2.1 Line arrangements and the operator

A line arrangement $\mathcal {C}=\ell _{1}+\dots +\ell _{n}$ is a union of finitely many distinct lines in $\mathbb {P}^{2}$ . A labeled line arrangement $\mathcal {C}=(\ell _{1},\dots ,\ell _{n})$ is a line arrangement with a numbering of the lines. We sometime put a superscript $^{\ell }$ (resp. $^{u}$ ) when we want to emphasize that an arrangement or related objects has (resp. does not have) a labeling.

For an integer $k\geq 2$ , a k-point of the line arrangement $\mathcal {C}$ is a point where exactly k lines of $\mathcal {C}$ meet. As in [Reference Roulleau9], for a subset $\mathfrak {n}$ of integers at least $2$ , let us denote by $\mathcal {P}_{\mathfrak {n}}(\mathcal {C})$ the set of k-points of $\mathcal {C}$ for all $k\in \mathfrak {n}$ . We denote by $t_{k}=t_{k}(\mathcal {C})=|\mathcal {P}_{\{k\}}(\mathcal {C})|$ the number of k-points of $\mathcal {C}$ . For a finite set of point $\mathcal {P}$ in $\mathbb {P}^{2}$ and $\mathfrak {n}$ as above, we denote by $\mathcal {L}_{\mathfrak {n}}(\mathcal {P})$ the set of lines which contain exactly n points in $\mathcal {P}$ for some $n\in \mathfrak {n}$ .

For subsets $\mathfrak {n,m}$ of integers at least $2$ , let us denote by the line arrangement that contains all lines of $\mathbb {P}^{2}$ containing exactly m points of $\mathcal {P}_{\mathfrak {n}}(\mathcal {C})$ for $m\in \mathfrak {m}$ . For example is the union of the lines that contain three or four double points of $\mathcal {C}$ . The arrangement could be the empty arrangement if no such lines exists.

2.2 Matroids and the period map of the moduli of a matroid.

A matroid is a fundamental and actively studied object in combinatorics. Matroids generalize linear dependency in vector spaces as well as forests in graphs. See, for example, [Reference Oxley8] for a comprehensive treatment of matroids. We just briefly mention a few concepts about matroids that are relevant for this article.

A matroid is a pair $M=(E,\mathcal {B})$ , where E is a finite ground set of elements called atoms and $\mathcal {B}$ is a nonempty collection of subsets of E, called bases, satisfying an exchange property reminiscent from linear algebra.

The prime examples of matroids arise by choosing a finite set E of vectors in a vector space and declaring the maximal linearly independent subsets of E as bases. In our case we obtain matroids through line arrangements: If $\mathcal {C}=(\ell _{1},\dots ,\ell _{m})$ is a labeled line arrangement, the subsets $\{i,j,k\}\subseteq \{1,\dots ,m\}$ such that the lines $\ell _{i},\ell _{j},\ell _{k}$ meet in three distinct points are the bases of a matroid $M(\mathcal {C})$ over the set $\{1,\dots ,m\}$ . We say that $M(\mathcal {C})$ is the matroid associated to $\mathcal {C}$ .

We denote by the automorphism group of the matroid M, that is, the set of isomorphisms from M to M.

A realization (over some field) of a matroid $M=(E,\mathcal {B})$ is a converse operation to the association $\mathcal {C}\to M(\mathcal {C})$ : it is a $3\times m$ -matrix with non-zero columns $C_{1},\dots ,C_{m}$ , which are considered up to a multiplication by a scalar (thus as point in the projective plane) such that a subset $\{i_{1},i_{2},i_{3}\}$ of E of size $3$ is a basis if and only if the $3\times 3$ minor $|C_{i_{1}},C_{i_{2}},C_{i_{3}}|$ is nonzero. We denote by $\ell _{i}$ the line with normal vector the point $C_{i}\in \mathbb {P}^{2}$ .

If $\mathcal {C}=(\ell _{1},\dots ,\ell _{m})$ is a realization of M and $\gamma \in PGL_{3}$ , then $(\gamma \ell _{1},\dots ,\gamma \ell _{m})$ is another realization of M; we denote by $[\mathcal {C}]$ the orbit of $\mathcal {C}$ under that action of $PGL_{3}$ . The moduli space $\mathcal {R}(M)$ of realizations of M parametrizes the orbits $[\mathcal {C}]$ of realizations. A more detailed introduction to these moduli spaces together with a description of a software package in OSCAR that can compute these spaces is given in [Reference Corey, Kühne and Schröter3].

In this article, we always assume that each subset of three elements of the first four atoms is a basis (otherwise, we replace M by a matroid isomorphic to it). Then in the moduli space $\mathcal {R}(M)$ , one can always map the first four vectors of $\mathcal {C}\in [\mathcal {C}]$ to a fixed projective basis, so that each element $[\mathcal {C}]$ of $\mathcal {R}(M)$ has a canonical representative, which we will identify with $[\mathcal {C}]$ .

A useful tool for the computations related to the moduli space $\mathcal {R}=\mathcal {R}(M)$ of realizations of a matroid M is what we call the period map: Let us denote by $\mathfrak {U}=\mathfrak {U}(M)$ the scheme of all realizations of M in $\mathbb {P}^{2}$ . By analogy with similar objects, we call the quotient map

$$\begin{align*}\mathfrak{q}:\mathfrak{U}(M)\to\mathcal{R}(M) \end{align*}$$

the period map; a point c of $\mathcal {R}=\mathcal {R}(M)$ is the class $c=[\mathcal {C}]$ of a realization $\mathcal {C}$ . Once a basis is fixed, each class c has a unique representative $\mathcal {C}_{0}$ and we can (and we will) identify c with that representative.

It often occurs that $\mathcal {R}$ is embedded in a space $\mathbb {S}=\mathbb {S}(y_{1},\dots ,y_{k})$ (affine or projective) of small dimension, like $\mathbb {P}^{3}$ . The coordinates of the normal vectors $n^{(j)}=(n_{1}^{(j)}:n_{2}^{(j)}:n_{3}^{(j)})$ of $\mathcal {C}_{0}$ are then polynomials $n_{1}^{(j)}=P_{1}^{(j)}(y),\dots ,n_{3}^{(j)}=P_{3}^{(j)}(y)$ in the coordinates $y_{1},\dots ,y_{k}$ of $\mathcal {R}$ in $\mathbb {S}$ .

One often arrives at the natural question on computing the point $y=(y_{1},\dots ,y_{k})$ in $\mathcal {R}$ from the knowledge of the normal vectors n. In other words, we need an explicit form of the period map $\mathfrak {q}$ as a map from $\mathfrak {U}$ to the scheme $\mathcal {R}$ embedded in the space $\mathbb {S}$ . The answer to that problem are polynomials (or rational functions) $Q_{1},\dots ,Q_{k}$ in the coordinates of the normal vectors $n^{(1)},\dots ,n^{(m)}$ etc.; here m is the number of lines in an arrangement.

