$
	$\displaystyle P_{2}+d_{7}P_{3},P_{11}+d_{8}P_{12},P_{14}+d_{9}P_{17},P_{4}+d_{10}P_{7},P_{20}+d_{11}P_{21},P_{22}+d_{12}P_{25})$

for some constants $d_{1},\ldots,d_{12}$. Moreover, we want a linear combination of the $g$-invariant quadrics (1) to vanish on the linear subspace of the cut. In view of the Cartan subgroup symmetry, it is reasonable to pick this linear combination to be the sum of the above quadrics. This gives $6$ simple equations on $d_{i}$, namely $d_{i}d_{13-i}=-1$. We have been able to verify by computer at a specific point that the resulting scheme is a smooth surface of degree $13$ and is thus a good candidate for our $X$.

Specifically, the $26$ quadrics that cut out $X$ are given by

(2)

$$\begin{array}[]{l}-t_{10}^{2}+d_{2}t_{2}t_{5}-t_{1}t_{6}-d_{2}t_{11}t_{9},\ t_{3}^{2}+t_{2}t_{4}-d_{1}t_{12}t_{7}+t_{11}t_{8},\ -d_{1}t_{1}t_{5}-d_{1}t_{12}t_{7}+d_{2}t_{11}t_{8}-t_{10}t_{9},\\ -t_{12}t_{4}-t_{11}t_{5}+t_{10}t_{6}-t_{7}t_{9},\ -t_{4}t_{6}-t_{3}t_{7}+d_{2}t_{2}t_{8}-d_{1}t_{1}t_{9},\ t_{3}t_{4}+t_{2}t_{5}+t_{1}t_{6}-t_{12}t_{8},\\ d_{1}t_{1}t_{2}+d_{1}t_{12}t_{4}+t_{10}t_{6}-t_{8}^{2},\ -d_{2}t_{11}t_{12}+t_{5}^{2}+t_{3}t_{7}+t_{1}t_{9},\ d_{2}t_{11}t_{2}-t_{10}t_{3}-t_{6}t_{7}+t_{4}t_{9},\\ t_{4}^{2}-t_{3}t_{5}+d_{2}t_{2}t_{6}+d_{1}t_{1}t_{7},\ -t_{12}t_{6}+t_{11}t_{7}-t_{10}t_{8}-t_{9}^{2},\ -d_{2}t_{2}t_{3}+t_{1}t_{4}+d_{2}t_{11}t_{7}-t_{10}t_{8},\\ t_{10}t_{12}+t_{4}t_{5}-t_{2}t_{7}-t_{1}t_{8},\ d_{1}t_{1}t_{3}-d_{1}t_{12}t_{5}+d_{2}t_{11}t_{6}-t_{8}t_{9},\ -t_{10}t_{11}+t_{3}t_{5}+t_{2}t_{6}-d_{1}t_{12}t_{9},\\ -d_{2}t_{11}^{2}+d_{1}t_{10}t_{12}-t_{4}t_{5}+t_{3}t_{6},\ d_{2}t_{2}^{2}+t_{1}t_{3}-t_{10}t_{7}-t_{8}t_{9},\ d_{1}t_{1}t_{12}+t_{6}t_{7}-t_{5}t_{8}-t_{4}t_{9},\\ -d_{1}t_{12}^{2}+t_{5}t_{6}+t_{3}t_{8}-t_{2}t_{9},\ d_{1}t_{1}^{2}-d_{2}t_{11}t_{4}-t_{10}t_{5}+t_{7}t_{8},\ -t_{12}t_{3}-t_{11}t_{4}+t_{7}t_{8}+t_{6}t_{9},\\ d_{1}d_{2}t_{12}t_{2}-d_{2}t_{11}t_{3}-t_{10}t_{4}-t_{6}t_{8},\ -t_{1}t_{11}-t_{10}t_{2}+t_{5}t_{7}-t_{3}t_{9},\ d_{1}t_{1}t_{10}+t_{5}t_{6}+t_{4}t_{7}+d_{2}t_{2}t_{9},\\ d_{2}t_{12}t_{2}+t_{10}t_{4}-t_{7}^{2}-t_{5}t_{9},\ d_{1}t_{1}t_{11}+t_{6}^{2}+t_{4}t_{8}+t_{3}t_{9}\end{array}$$

in the homogeneous coordinates $(t_{1}:\ldots:t_{12})$ of $\mathbb{P}^{11}$. The action of $g$ is $t_{i}\mapsto\zeta_{13}^{i}t_{i}$.