2.3 Degree two K3 surfaces semi-conjugated to the plane.

Let $C_{1}:Q_{1}=0$ be a sextic curve with at most ADE singularities, so that the desingularization $X^{s}$ of the associated double cover

$$\begin{align*}X=\{y^{2}=Q_{1}(z_{1},z_{2},z_{3})\}\hookrightarrow\mathbb{P}(3,1,1,1){,} \end{align*}$$

is a K3 surface. Let $F:\mathbb {P}^{2}\to \mathbb {P}^{2}$ be a rational self-map defined by coprime homogeneous polynomials $(F_{1},F_{2},F_{3})$ of degree m. Suppose that $F^{*}C_{1}=C_{1}+2D$ , for an effective divisor D; in algebraic terms, that means that we assume that

$$\begin{align*}Q_{1}(F_{1},F_{2},F_{3})=Q_{1}\cdot R^{2}, \end{align*}$$

for some polynomial R. Then the following relation holds

$$\begin{align*}(y\,R(z))^{2}=Q_{1}(z)\,R(z)^{2}=Q_{1}(F_{1}(z),F_{2}(z),F_{3}(z)), \end{align*}$$

where $z=(z_{1}:z_{2}:z_{3})\in \mathbb {P}^{2}$ . Hence, the rational map

$$\begin{align*}\tilde{F}:(y;z)\dashrightarrow(y\,R(z);F_{1}(z):F_{2}(z):F_{3}(z)), \end{align*}$$

is a rational self-map acting on the K3 surface $X^{s}$ . Let $\pi :X\to \mathbb {P}^{2}$ be the double cover map. The following diagram

(2.1) $$ \begin{align} \begin{array}{ccc} X & \stackrel{\tilde{F}}{\dashrightarrow} & X\\ \pi\downarrow & & \pi\downarrow\\ \mathbb{P}^{2} & \stackrel{F}{\dashrightarrow} & \mathbb{P}^{2,} \end{array} \end{align} $$

is commutative and, by analogy with other dynamical systems, we say that the dynamical system $(X,\tilde {F})$ is semi-conjugated to $(\mathbb {P}^{2},F)$ .

3.1 is a rational self-map on $ \mathcal {R}_{7}$ and $ \mathfrak {U}_7.$

3.1.1 Definition of the matroid $M_{7}.$

The matroid $M_{7}$ has $14$ atoms $1,\dots ,7,1',\dots ,7'$ and the bases are the triples $\{a,b,c\}$ with $\{a,b\}\subset \{1,\dots ,7\}$ and $c\in \{1',\dots ,7'\}$ such that $a+b\neq 2c\ \mod 7$ . A sketch of $M_7$ is described in Figure 1, where the atoms $i\in \{1,\dots ,7\}$ and $j\in \{1',\dots ,7'\}$ correspond to the lines $\ell _i$ and $\ell _j'$ , resp., and three lines form a basis if they do not meet in one point. Note that the central singularity of arrangement in Figure 1 is not part of the matroid and therefore removed.

Figure 1

The matroid $M_{7}$ whose construction is based on the regular heptagon.

Let $\mathcal {A}_{1}$ be a generic line arrangement realizing the matroid $M_{7}$ . We write $\mathcal {A}_{1}=\mathcal {C}_{0}\cup \mathcal {C}_{1}$ where $\mathcal {C}_{0}$ are the first seven lines and $\mathcal {C}_{1}$ are the seven last ones. By the combinatorics of the matroid $M_{7}$ and the genericity assumption, the property holds, and – that will be important for us – the image of $\mathcal {C}_{0}$ by the operator has a natural labeling: for any $j\in \{1,\dots ,7\}$ , the six line arrangement

(3.1) $$ \begin{align} H_{j}=\sum_{k\in\{1,\dots,7\},\,k\neq j}\ell_{k{,}} \end{align} $$

is such that the line arrangement is a unique line $\ell _{j}'$ , moreover:

$$\begin{align*}\mathcal{C}_{1}=(\ell_{1}',\dots,\ell_{7}'). \end{align*}$$

Since , to shorten our notations, we will often speak of $\mathcal {C}_{0}$ as a realization of $M_{7}$ instead of $\mathcal {C}_{0}\cup \mathcal {C}_{1}$ .

The singularities of $\mathcal {C}_{0}$ (resp. $\mathcal {C}_{1}$ ) are $21$ double points. The $21$ singularities on $\mathcal {C}_{0}$ become the triple points on $\mathcal {C}_{0}\cup \mathcal {C}_{1}$ , moreover $t_{2}(\mathcal {C}_{0}\cup \mathcal {C}_{1})=28$ .

3.1.2 Equation of the quartic surface $Z_7$ and realization space of $M_{7.}$

Consider $Z_7$ , the quartic surface in $\mathbb {P}^{3}$ given by the equation

(3.2) $$ \begin{align} y_{1}^{2}y_{2}^{2}+y_{1}^{2}y_{2}y_{3}-y_{1}y_{2}^{2}y_{3}-y_{1}y_{2}y_{3}^{2}-y_{1}^{2}y_{2}y_{4}-y_{1}y_{2}^{2}y_{4}+y_{1}y_{2}y_{3}y_{4}-y_{2}y_{3}^{2}y_{4}+y_{1}y_{2}y_{4}^{2}+y_{3}^{2}y_{4}^{2}=0. \end{align} $$

The eight singularities of $Z_7$ are of type $4A_{1}+A_{2}+3A_{3},$ at the points respectively

$$\begin{align*}\begin{array}{c} s_{1}=(0:0:0:1),s_{2}=(1:0:0:1),s_{3}=(0:0:1:0),s_{4}=(1:0:1:0),\\ s_{5}=(0:1:0:0),s_{6}=(0:1:0:1),s_{7}=(1:-1:1:0),s_{8}=(1:0:0:0). \end{array} \end{align*}$$

The minimal desingularization of $Z_7$ is a K3 surface which we denote by $Z_7^{s}$ . Let $x_{1},x_{2},x_{3}$ be the coordinates on the affine chart $y_{4}\neq 0$ . For a generic point $x=(x_{1},x_{2},x_{3})$ on the surface $Z_7$ in the chart $y_{4}\neq 0$ , let us define the labeled arrangement of seven lines $\mathfrak {\mathcal {C}}_{0}=\mathfrak {\mathcal {C}}_{0}(x)$ with normal vectors the points $p_{1},\dots ,p_{7}$ respectively defined by

(3.3) $$ \begin{align} \begin{array}{c} (1:0:0),(0:1:0),(0:0:1),(-1:1:1)\\ (-x_{1}x_{2}^{2}-x_{1}x_{2}x_{3}+x_{1}x_{2}-x_{2}x_{3}+x_{3}\,:\,x_{1}x_{2}+x_{1}x_{3}-x_{1}\,:\,x_{2}-1)\\ (-x_{1}x_{2}^{2}-x_{1}x_{2}x_{3}+x_{1}x_{2}-x_{2}x_{3}+x_{3}\,:\,x_{1}x_{2}+x_{1}x_{3}-x_{1}+x_{2}^{2}\\ +x_{2}x_{3}-2x_{2}-x_{3}+1\,:\,x_{2}^{2}+x_{2}x_{3}-x_{2}-x_{3})\\ (-x_{1}x_{2}^{2}-x_{1}x_{2}x_{3}+x_{1}x_{2}+x_{3}^{2}\,:\,x_{1}x_{2}+x_{1}x_{3}-x_{1}-x_{2}x_{3}\\ -x_{3}^{2}+x_{3}\,:\,x_{2}^{2}+x_{2}x_{3}-x_{2}-x_{3}). \end{array} \end{align} $$

Let us also define the lines arrangement $\mathcal {C}_{1}=\mathcal {C}_{1}(x)$ with normal vectors

(3.4) $$ \begin{align} \begin{array}{c} (-x_{1}x_{2}^{2}-x_{1}x_{2}x_{3}+x_{1}x_{2}+x_{3}^{2}\,:\,x_{1}x_{2}^{2}+2x_{1}x_{2}x_{3}-x_{1}x_{2}+x_{1}x_{3}^{2}\\ -x_{1}x_{3}-x_{2}^{2}x_{3}-2x_{2}x_{3}^{2}+x_{2}x_{3}-x_{3}^{3}+x_{3}^{2}\,:\,x_{2}^{2}+x_{2}x_{3}-x_{2}-x_{3}),\\ (-x_{1}x_{2}-x_{1}x_{3}+x_{1}\,:\,x_{1}x_{2}+x_{1}x_{3}-x_{1}\,:\,x_{2}-1),(-x_{2}:1:0),\\ (-x_{1}x_{2}^{3}-2x_{1}x_{2}^{2}x_{3}+x_{1}x_{2}^{2}-x_{1}x_{2}x_{3}^{2}+x_{1}x_{2}x_{3}-x_{2}^{2}x_{3}-x_{2}x_{3}^{2}+x_{2}x_{3}\\ +x_{3}^{2}\,:\,x_{1}x_{2}+x_{1}x_{3}-x_{1}+x_{2}^{2}+x_{2}x_{3}-2x_{2}-x_{3}+1\,:\,x_{2}^{2}+x_{2}x_{3}-x_{2}-x_{3}),\\ (-x_{1}x_{2}^{2}-x_{1}x_{2}x_{3}+x_{1}x_{2}-x_{2}x_{3}+x_{3}\,:\,0\,:\,x_{2}^{2}+x_{2}x_{3}-x_{2}-x_{3}),\\ (-x_{2}^{2}-x_{2}x_{3}+x_{2}+x_{3}\,:\,x_{1}x_{2}+x_{1}x_{3}-x_{1}-x_{2}x_{3}-x_{3}^{2}+x_{3}\,:\,x_{2}^{2}+x_{2}x_{3}\\ -x_{2}-x_{3}),(0:1:1). \end{array} \end{align} $$

A computation in OSCAR yields the following concrete description of the moduli space $\mathcal {R}_{7}=\mathcal {R}(M_7)$ .

Proposition 3.1. The moduli space $\mathcal {R}_{7}$ is an open sub-scheme of $Z_7$ : for $x\in \mathcal {R}_{7}$ , the line arrangement $\mathcal {A}=\mathcal {C}_{0}(x)\cup \mathcal {C}_{1}(x)$ is a realization of $M_{7}$ , and conversely any realization of $M_{7}$ is projectively equivalent to a unique such line arrangement.

The complement of $\mathcal {R}_{7}$ in $Z_7$ is the union of $20$ irreducible curves described in §3.2.

From the definition of the matroid $M_{7},$ if $\mathcal {A}=\mathcal {C}_{0}\cup \mathcal {C}_{1}$ is a realization of $M_{7}$ , one has , but the following result on is unexpected

Theorem 3.2. Let $\mathcal {A}_{0}=\mathcal {C}_{0}\cup \mathcal {C}_{1}$ be a generic realization of $M_{7}$ and define . The labeled line arrangement $\mathcal {A}_{1}=\mathcal {C}_{1}\cup \mathcal {C}_{2}$ is again a realization of $M_{7}$ . The operator induces a rational self-map on the schemes $\mathfrak {U}_7$ of all realizations of $M_{7}$ and its moduli space $\mathcal {R}_{7}$ .

We denote by the rational self-map on $Z_7$ induced by .

Proof. Up to projective automorphism, one can suppose that the line arrangement $\mathcal {A}_{0}$ is of the form $\mathcal {A}_{0}=\mathcal {C}_{0}(x)\cup \mathcal {C}_{1}(x)$ for x generic in $Z_7$ : concretely, we use $x=(x_{1},x_{2},x_{3})$ , where $x_{1},x_{2},x_{3}\in \mathbb {C}(Z_7)$ are considered as rational functions. A direct computation (with Magma) then shows that is a line arrangement of seven lines. It has a canonical labeling as described in the previous Subsection and we then check that the matroid associated to $\mathcal {C}_{1}\cup \mathcal {C}_{2}$ is equal to $M_{7}$ , so that $\mathcal {C}_{1}\cup \mathcal {C}_{2}$ is a realization of $M_{7}$ . Using the period map, one computes and obtain that it is a dominant rational map. The reader can find the polynomials defining in an ancillary file of this paper on arXiv; it can be also retrieved from the polynomials given in §3.6. That describes action of on the space of realization $\mathfrak {U}_7$ and on the moduli space $\mathcal {R}_{7}$ .

3.2 The open surface $ \mathcal {R}_{7}$ inside $Z_7.$

The scheme $Z_7\setminus \mathcal {R}_{7}$ is the union of the following curves:

• • The $12$ lines

  $$\begin{equation*}\begin{array}{lll} L_{1}: y_{2}=y_{3}=0, &L_{2}:y_{1}=y_{3}=0, &L_{3}:y_{2}=y_{4}=0,\\ L_{4}:y_{1}-y_{3}=y_{4}=0, &L_{5}: y_{1}=y_{4}=0, &L_{6}:y_{2}-y_{4}=y_{3}=0,\\ L_{7}:y_{1}-y_{3}-y_{4}=y_{2}+y_{3}=0,\, &L_{8}: y_{1}-y_{3}=y_{2}+y_{3}=0, &L_{9}:y_{2}+y_{3}=y_{4}=0,\\ L_{10}:y_{1}-y_{4}=y_{3}=0, &L_{11}: y_{1}-y_{3}=y_{2}-y_{4}=0,\, &L_{12}:y_{1}=y_{2}-y_{4}=0. \end{array} \end{equation*}$$
  These lines are also the lines contained in the quartic surface $Z_7$ that contain at least two double points of $Z_7$ .

• • The conic $C_{o}$ defined by $y_{1}y_{3}-y_{3}^{2}-y_{1}y_{4}=y_{2}+y_{3}-y_{4}=0$ .

• • Seven curves $E_{1},\dots ,E_{7}$ of geometric genus one. For example, one of these curves is given by

  $$\begin{align*}y_{1}^{2}-2y_{1}y_{3}+y_{3}^{2}-y_{1}y_{4}=y_{2}^{2}+y_{2}y_{3}+y_{1}y_{4}-y_{3}y_{4}-y_{4}^{2}=0. \end{align*}$$

The j-invariant of the normalizations of the curves $E_{i}$ is equal to $-5^{6}/28$ . The elliptic curve with this j-invariant is known as the modular curve $X_{1}(14)$ parameterizing pairs $(E,t)$ where E is an elliptic curve and t is an order $14$ torsion element of E. For a generic point p on the curves $E_{1},\dots ,E_{7}$ , the line arrangement $\mathcal {C}_{0}(p)$ with normal vectors as in (3.3) is well-defined. The line arrangement has seven lines, but its singularities are $t_{2}=6,t_{3}=5$ , and one has . Moreover, the singularities of $\mathcal {C}_{0}\cup \mathcal {C}_{1}$ are $t_{2}=13,t_{3}=26$ .

The image of the curves $C_{o},E_{1},\dots ,E_{7}$ under the map are lines $L_{k}$ ; when defined, the image of the lines $L_{k}$ are lines $L_{k'}$ or points.

3.3 The degree of

Recall that denotes the action of the operator on the K3 surface $Z_7$ . One has:

In order to prove Theorem 3.3, let us describe the period map: Let $\ell _{1},...,\ell _{7}$ be the lines of $\mathcal {C}_{0}$ with normal vectors as in Equation (3.3). Let us denote by $p_{i,j}$ the intersection point of the lines $\ell _{i}$ and $\ell _{j}$ . The point $p_{5,7}$ is $(1:x_{2}:x_{3})$ , so that one may recover $x_{2},x_{3}$ from the knowledge of that point. Also the point $p_{1,7}$ is

(3.5) $$ \begin{align} (0:-x_{1}x_{2}{}^{2}-x_{1}x_{2}x_{3}+x_{1}x_{2}+x_{2}{}^{2}-x_{2}:x_{2}x_{3}+x_{3}{}^{2}-x_{3}), \end{align} $$

this is linear in $x_{1}$ , so that from the knowledge of $p_{5,7}$ and $p_{1,7}$ , one may recover the point $(x_{1},x_{2},x_{3})\in Z_7$ .