Our method of constructing a fake projective plane largely followed the blueprint of [BF20].

We set $d_{3}=d_{4}=d_{5}=d_{6}=1$ and tried to find out which $(d_{1},d_{2})$ give singular cuts. In order to achieve this, we worked on an affine coordinate chart of $\mathbb{O}\mathbb{P}^{2}$ which can be obtained by solving the equations of $\mathbb{O}\mathbb{P}^{2}$ for eleven of the variables as follows.

$$\begin{array}[]{l}P_{4}=P_{10}P_{16}+P_{2}P_{24}+P_{12}P_{26}+P_{14}P_{9},P_{6}=-P_{14}P_{17}+P_{2}P_{20}-P_{10}P_{27}-P_{12}P_{7},\\ P_{8}=-P_{1}P_{14}-P_{16}P_{20}-P_{24}P_{27}+P_{12}P_{3},P_{11}=P_{15}P_{24}-P_{1}P_{26}-P_{16}P_{5}-P_{3}P_{9},\\ P_{13}=P_{15}P_{20}+P_{17}P_{3}+P_{27}P_{5}+P_{1}P_{7},P_{18}=-P_{20}P_{26}-P_{10}P_{3}+P_{14}P_{5}-P_{24}P_{7},\\ P_{19}=-P_{14}P_{15}-P_{26}P_{27}-P_{2}P_{3}+P_{16}P_{7},P_{21}=-P_{10}P_{15}+P_{17}P_{26}-P_{2}P_{5}-P_{7}P_{9},\\ P_{22}=1,P_{23}=-P_{12}P_{15}-P_{16}P_{17}-P_{1}P_{2}+P_{27}P_{9},P_{25}=P_{1}P_{10}-P_{17}P_{24}-P_{12}P_{5}-P_{20}P_{9}\end{array}$$

We obtained this chart by connecting the formulas for the Cartan cubic from [GE96] and [L11]. We then further solved for five of the variables to reduce their number while still keeping the equations relatively short. Then we looked for tangent vectors for the surfaces with $d_{1}=1$ that lie in a codimension three subspace, by a multivariable Newton method starting at random points. The idea is that some of these would happen at values of $d_{2}$ where the surface $X=X_{1,d_{2}}$ acquires a node. After some trial and error we saw that solutions to

$$-27-34d_{2}-397d_{2}^{2}-172d_{2}^{3}-821d_{2}^{4}+190d_{2}^{5}-83d_{2}^{6}+16d_{2}^{7}=0$$

give singular surfaces. As in [BF20], we then perturbed $d_{1}$ slightly to $1+10^{-20}$ to find a nearby point on the locus of singular surfaces. This lead us to conjecture that generic points $(d_{1},d_{2})$ on the curve