Proof of Theorem 3.3

Let $A\in PGL_{3}(\mathbb {C})$ be the projective transformation that sends the first four lines of $\mathcal {C}_{1}$ to the four lines having the same normal vectors as the one of $\mathcal {C}_{0}$ . Let $\mathcal {C}_{1}'=(\ell _{1}',\dots ,\ell _{7}')$ be the image of $\mathcal {C}_{1}$ by A. Using the period map, one can determine the points $p_{5,7}'$ and $p_{1,7}'$ and we obtain a point $x'=(x_{1}',x_{2}',x_{3}')$ (in the function field of $Z_7$ ). The line arrangements $\mathcal {C}_{0}(x_{1}',x_{2}',x_{3}')$ and $\mathcal {C}_{1}'$ are equal, and the action of on $Z_7$ is through the map

The rational self-map is studied in §3.6.

Let us compute the degree of ; we apply the method from [Reference Voisin15]. Let $f(x_{1},x_{2},x_{3})$ be the equation of the quartic $Z_7$ in the chart $U_{4}:y_{4}\neq 0$ . The space of global non-vanishing differential $2$ -forms is generated by a form $\unicode{xf8} $ , which one can choose so that on an open set of $U_{4}$ one has:

The rational self-map preserves $U_{4}$ , and by a direct computation one obtains that

The above expression shows that when applying , the volume form is multiplied by $4$ , which gives the degree of .

3.4 Action of on the K3 surface $Z_7^{s}.$

The automorphism group of $M_{7}$ is generated by the order $7$ and $6$ permutations

$$\begin{align*}\sigma_{1}=(1,7,4,3,6,5,2)(8,14,11,10,13,12,9)\text{ and }\sigma_{2}=(1,3,5,6,7,2)(8,10,12,13,14,9). \end{align*}$$

with $1'=8,\dots ,7'=14$ . This group is isomorphic to the the Frobenius group $F_{7}=\mathbb {Z}/6\mathbb {Z}\rtimes \mathbb {Z}/7\mathbb {Z}$ . These automorphisms act on the K3 surface Z.

Proof. As in the proof of Theorem 3.3, let $\mathcal {C}_{0}=\mathcal {C}_{0}(x_{1},x_{2},x_{3})$ be the generic line arrangement in $\mathcal {R}_{7}$ , where $x_{1},x_{2},x_{3}\in \mathbb {C}(Z_7)$ are considered as rational functions.

For let $\mathcal {C}_{0}^{\sigma }$ be the image of $\mathcal {C}_{0}$ under the action of $\sigma $ (that is just the permutation of the lines under $\sigma $ ). We apply the period map to the line arrangement $\mathcal {C}_{0}^{\sigma }$ , where $\mathcal {C}_{0}=\mathcal {C}_{0}(x)$ . Using the period map, we obtain the point $\sigma (x)=(x_{1}',x_{2}',x_{3}')$ which is a zero of the equation of $Z_7$ and such that $\mathcal {C}_{0}(\sigma (x))$ is projectively equivalent to $\mathcal {C}_{0}^{\sigma }$ .

When $\sigma =\sigma _{1}$ , the automorphism $\sigma _{1}$ acts on $Z_7$ through the map in $\mathbb {P}^{3}$ given by the ring homomorphism which to $(y_{1},y_{2},y_{3},y_{4})$ associates

$$\begin{align*}\begin{array}{c} (y_{1}y_{2}^{2}y_{3}+y_{1}y_{2}y_{3}^{2}-y_{2}^{2}y_{3}^{2}-y_{2}y_{3}^{3}-y_{1}y_{2}y_{3}y_{4}-y_{2}^{2}y_{3}y_{4}+y_{2}y_{3}y_{4}^{2}+y_{3}^{2}y_{4}^{2},\\ y_{1}y_{2}^{3}+y_{1}y_{2}^{2}y_{3}+y_{2}^{2}y_{3}^{2}+y_{2}y_{3}^{3}-2y_{1}y_{2}^{2}y_{4}-y_{1}y_{2}y_{3}y_{4}-2y_{2}y_{3}^{2}y_{4}-y_{3}^{3}y_{4}+y_{1}y_{2}y_{4}^{2}+y_{3}^{2}y_{4}^{2},\\ y_{1}y_{2}^{2}y_{3}+y_{1}y_{2}y_{3}^{2}-y_{2}^{2}y_{3}^{2}-y_{2}y_{3}^{3}-y_{1}y_{2}y_{3}y_{4}+y_{2}y_{3}^{2}y_{4},\\ y_{2}^{3}y_{3}+2y_{2}^{2}y_{3}^{2}+y_{2}y_{3}^{3}-2y_{2}^{2}y_{3}y_{4}-3y_{2}y_{3}^{2}y_{4}-y_{3}^{3}y_{4}+y_{2}y_{3}y_{4}^{2}+y_{3}^{2}y_{4}^{2}). \end{array} \end{align*}$$

For $\sigma _{2}$ , we obtain that it acts on the surface $Z_7$ through the map which to $(y_{1},y_{2},y_{3},y_{4})$ associates

$$\begin{align*}\begin{array}{c} (-y_{2}^{2}y_{3}-y_{2}y_{3}^{2}+y_{2}y_{3}y_{4},-y_{1}y_{2}y_{3}+y_{2}y_{3}^{2}+y_{2}y_{3}y_{4}-y_{3}y_{4}^{2},\\ y_{1}y_{2}^{2}+y_{1}y_{2}y_{3}-y_{2}^{2}y_{3}-y_{2}y_{3}^{2}-y_{1}y_{2}y_{4}+y_{2}y_{3}y_{4},y_{2}y_{3}y_{4}-y_{3}y_{4}^{2}); \end{array} \end{align*}$$

this map is a birational transformation of $\mathbb {P}^{3}$ . In order to check that the action of is faithful on $Z_7$ , it is then enough to check that the orbit of one point (for example the point $(-6:-25/8:5:1)$ in $Z_7$ ) has $42$ elements, which is a direct computation.

The fixed points under the order seven element $\sigma _{1}$ are the singularities $s_{5},s_{7},s_{8}$ ; (there is a unique conjugacy class of elements of order $7$ in $F_{7}$ ).

The fixed points locus of $\sigma _{2}$ and the order $3$ automorphism $\sigma _{2}^{2}$ acting on $Z_7$ are:

1. 1. The $A_{3}$ singularity $(0:1:0:1)$ ,

2. 2. The four points $p=(r^{2}+1:r{}^{2}-r+2:r:1)$ where r is any complex root of $X^{4}-X^{3}+3X^{2}-X+1$ . These points are in $Z_7\setminus \mathcal {R}_{7}$ ; they are periodic of period $2$ for the rational self-map , moreover the (unlabeled) line arrangements have $7,10,$ and $37$ lines, respectively. It seems likely that the number of lines of the sequence goes to infinity.

3. 3. The points $(w+1:-w:w:1)$ where $w^{2}+w+1=0$ , which are fixed by the rational self-map ; these two points are in $Z_7\setminus \mathcal {R}_{7}$ .

The fixed-point locus of the involution $\sigma _{2}^{3}$ acting on $Z_7$ is the union of the line $L_{7}$ and a curve $E_{j}$ , which is in $Z_7\setminus \mathcal {R}_{7}$ (see §3.2). There is a unique conjugacy class of involutions in $F_{7}$ , so that similarly, any involution from fixes a curve and a line.