$$\begin{array}[]{l}0=-4d_{1}^{3}+8d_{1}^{4}-4d_{1}^{5}-12d_{1}^{2}d_{2}-16d_{1}^{3}d_{2}+28d_{1}^{4}d_{2}-39d_{1}^{5}d_{2}+12d_{1}^{6}d_{2}-12d_{1}d_{2}^{2}-28d_{1}^{2}d_{2}^{2}-54d_{1}^{3}d_{2}^{2}\\ +78d_{1}^{4}d_{2}^{2}-34d_{1}^{5}d_{2}^{2}+28d_{1}^{6}d_{2}^{2}-12d_{1}^{7}d_{2}^{2}-4d_{2}^{3}-39d_{1}d_{2}^{3}-34d_{1}^{2}d_{2}^{3}-277d_{1}^{3}d_{2}^{3}+192d_{1}^{4}d_{2}^{3}-277d_{1}^{5}d_{2}^{3}\\ +54d_{1}^{6}d_{2}^{3}-16d_{1}^{7}d_{2}^{3}+4d_{1}^{8}d_{2}^{3}-8d_{2}^{4}-28d_{1}d_{2}^{4}-78d_{1}^{2}d_{2}^{4}-192d_{1}^{3}d_{2}^{4}+192d_{1}^{5}d_{2}^{4}-78d_{1}^{6}d_{2}^{4}+28d_{1}^{7}d_{2}^{4}\\ -8d_{1}^{8}d_{2}^{4}-4d_{2}^{5}-16d_{1}d_{2}^{5}-54d_{1}^{2}d_{2}^{5}-277d_{1}^{3}d_{2}^{5}-192d_{1}^{4}d_{2}^{5}-277d_{1}^{5}d_{2}^{5}+34d_{1}^{6}d_{2}^{5}-39d_{1}^{7}d_{2}^{5}+4d_{1}^{8}d_{2}^{5}+12d_{1}d_{2}^{6}\\ +28d_{1}^{2}d_{2}^{6}+34d_{1}^{3}d_{2}^{6}+78d_{1}^{4}d_{2}^{6}+54d_{1}^{5}d_{2}^{6}-28d_{1}^{6}d_{2}^{6}+12d_{1}^{7}d_{2}^{6}-12d_{1}^{2}d_{2}^{7}-39d_{1}^{3}d_{2}^{7}-28d_{1}^{4}d_{2}^{7}-16d_{1}^{5}d_{2}^{7}\\ +12d_{1}^{6}d_{2}^{7}+4d_{1}^{3}d_{2}^{8}+8d_{1}^{4}d_{2}^{8}+4d_{1}^{5}d_{2}^{8}\end{array}$$

give nodal $X_{d_{1},d_{2}}$.

We then looked for singular points of this curve. There were several such points, one of which was a cusp of the curve. We focused our attention on it and discovered a surface $X_{d_{1},d_{2}}$ with $39$ $A_{2}$ singularities. Specifically, both $d_{1}$ and $d_{2}$ can be given given as roots of

$$0=2187+7290d+23433d^{2}+21640d^{3}+66393d^{4}-21640d^{5}+23433d^{6}-7290d^{7}+2187d^{8}$$

approximately given by $(d_{1},d_{2})\approx(1.93+2.30\,\rm i,0.0125-0.515\,\rm i)$. Of course, the same is true for all of the Galois conjugates of this pair. From now on we will call this surface $X_{0}$.

We observed that four of the Galois conjugate pairs of $(d_{1},d_{2})$ give isomorphic surfaces. To see that, we noticed that scheme $X_{d_{1},d_{2}}$ cut out by (3) is isomorphic to $X_{-1/d_{2},d_{1}}$ under the coordinate change

(4)

$$(t_{1},\ldots,t_{12})\mapsto(d_{2}t_{5},t_{10},d_{2}t_{2},-d_{1}t_{7},t_{12},t_{4},-d_{1}d_{2}t_{9},-d_{1}d_{2}t_{1},d_{2}t_{6},-d_{1}t_{11},-d_{1}d_{2}t_{3},-d_{1}t_{8})$$

The idea behind it was to use $i\to 5i\hskip-4.0pt\mod 13$, and we heavily relied on Mathematica computations, see [BBF20+, Section2.nb].

We then used the symmetry (4) to average the $C_{13}$-invariants of the coordinate ring of the surface $X_{0}$ suspected to have $A_{2}$ singularities, to get $X_{0}/C_{13}$ defined in the $13$-dimensional weighted projective space $W\mathbb{P}(2^{4},3^{10})$ by $9$ equations of degree five and $29$ equations of degree six. Due to the above symmetrization, the coefficients were in the field $\mathbb{Q}(\sqrt{-2})$.