There is an open set in the quotient surface which parametrizes unlabeled line arrangements $\mathcal {C}_{0}^{o}$ associated to $\mathcal {C}_{0}$ in $\mathcal {R}_{7}$ . One has:

For a labeled line arrangement $\mathcal {C}_{0}=(\ell _{1},\dots ,\ell _{k})\in \mathfrak {U}_7$ and $j\in \{1,\dots ,7\}$ , let us denote by $H_{j}(\mathcal {C}_{0})$ the line arrangement $H_{j}=\sum _{k\neq j}\ell _{k}$ . The labeled line arrangement is

An element permutes the lines of $\mathcal {C}_{0}$ : it can also be seen as a permutation of $\{1,\dots ,7\}$ . The $\sigma (j)^{th}$ line of $\sigma .\mathcal {C}_{1}$ is . Since

$$\begin{align*}H_{\sigma(j)}(\mathcal{C}_{0})=\sum_{k\neq j}\ell_{\sigma(k)}=H_{j}(\sigma.\mathcal{C}_{0}), \end{align*}$$

the $\sigma (j)^{th}$ line of $\sigma .\mathcal {C}_{1}$ is . Thus

and we obtain that:

Proposition 3.6. The action of commutes with the action of , that is for all it holds that

3.5 Fibration preserved by and the elliptic modular surface $\Xi _{1}(7).$

The line $L_{6}:\,y_{2}-y_{4}=y_{3}=0$ is contained in the surface $Z_7$ . Let $\gamma :Z_{7}\to \mathbb {P}^{1}$ be the elliptic fibration induced by the projection from that line. One obtains a smooth cubic affine model A in of that elliptic fibration by substituting $(x,1+ty,y,1)$ in the equation of $Z_{7}$ . A computation shows that $Z_{7}^{s}\to \mathbb {P}^{1}$ is (isomorphic to) the elliptic surface Y associated to the elliptic curve $E_{/\mathbb {Q}(t)}$ with Weierstrass model

$$\begin{align*}E:\,y^{2}=x^{3}+\tfrac{(t^{4}-2t^{3}+3t^{2}+6t+1)}{(t+1)^{2}}x^{2}+\tfrac{8t^{3}(t^{2}-t-1)}{(t+1)^{3}}x+16\tfrac{t^{6}}{(t+1)^{4}}. \end{align*}$$

The map between A and Y sends $(0,0)$ to the zero section. The elliptic fibration $Y\to \mathbb {P}^{1}$ has singular fibers $3I_{7}+3I_{1}$ at the points

$$\begin{align*}\infty,0,-1,t^{3}-5t^{2}-8t-1=0, \end{align*}$$

respectively.

We recall that the curve $X_{1}(7)$ parametrizes (up to isomorphisms) the pairs $(E,p)$ where E is an elliptic curve and p is a torsion point of order $7$ on E. A Weierstrass model $E'$ of the elliptic modular surface $\Xi _{1}(7)$ over the curve $X_{1}(7)\simeq \mathbb {P}^{1}$ is computed in [Reference Top and Yui13]. The j-invariant maps $j_{E}(t),j_{E'}(t)\in \mathbb {Q}$ of E and $E'$ are related by the equality $j_{E'}(t)=j_{E}(-\tfrac {1}{t})$ , which shows that E is isomorphic to $E'$ and $Z_7^{s}$ is isomorphic to the elliptic modular surface $X_{1}(7)$ .

The Mordell–Weil group of E is isomorphic to $\mathbb {Z}/7\mathbb {Z}$ ; it is generated by the point

$$\begin{align*}p_{t}=(0:4t^{3}:(t+1)^{2})\in E. \end{align*}$$

We thus obtained the first part of the following theorem

The last property implies that the operator preserves the moduli interpretation of $X_{1}(7)$ .

Proof. Using the period map and the function field of A, one computes that the action of on the base $\mathbb {P}^{1}$ of the fibration $A\to \mathbb {P}^{1}$ is through the map $t\to -1/(t+1)$ .

An automorphism acts on the surface $Z_7\cap \{y_{4}\neq 0\}$ and on the affine model A. Using the period map and again the generic point of A, one computes the action of the rational self-maps () on A. For $14$ of these maps, the action on the base curve $\mathbb {P}^{1}$ is trivial. This is the case for example for

$$\begin{align*}\sigma_{0}=(1,2,4)(3,6,7)(8,9,11)(10,13,14). \end{align*}$$

The map also acts on E, one can thus compute its action on the generic point of E. Knowing that action, we are now able to compute the pull-back of a non-zero holomorphic one-form by $\mu $ , which is: . Using the seven torsion points, one computes that $\mu $ fixes the origin, thus $\mu =[2]$ .

Among the $12$ lines contained in $Z_7$ , in the complement of $\mathcal {R}_{7}$ , eight are contained in the singular fibers of the fibration $\gamma $ , and $4$ are sections.

Using the pull-back to $Z_7^{s}$ of the lines contained in $Z_7$ and the $(-2)$ -curves of the desingularization, one may compute the Néron–Severi lattice of $Z_7^{s}$ , and obtain that it has discriminant $-7$ and rank $20$ . The modular elliptic surface $\Xi _{1}(7)$ is well-known and studied; it is known as the unique K3 surface with Néron–Severi lattice of rank $20$ and discriminant $-7$ : we obtain that way another proof that $Z^s$ is isomorphic to $\Xi _1(7)$ . The inequivalent fibrations of $\Xi _{1}(7)$ have been classified (see [Reference Lecacheux6]). Another remarkable fact is that $\Xi _{1}(7)$ is a ball-quotient surface: there exists a co-compact lattice $\Gamma $ in the automorphism group of the unit ball $\mathbb {B}_{2}$ such that $\Xi _{1}(7)\simeq \mathbb {B}_{2}/\Gamma $ [Reference Naruki7]. The automorphism group of $\Xi _{1}(7)$ is studied in [Reference Ujikawa14].

3.6 The K3 surface $Z_7^{s}$ is semi-conjugated to the plane.

The rational self-map acting on the quartic $Z_7\hookrightarrow \mathbb {P}^{3}(y_{1},\dots ,y_{4})$ is defined by

where $P_{1},\dots ,P_{4}$ are four homogeneous degree $11$ polynomials computed via the period map. These polynomials are given in the ancillary file in the arXiv version of this paper; they also may be obtained from the polynomials $Q_{1},Q_{2},Q_{3}$ and R below. A remarkable fact about the polynomials $P_{1},\dots ,P_{4}$ is that

(3.6) $$ \begin{align} \deg_{y_{1}}(P_{1})=1,\,\deg_{y_{1}}(P_{2})=\deg_{y_{1}}(P_{3})=\deg_{y_{1}}(P_{4})=0, \end{align} $$

where $\deg _{y_{1}}$ denote the degree relative to the variable $y_{1}$ .

Let us define the polynomials $\tilde {P_{k}}=P_{k+1}(0,z_{1},z_{2},z_{3})$ for $k\in \{1,2,3\}$ (where $z_{1},z_{2},z_{3}$ are the three coordinates on the plane $\mathbb {P}^{2}:y_{1}=0$ ). The polynomials $\tilde {P}_{k}$ , $k\in \{1,2,3\}$ define a rational self-map $F:\mathbb {P}^{2}\dashrightarrow \mathbb {P}^{2}$ ; the base locus of the linear system generated by $\tilde {P_{1}},\tilde {P_{2}},\tilde {P_{3}}$ is the quintic curve B defined by

$$\begin{align*} Q&=z_{1}^{3}z_{2}^{2}+2z_{1}^{2}z_{2}^{3}+z_{1}z_{2}^{4}+2z_{1}^{3}z_{2}z_{3}+4z_{1}^{2}z_{2}^{2}z_{3}+2z_{1}z_{2}^{3}z_{3}+z_{1}^{3}z_{3}^{2}\\ &\quad-4z_{1}^{2}z_{2}z_{3}^{2}-9z_{1}z_{2}^{2}z_{3}^{2}-4z_{2}^{3}z_{3}^{2}-2z_{1}^{2}z_{3}^{3}+2z_{1}z_{2}z_{3}^{3}+4z_{2}^{2}z_{3}^{3}+z_{1}z_{3}^{4}. \end{align*}$$

That curve is irreducible, has geometric genus $1$ and its normalization has j-invariant $-5^{6}/28$ . By removing the base locus B, one obtains that the rational self-map F is defined by the following degree $6$ polynomials

$$ \begin{align*} Q_{1}=&z_{1}Q,\\ Q_{2}=&-z_{1}^{5}z_{2}-3z_{1}^{4}z_{2}^{2}-3z_{1}^{3}z_{2}^{3}-z_{1}^{2}z_{2}^{4}+z_{1}^{4}z_{2}z_{3}+2z_{1}^{3}z_{2}^{2}z_{3}+z_{1}^{2}z_{2}^{3}z_{3}\\ &+z_{1}^{3}z_{2}z_{3}^{2}+2z_{1}^{2}z_{2}^{2}z_{3}^{2}+z_{1}z_{2}^{3}z_{3}^{2}-z_{1}^{2}z_{2}z_{3}^{3}+z_{2}^{3}z_{3}^{3}-z_{2}^{2}z_{3}^{4},\\ Q_{3}=&2z_{1}^{4}z_{2}z_{3}+4z_{1}^{3}z_{2}^{2}z_{3}+2z_{1}^{2}z_{2}^{3}z_{3}+z_{1}^{4}z_{3}^{2}-4z_{1}^{3}z_{2}z_{3}^{2}-8z_{1}^{2}z_{2}^{2}z_{3}^{2}-3z_{1}z_{2}^{3}z_{3}^{2}\\ &-2z_{1}^{3}z_{3}^{3}+2z_{1}^{2}z_{2}z_{3}^{3}+4z_{1}z_{2}^{2}z_{3}^{3}+z_{2}^{3}z_{3}^{3}+z_{1}^{2}z_{3}^{4}, \end{align*} $$

and the indeterminacy locus of $F=(Q_{1}:Q_{2}:Q_{3})$ are the $8$ points

$$\begin{align*}q_{1}&=(0:0:1),q_{2}=(1:0:1),q_{3}=(0:1:0),q_{4}=(-1:1:0),\\q_{5}&=(1:0:0),\,q_{r}=(-r{}^{2}+2r:r:1) \end{align*}$$

where r is any root of $X^{3}-4X^{2}+3X+1$ (the field $\mathbb {Q}(r)$ is the degree $3$ real subfield of $\mathbb {Q}(\zeta _{7})$ ).

Let us define the projection map $\pi _{1}:Z_7\to \mathbb {P}^{2}(z_{1},z_{2},z_{3})$ from the point $s_{8}:y_{2}=y_{3}=y_{4}=0$ contained in surface $Z_7$ . This point is an $A_{3}$ singularity on $Z_7$ , in particular it has multiplicity $2$ , thus the map $\pi _{1}$ from the quartic to the plane has degree $2$ . One has:

The curve B has singularities of type $A_{4},A_{4},A_{2}$ at the points $q_{2},q_{4},q_{5}$ , respectively. The union $L+B$ has singularities of type $A_{1},A_{3},A_{4},A_{3},A_{4},A_{2}$ at the points $q_{0}=(0:1:1),q_{1},q_{2},q_{3},q_{4},q_{5}$ , respectively.

A direct computation shows that

$$\begin{align*}Q_{1}(Q_{1},Q_{2},Q_{3})=Q_{1}\,R^{2} \end{align*}$$

for $R=\tfrac {1}{8}z_{2}^{2}(z_{1}-z_{3})^{2}R_{4}R_{7}$ , where

$$ \begin{align*} R_{4}=&z_{1}^{4}+2z_{1}^{3}z_{2}+z_{1}^{2}z_{2}^{2}-z_{1}^{2}z_{3}^{2}-z_{1}z_{2}z_{3}^{2}-z_{2}z_{3}^{3},\\ R_{7}=&z_{1}^{6}z_{2}+4z_{1}^{5}z_{2}^{2}+6z_{1}^{4}z_{2}^{3}+4z_{1}^{3}z_{2}^{4}+z_{1}^{2}z_{2}^{5}+z_{1}^{6}z_{3}-7z_{1}^{4}z_{2}^{2}z_{3}-11z_{1}^{3}z_{2}^{3}z_{3}-6z_{1}^{2}z_{2}^{4}z_{3}-z_{1}z_{2}^{5}z_{3}\\ &-z_{1}^{5}z_{3}^{2}+3z_{1}^{3}z_{2}^{2}z_{3}^{2}+2z_{1}^{2}z_{2}^{3}z_{3}^{2}+3z_{1}^{2}z_{2}^{2}z_{3}^{3}+5z_{1}z_{2}^{3}z_{3}^{3}+2z_{2}^{4}z_{3}^{3}-2z_{1}z_{2}^{2}z_{3}^{4}-2z_{2}^{3}z_{3}^{4}-z_{1}z_{2}z_{3}^{5}. \end{align*} $$

The images of the curves $z_{2}=0,\,z_{1}-z_{3}=0$ and $R_{4}=0$ by the rational self-map $F=(Q_{1}:Q_{2}:Q_{3})$ of $\mathbb {P}^{2}$ are the indeterminacy points $q_{2}, q_{4}, q_{2}$ , respectively. The image of the curve $R_{7}=0$ under the map F is the quintic curve B. The image of the quintic curve B under the map F is the line $L:z_{1}=0$ . The rational map F preserves L and the action of F on L is through the map $(z_{2}:z_{3})\to (z_{2}-z_{3}:z_{3})$ .

From the above description and §2.3, the surface $Z_7^{s}$ is the minimal desingularization of the double cover

$$\begin{align*}X:\{y^{2}=Q_{1}(z_{1},z_{2},z_{3}\}\hookrightarrow\mathbb{P}(3,1,1,1) \end{align*}$$

branched over $L+B$ . The birational map between X and $Z_7$ is given by the equalities $y_{i+1}=z_{i}$ for $i\in \{1,2,3\}$ and

$$\begin{align*}y_{1}=\tfrac{1}{2}(y+z_{2}^{2}z_{3}+z_{2}z_{3}^{2}+z_{2}^{2}z_{4}-z_{2}z_{3}z_{4}-z_{2}z_{4}^{2})/(z_{2}^{2}+z_{2}z_{3}-z_{2}z_{4}). \end{align*}$$

We continue to denote by the rational self-map

$$\begin{align*}(y;z)\to(yR(z);F(z)). \end{align*}$$

Applying the results of §2.3, we obtain that:

4.1 The matroid $M_{8}$ constructed from the regular octagon.

Consider the $16$ lines in Figure 2: the black lines $\ell _{1},\dots ,\ell _{8}$ are the $8$ lines of the regular octagon $\mathcal {C}_{1}$ and the blue lines $\ell _{1}',\dots ,\ell _{8}'$ are the 8 lines symmetries of $\mathcal {C}_{0}$ . The image of $\mathcal {C}_{0}=\ell _{1}+\dots +\ell _{8}$ by the operator is the line arrangement $\mathcal {C}_{1}=\ell _{1}'+\dots +\ell _{8}'$ .

Figure 2

Matroid $M_8$ ( $\ell _i$ and $\ell _{i+4}$ meet at infinity at point $p_{i,i+4}$ ).

The $8$ lines $\ell _{i},\ell _{j}$ of $\mathcal {C}_{0}=(\ell _{1},\dots \ell _{8})$ meet in $28$ double points denoted by $p_{i,j}$ (some points are at infinity). The lines $\ell _{1}',\dots ,\ell _{8}'$ are the lines containing the points in sets $S_{1},\dots ,S_{8}$ which are respectively

(4.1) $$ \begin{align} \begin{array}{c} \{p_{1,8},p_{2,7},p_{3,6},p_{4,5}\},\{p_{1,7},p_{2,6},p_{3,5}\},\{p_{1,6},p_{2,5},p_{3,4},p_{7,8}\},\{p_{1,5},p_{2,4},p_{6,8}\},\\ \{p_{1,4},p_{2,3},p_{5,8},p_{6,7}\},\{p_{1,3},p_{4,8},p_{5,7}\},\{p_{1,2},p_{3,8},p_{4,7},p_{5,6}\},\{p_{2,8},p_{3,7},p_{4,6}\}. \end{array} \end{align} $$

These sets $S_{k},\,k=1,\dots ,8$ form a partition of the $28$ double points of $\mathcal {C}_{0}$ ; these $28$ points are the triple points of $\mathcal {C}_{0}\cup \mathcal {C}_{1}$ . One has the relation (as unlabeled line arrangements).