It was not entirely trivial to find the singular points of $X_{0}/C_{13}$. We did it by calculating a degree $12$ equation in the first four variables which gives a (non-normal) image of $X_{0}/C_{13}$ in $\mathbb{P}^{3}$. Then we looked for its curves of singularities by finding multiple singular points on random hyperplane cuts. Then we have looked for singular points outside of the curve of singularities, and indeed hit upon $A_{2}$ singularities. We were then able to verify that these were the only singularities by computing the degree of the singular locus over a finite field. As in the case of [BF20], the $A_{2}$ singularities were not defined over the quadratic extension of $\mathbb{Q}$, but a coordinate change gave us a model of $X_{0}/C_{13}\subseteq W\mathbb{P}^{13}$ still defined over $\mathbb{Q}(\sqrt{-2})$ and with three singular points defined over $\mathbb{Q}$.

By the work of Keum [Ke12], the surface $X_{0}/C_{13}$ admits a Galois triple cover which is a fake projective plane. In this, it is very similar to the situation in [BF20] and we employed the same general method. It was useful that in both cases there was an additional order three automorphism $\sigma$ because the FPP had a $C_{3}\times C_{3}$ group of automorphisms. Specifically, we looked for sections $f$ and $d$ of $4K_{X_{0}/C_{13}}$ which satisfy

$$f\,\sigma(f)\sigma^{2}(f)=d^{3}.$$

Moreover $f$ and $d$ should have certain vanishing on the exceptional lines at the blowup of $A_{2}$ singularities. We refer the reader to [BF20] for details.

The nature of $X_{0}/C_{13}$ made the computations more challenging. In particular, at some point we had to work with random points on the surface computed with $6\times 10^{4}$ digits of accuracy. The equations for $f$ and $d$ had coefficients in $\mathbb{Q}(\sqrt{-2})$ which were about $1.5\times 10^{4}$ digits long. As in [BF20], we solved it over a finite field of $19$ elements, but now we used a p-adic version of the Newton’s method to quickly gain the needed accuracy.

Once the triple cover was constructed, we used the fixed points of the automorphisms of $\mathbb{P}^{2}_{fake}$ to get a basis with nicer equations, only about 100 digits long coefficients, see [BBF20+, Section3.nb] for details. This surface is labeled by $(\operatorname{C18},p=3,\emptyset,d_{3}D_{3})$ in the classification of [CS11+], since it is the only one with an automorphism group that contains $(C_{3})^{2}$ and Picard group that contains $C_{13}$.

The details of the above process are in [BBF20+, Section3.nb].

Let us now discuss open problems related to this construction.

The first question is how to verify that the special cuts $X_{d_{1},d_{2}}$ of $\mathbb{O}\mathbb{P}^{2}$ are simply connected. Since these are not complete intersections, the Lefschetz Hyperplane theorem can not be applied, so other methods are needed. It might perhaps follow from our construction and [CS11+], but a more direct argument is desirable.

A related question is how to construct non-$C_{3}$-invariant deformations of $X_{d_{1},d_{2}}$. It looks like they will no longer be cuts of $\mathbb{O}\mathbb{P}^{2}$ but perhaps one can get them by carefully examining the equations (2).

The quotient of the fake projective plane $(\operatorname{C18},p=3,\emptyset,d_{3}D_{3})$ by $(C_{3})^{2}$ is also covered by $(\operatorname{C18},p=3,\{2I\})$. This fake projective plane in turn covers a surface $(\operatorname{C18},p=3,\{2\})$, which is covered by three other fake projective planes. The method of [BF20] is, unfortunately, not quite applicable here, so how do we find (the equations of) these other surfaces?

It is known from [CS11+] that $(\operatorname{C18},p=3,\emptyset,d_{3}D_{3})$ has Picard group $C_{2}\times C_{2}\times C_{13}$. While the $C_{13}$ part can be inferred from our construction (even though it may not be entirely trivial to follow), the other two factors are mysterious. It would be interesting to see them explicitly, and they may be useful in answering both the previous and the next questions.

A perennial question is how one can reduce the size of the coefficients in the equations. There are currently only ad hoc tools that are not very successful, except in [BK19] case.