Let $M_{8}$ be the matroid associated to the incidences between the $16$ labeled lines $\ell _{1},\dots ,\ell _{8},\ell _{1}',\dots ,\ell _{8}'$ and the $28$ triple points: it is obtained from the matroid associated to the labeled line arrangement $\mathcal {C}_{0}\cup \mathcal {C}_{1}$ , but we discard all non-bases coming from the central point, so that $M_{8}$ has $16$ atoms and only $28$ non-bases. We denote by $\mathcal {R}_{8}$ the moduli space of realizations of $M_{8}$ (over $\mathbb {C})$ .

4.2 The moduli space $ \mathcal {R}_{8}$ of $M_{8.}$

A direct computation in OSCAR shows that the moduli space $\mathcal {R}_{8}$ is two-dimensional, and an open sub-set of the quartic surface $Z_{8}$ in $\mathbb {P}^{3}$ with the equation

$$\begin{align*}y_{1}y_{2}^{2}y_{3}-y_{1}^{2}y_{2}y_{4}+y_{1}y_{2}^{2}y_{4}+y_{1}^{2}y_{3}y_{4}-2y_{1}y_{2}y_{3}y_{4}-y_{1}y_{3}^{2}y_{4}+y_{1}y_{3}y_{4}^{2}-y_{2}y_{3}y_{4}^{2}+y_{3}^{2}y_{4}^{2}=0. \end{align*}$$

The surface $Z_{8}$ has singularities $A_{2},A_{2},A_{3},A_{4},A_{3},A_{1}$ at the respective points

$$\begin{align*}(1:0:0:0):(0:1:0:0):(0:0:1:0):(0:0:0:1):(1:1:1:1):(1:0:1:0). \end{align*}$$

Its minimal desingularization $Z_{8}^{s}$ is a K3 surface. The realization $\mathcal {A}(x)$ corresponding to a generic point $x=(x_{1},x_{2},x_{3})$ of $Z_{8}$ in the affine chart $\mathbb {A}^{3}=\{y_{4}\neq 0\}$ is the union $\mathcal {A}(x)=\mathcal {C}_{0}(x)\cup \mathcal {C}_{1}(x)$ , where $\mathcal {C}_{0}(x)$ is the line arrangement with eight lines with normal vectors the four vectors of the canonical basis and the following four vectors

$$\begin{align*}\begin{array}{c} (x_{1}-x_{2}:x_{1}^{2}-x_{1}x_{2}-x_{1}x_{3}+x_{1}-x_{2}+x_{3}:x_{1}-x_{2}x_{3}-x_{2}+x_{3}),\\ (x_{1}x_{2}-x_{1}x_{3}-x_{2}+x_{3}:x_{1}x_{2}^{2}-x_{1}x_{2}-x_{1}x_{3}+x_{1}-x_{2}+x_{3}:x_{1}x_{2}x_{3}-2x_{1}x_{3}+x_{1}-x_{2}+x_{3})\\ (x_{1}-1:x_{1}x_{2}-x_{2}:x_{1}-x_{2}),(1:x_{1}:x_{3}). \end{array} \end{align*}$$

Moreover, $\mathcal {C}_{1}(x)$ is the line arrangement with normal vectors

$$\begin{align*}\begin{array}{c} (x_{1}x_{2}-x_{1}x_{3}-x_{2}+x_{3}:x_{1}x_{2}^{2}-x_{1}x_{2}-x_{1}x_{3}+x_{1}-x_{2}+x_{3}:x_{1}x_{2}-x_{1}x_{3}-x_{2}^{2}+x_{2}x_{3}),\\ (x_{1}^{2}x_{2}-x_{1}^{2}x_{3}-x_{1}x_{2}^{2}+x_{1}x_{2}x_{3}-x_{1}x_{2}+x_{1}x_{3}+x_{2}^{2}-x_{2}x_{3}:x_{1}^{3}x_{2}-x_{1}^{3}x_{3}-x_{1}^{2}x_{2}^{2}+x_{1}^{2}x_{3}^{2}\\ +2x_{1}x_{2}x_{3}-x_{1}x_{2}-2x_{1}x_{3}^{2}+x_{1}x_{3}+x_{2}^{2}-2x_{2}x_{3}+x_{3}^{2}:x_{1}^{2}x_{2}x_{3}-x_{1}^{2}x_{2}-x_{1}^{2}\\ x_{3}+x_{1}^{2}+x_{1}x_{2}^{2}-2x_{1}x_{2}-x_{1}x_{3}^{2}+2x_{1}x_{3}+x_{2}^{2}-2x_{2}x_{3}+x_{3}^{2}),\\ (x_{1}-x_{2}:x_{1}-x_{2}:x_{1}-x_{2}x_{3}-x_{2}+x_{3}),(x_{3}:x_{1}x_{2}:x_{3}),(0:1:1),\\ (x_{1}-1:0:x_{1}-x_{3}),(1:x_{1}:0),(1:x_{2}:x_{3}). \end{array} \end{align*}$$

From the definition of the matroid $M_{8}$ , if $\mathcal {A}_{0}=\mathcal {C}_{0}\cup \mathcal {C}_{1}$ is a realization of $M_{8}$ and $\mathcal {C}_{0}$ (resp. $\mathcal {C}_{1}$ ) denotes its first (resp. last) eight lines then, as unlabeled line arrangements. The following operator gives a labeling to :

For a generic arrangement $L_{8}$ of eight lines, one has . The operator is constructed so that if $\mathcal {A}_{0}$ is any realization of $M_{8}$ and $\mathcal {C}_{0}$ (resp. $\mathcal {C}_{1}$ ) denotes its first (resp. last) eight lines then as labeled line arrangements (and of course if one forgets the labels).

4.3 The operator acts as a rational self-map on $ \mathcal {R}_{8.}$

A priori the line arrangement could be empty, however:

The operator acts on realizations of $M_{8}$ , sending $\mathcal {A}_{0}=\mathcal {C}_{0}\cup \mathcal {C}_{1}$ to $\mathcal {A}_{1}=\mathcal {C}_{1}\cup \mathcal {C}_{2}$ , where . It therefore acts on the moduli space $Z_{8}$ : we denote by

that action. In order to obtain the explicit polynomials defining , we remark that one may recover the coordinates $x_{1},x_{2},x_{3}$ of the line arrangement $\mathcal {A}_{0}(x)$ from the two last normal vectors $(1:x_{1}:0),(1:x_{2}:x_{3})$ of $\mathcal {C}_{1}(x)$ . Then one computes the unique line arrangement $\tilde {\mathcal {C}_{1}}\cup \tilde {\mathcal {C}_{2}}$ projectively equivalent to $\mathcal {A}_{1}(x)=\mathcal {C}_{1}(x)\cup \mathcal {C}_{2}(x)$ such that the first four normal vectors are the canonical basis. The image of x by is the point $x'=(x_{1}',x_{2}',x_{3}')$ such that the two last normal vectors of $\tilde {\mathcal {C}_{2}}$ are $(1:x_{1}':0),(1:x_{2}':x_{3}')$ (and $\tilde {\mathcal {C}_{1}}\cup \tilde {\mathcal {C}_{2}}=\mathcal {A}_{0}(x')$ ). Taking the homogenization to $\mathbb {P}^{3}$ , one obtains that the map is defined by the four degree $10$ coprime polynomials $P_{1},\dots ,P_{4}$ given in the ancillary file of the arXiv version of this paper. The base points of are

$$\begin{align*}\begin{array}{c} (-\sqrt{2}-1:\sqrt{2}+2:2\sqrt{2}+3:1),(\sqrt{2}-1:-\sqrt{2}+2:-2\sqrt{2}+3:1),\\ (i:0:1:1),(-i:0:1:1),(1:1:0:1),(0:1:1:0),(0:1:0:1). \end{array} \end{align*}$$

The line arrangements $\mathcal {C}_{0}\cup \mathcal {C}_{1}$ associated to the first two points are the regular octagon and its lines of symmetries. The line arrangements $\mathcal {C}_{0}$ associated to the third and fourth points are such that is the Ceva line arrangement with $12$ lines; it contains $\mathcal {C}_{0}$ .

Using the explicit polynomials $P_{1},\dots ,P_{4}$ , we obtain that:

4.4 The dynamical system is semi-conjugated to the plane.

The four polynomials $P_{1},\dots ,P_{4}$ such that verify $\deg _{y_{1}}(P_{1})=1$ and $\deg _{y_{k}}(P_{1})=0$ for $k\geq 2$ . Let $\pi :Z_{8}\dashrightarrow \mathbb {P}^{2}$ be the double cover obtained by projecting from the double point $(1:0:0:0)$ of $Z_{8}$ .

Lemma 4.5. The branch curve B of $\pi $ is the union of the conic $C=\{z_{1}^{2}-z_{2}z_{3}=0\}$ and the quartic curve

$$\begin{align*}Q=\{z_{1}^{2}z_{2}^{2}+2z_{1}^{2}z_{2}z_{3}-4z_{1}z_{2}^{2}z_{3}-z_{2}^{3}z_{3}+z_{1}^{2}z_{3}^{2}-4z_{1}z_{2}z_{3}^{2}+6z_{2}^{2}z_{3}^{2}-z_{2}z_{3}^{3}\}. \end{align*}$$

The quartic Q has geometric genus $0$ and is singular at the points $(1:0:0),(1:1:1)$ with singularities $A_{3}$ and $A_{1}$ . The curve $B=C+Q$ is singular at the points

$$\begin{align*}(1:0:0),(0:1:0),(0:0:1),(1:1:1) \end{align*}$$

with singularities $A_{3},A_{5},A_{5},D_{4}$ , respectively.

Let us define the polynomials $Q_{k}=P_{k+1}(0,z_{1},z_{2},z_{3})$ $(k=1,2,3$ ) and the rational self-map $\mu :\mathbb {P}^{2}\dashrightarrow \mathbb {P}^{2}$ , $\mu =(Q_{1}:Q_{2}:Q_{3})$ . One has $ \mu ^{*}(B)=B+2D $ for a degree $27$ curve D. Using §2.3, the double cover of $\mathbb {P}^{2}$ branched over $B=C+Q$ is birational to the surface $Z_{8}$ and is semi-conjugated to $(\mathbb {P}^{2},\mu )$ .

The indetermination points of $\mu $ are the $9$ points

$$\begin{align*}\begin{array}{c} (1:0:0),(0:1:0),(0:0:1),(1:1:1),(1:0:1),(1:1:0),\\ (0:1:1),(-\sqrt{2}+2:-2\sqrt{2}+3:1),(\sqrt{2}+2:2\sqrt{2}+3:1). \end{array} \end{align*}$$

The image by $\mu $ of Q is the conic C; the rational map $\mu $ restricts to the identity on C.

4.5 The K3 surface $Z_{8}$ and the modular surface $\Xi _{1}(8).$

One has:

Proof. The eight lines with equations

$$\begin{align*}\begin{array}{c} (y_{1}=y_{3}=0),(y_{1}=y_{4}=0),(y_{2}=y_{3}=0),(y_{2}=y_{4}=0),(y_{3}=y_{4}=0),\\ (y_{1}-y_{4}=y_{2}-y_{4}=0),(y_{1}-y_{3}=y_{2}-y_{4}=0),(y_{2}-y_{4}=y_{3}-y_{4}=0) \end{array} \end{align*}$$

are contained in the surface $Z_8$ . Using Magma, one can compute that their strict transforms on $Z_8^{s}$ together with the $15 (-2)$ -curves coming from the resolutions of the singularities of $Z_8$ , generate a rank $20$ lattice with discriminant $-8$ . There is no K3 surface with Picard number $20$ and discriminant $-2$ and there is a unique K3 surface with Picard number 20 and discriminant −8 (see, e.g., [Reference Schütt10]) which yields the conclusion.

Proof. The projection map from the line $y_{2}-y_{4}=y_{3}-y_{4}=0$ induces a fibration $Z_8\to \mathbb {P}^{1}$ . By evaluating the Equation of $Z_8$ at $(X,1+t(Y-1),Y,1)$ , one gets the cubic affine model

$$\begin{align*}(t-1)X^{2}-t^{2}XY^{2}+XY+(t-1)^{2}X+(t-1)Y=0, \end{align*}$$

of the generic fiber, where t is the parameter of $\mathbb {P}^{1}$ . One computes that the Weierstrass model of it is the elliptic curve

$$\begin{align*}\begin{array}{c} E:y^{2}=x^{3}+(4t^{4}-8t^{3}+4t^{2}+1)/t^{4}x^{2}+8(t-1)^{2}/t^{6}x+16(t-1)^{4}/t^{8}.\end{array} \end{align*}$$

The associated elliptic surface is a smooth model of the K3 surface $Z_8$ : it is isomorphic to $Z_8^{s}$ . One computes that the singular fibers of the fibration are $ 2I_{8}+I_{4}+I_{2}+2I_{1}, $ at the points $1,0,\infty ,1/2,t^{2}-t-1/4=0$ , respectively.

By [Reference Top and Yui13, §2.3.3], the equation of a Weierstrass model of the elliptic surface $\Xi _{1}(8)$ above the modular curve $X_{1}(8)$ is

$$\begin{align*}E':\,\,\eta^{2}=\xi^{3}+(2-s^{2})\xi^{2}+\xi, \end{align*}$$

where $s=2t^{2}/(t^{2}-1)$ . To check that $\Xi _{1}(8)$ is isomorphic to $Z_8^{s}$ , one just has to compare the two j-invariants $j(E)(t)\in \mathbb {Q}(t)$ and $j(E')(t)\in \mathbb {Q}(t)$ . We compute that $j(E)(\tfrac {1}{2}(1-\tfrac {1}{t}))=j(E')(t)$ , therefore E is isomorphic to $E'$ , and $\Xi _{1}(8)\simeq Z_8^{s}$ .

4.6 Action of

The automorphism group of $M_{8}$ is generated by the involutions

$$\begin{align*}\begin{array}{c} s_{1}=(2,4)(3,7)(6,8)(9,11)(10,14)(13,15),s_{2}=(2,6)(4,8)(9,13)(11,15),\\ s_{3}=(1,2)(3,8)(4,7)(5,6)(9,13)(10,12)(14,16). \end{array} \end{align*}$$

The group is the semi-direct product $\mathbb {Z}/8\mathbb {Z}\rtimes (\mathbb {Z}/2\mathbb {Z})^{2}$ . One computes that it acts faithfully on the K3 surface $Z_{8}$ . The map $s_{2}$ (acting on $Z_8$ ) is given in the ancillary file of the arXiv version of this paper. It is a birational involution of $\mathbb {P}^{3}$ .

The group of elements $\sigma $ commuting with the action of is isomorphic to $(\mathbb {Z}/2\mathbb {Z})^{2}$ . The involution $s=(1,5)(2,6)(3,7)(4,8)$ is the unique automorphism of such that

4.7 Periodic line arrangements.

Let us prove: