Comparison of different models of the moduli space marked cubic surfaces

E. Freitag Mathematisches Institut Universität Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg E-mail: freitag@mathi.uni-heidelberg.de home-page: http://www.rzuser.uni-heidelberg.de/∼t91/

July , 2001

1. Introduction

The moduli space of marked cubic surfaces admits several quite different descriptions. The most important are based on

1. cross ratios in the sense of Cayley,

2. Coble-polynomials,

3. Picard modular forms.

The most complete result seem to have been derived by means of the third method. The moduli space M of marked cubic surfaces is the Baily-Borel compactification \overline{B_{4}/Γ} of the quotient of the 4-ball by a certain Picard modular group, namely the principal congruence subgroup Γ of level \sqrt{−3} of the group of all matrices in U(1, 4), which belong to the ring of Eisenstein integers E. This description has been derived in [ACT1,2] extending earlier work of Griffith and Clemens. This work reduces the determination of M to that of the graded algebra of modular forms A(Γ), because one knows \overline{B_{4}/Γ} = proj(A(Γ)). The ring of modular forms has been completely determined in the papers [AF], [Fr2]. It is generated by 10 modular forms of equal weight. Hence M appears as subvariety of P^{9}(ℂ). The equations of this subvariety are known.

The second description (historically the first one) uses the cross-ratios of Cayley. The cross ratio variety has been described as an algebraic variety by Naruki [N] in 1980. As in [Na], we denote this variety by C. It is a smooth model and related to M as follows. There is a proper regular map C → M which identifies C with the blow-up of M at its 40 singular points (the cusps). Hence M ≅ Č in Naruki’s notation.

Following Cayley, Naruki uses 4 basic rational functions on C and denotes them by λ, ρ, μ, ν. They generate the function field of C (which is a rational variety). This means that the map C → M is known as soon as one has explicit expressions of λ, ρ, μ, ν as modular functions. One goal of this paper is to give these expressions.

There is a completely different approach which bases on the fact that every cubic surface is isomorphic to the blow-up of P^{2}(ℂ) at six points. Hence the configuration space of 6 points in the plane can be taken as a starting point in the description of the moduli space of marked cubic surfaces. Coble [Co] constructed 80 polynomials in cartain parameters of the configuration space which lead to an embedding of M into P^{80−1}(ℂ). Recently this embedding has been recovered by M. Yoshida and we will use here his description [Yo], which has the advantage of being short and rigorous. There are linear relations between the Coble polynomials, hence Yoshida’s model —we call it Y— should be considered as variety in P^{9}(ℂ). We will show that the isomorphism Y → proj(A(Γ)) extends to a projective transformation P^{9}(ℂ) → P^{9}(ℂ). To define this transformation we need an isomorphism of the space generated by the Coble polynomials with the space [Γ, 3] of modular forms of lowest weight, which is 3, because the weight of any non-vanishing modular form is divisible by 3. It is very remarkable that the modular forms which correspond to the Coble polynomials are infinite products in the sense of Borcherds. Actually the space [Γ, 3] contains two different kinds of Borcherds products. As in [AF] has been described in detail there exist 135 infinite products —called crosses— in [Γ, 3], such that the classical cross ratios arise as ratios of crosses. So we see that the classical cross ratios as well as the Coble polynomials can be considered as Borcherds products!

The birational isomorphisms

Č ≅ proj(A(Γ)) ≅ Y

lead to the (birational) identification C ≅ Y which also will be given explicitly.

There is another important new approach to Coble’s theory due to van Geemen [vG]. He relates directly (without using modular forms) Coble’s and Naruki’s models using a certain 10-dimensional space of rational functions in the Naruki coordinates λ, ρ, μ, ν. We will construct the explicit isomorphism between [Γ, 3] and van-Geemens 10-dimensional system.

There is a different approach of Matsumoto and Terasoma [MT] to this theory. They also parametrize the classical moduli space of marked cubic surfaces by modular forms [MT]. But their method is completely different. They use theta series whereas in [AF] Borcherds liftings are used. We will give some more comments and the end of section 4.

§2. Realizations of the Weyl group of the lattice E_{6} 3

2. Realizations of the Weyl group of the lattice E_{6}

The Weyl group W (E_{6}) enters the theory of cubic surfaces as the symmetry group of the configurations of the 27 lines. As concrete realization of the lattice E_{6} we take the subgroup E_{6} ⊂ ℝ^{8} (equipped with the standard Euclidean metric) generated by

e_{1}=( 0, −1, 1, 0, 0, 0, 0, 0)^{e_{2}=( 0, 0,−1, 1, 0, 0, 0, 0)}^{e_{3}=( 0, 0, 0,−1, 1, 0, 0, 0)}^{e_{4}=( 0, 0, 0, 0, −1, 1, 0, 0)}^{e_{5}=( 0, 0, 0, 0, 0, −1, 1, 0)} e_{6}=(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2},−\frac{1}{2},−\frac{1}{2},−\frac{1}{2},−\frac{1}{2})

This is a system of simple roots (vectors of minimal norm (a, a) = 2) with Dynkin diagram

e_{1}  e_{2}  e_{3}  e_{4}  e_{5}
•  •  •  •  •
•
e_{6}

The Weyl group is the group which is generated by reflections along the 72 roots a (the simple ones are enough). It is the subgroup of index two of the whole group of automorphisms Aut(E_{6}), which acts trivial on E′_{6}/E_{6} ≅ ℤ/3ℤ. Here E′_{6} denotes the dual lattice of E_{6}. It is generated by E_{6} and the vector

v = (0, −\frac{2}{3}, −\frac{2}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, 0).

It is known that Aut(E_{6}) is isomorphic to the orthogonal group O(5, 𝔽_{3}) with respect to the quadratic form q(x) = x_{1}^{2} − x_{2}^{2} − x_{3}^{2} − x_{4}^{2} − x_{5}^{2} on 𝔽_{3}^{5}. This can be seen as follows: The lattice 3E′_{6} is contained in E_{6}. The kernel of the natural homomorphism E_{6}/3E_{6} → E_{6}/3E′_{6} is generated by 3v, which shows

E_{6}/3E′_{6} ≅ 𝔽_{3}^{5}.

The quadratic form b(x, y) = ∑_{i=1}^{8} x_{i}y_{i} on E_{6} induces a bilinear form b̄ on V := E_{6}/3E′_{6}

b̄: V × V ⟶ 𝔽_{3} = ℤ/3ℤ, b̄(x̄, ȳ) := −b(x, y) mod 3.

(We use the minus-sign to get compatibility with notations used in [AF].) We call an element a ∈ V short if b̄(a, a) = −1 and long if b̄(a, a) = 1. The equality −2 = 1 may taken as justification for the notions short and long. A better justification comes from

An element a ∈ V is short if and only if a is the image of a root of E_{6}.

We need a concrete identification E_{6}/3E′_{6} ≅ 𝔽_{3}^{5}. For this we use the elements

f_{1}=(0, 1, 1,−1,−1, 0, 0, 0)^{f_{2}=(1, 0, 0, 0, 0, 0, 0,−1)}^{f_{3}=(0, 1,−1, 0, 0, 0, 0, 0)}^{f_{4}=(0, 0, 0, 1,−1, 0, 0, 0)} f_{5}=(0, 0, 0, 0, 0, 1,−1, 0)

They are contained in E_{6} and their images in V perform a basis. This basis is used to identify V and 𝔽_{3}^{5}. The bilinear form then is

b̄(x, y) = x_{1}y_{1} − x_{2}y_{2} − x_{3}y_{3} − x_{4}y_{4} − x_{5}y_{5}

(the same as is used in [AF].)

2.1 Remark. The images of the roots simple roots of E_{6} in 𝔽_{3}^{5} are

\begin{matrix}e_{1} : & (0, 0, 2, 0, 0)  e_{2} : (1, 0, 2, 2, 0)  e_{3} : (0, 0, 0, 2, 0) \\ e_{4} : & (1, 0, 0, 2, 2)  e_{5} : (0, 0, 0, 0, 2)  e_{6} : (1, 2, 0, 2, 0) \end{matrix}

The restriction map E_{6} → V induces a homomorphism

Aut(E_{6}) ⟶ O(5, 𝔽_{3}) := O(V, b̄).

Its is known that this is an isomorphism. The image of W (E_{6}) is a subgroup of index two O(5, 𝔽_{3})^{+} ⊂ O(5, 𝔽_{3}). This subgroup is generated by the reflections along short vectors. The intersection with the special orthogonal group defines a subgroup of index two SO(5, 𝔽_{3})^{+} ⊂ O(5, 𝔽_{3})^{+}, which is copy of the simple group of order 25 920. We need also a realization of W (E_{6}) as a Cremona group. We use the notations of Naruki [Na] (which also are used by van Geemen [vG]). Naruki considers the Coxeter group which is generated by the following 6 involutions s_{1}, …, s_{6}.

s_{2} :(λ, μ, ν, ρ) ⟼ (λ, μ^{−1}, ν, μρ)
s_{3} :(λ, μ, ν, ρ) ⟼ (λ^{−1}, μ, ν, λρ)
s_{5} :(λ, μ, ν, ρ) ⟼ (λ, μ, ν^{−1}, νρ)
s_{4} :(λ, μ, ν, ρ) ⟼ (λρ, μρ, νρ, ρ^{−1})

s_{1} : \{ \begin{matrix}λ ⟼ \frac{λμνρ^{2}(1 − λ)}{λμνρ^{2} − 1} \\ μ ⟼ \frac{(λμρ − 1)(λμνρ − 1)}{μ(λρ − 1)(λνρ − 1)} \\ μ ⟼ \frac{(λνρ − 1)(λμνρ − 1)}{ν(λρ − 1)(λμρ − 1)} \\ ρ ⟼ \frac{λρ − 1)(λμνρ^{2} − 1)}{ρ(λ − 1)(λμνρ − 1)} \end{matrix}

s_{6} = τs_{1}τ  where τ interchanges λ and ν

The relations are given by the Coxeter graph

s_{1} — s_{3} — s_{4} — s_{5} — s_{6}
|
s_{2}

This means that the group W_{Nar} generated by the s_{1}, …, s_{6} is isomorphic to W (E_{6}), the concrete isomorphism, which we choose is

W_{Nar} ⟶^{∼} W (E_{6}), \begin{matrix}s_{1} ↦ R(e_{1}) \\ s_{3} ↦ R(e_{2}) \\ s_{4} ↦ R(e_{3}) \\ s_{5} ↦ R(e_{4}) \\ s_{6} ↦ R(e_{5}) \\ s_{2} ↦ R(e_{6}) . \end{matrix}

(We denote be R(a) ∈ W (E_{6}) the reflection along a root a).

2.2 Remark. Let W_{Nar} be the group generated by the transformations s_{1}, …, s_{6} There is an isomorphism

W_{Nar} ⟶^{∼} O^{+}(5, 𝔽_{3})

which maps the transformations s_{i} to reflections along vectors in 𝔽_{3}^{5} given in the following list:

\begin{matrix}s_{1} : & (0, 0, 2, 0, 0), s_{3} : (1, 0, 2, 2, 0), s_{4} : (0, 0, 0, 2, 0), \\ s_{5} : & (1, 0, 0, 2, 2), s_{6} : (0, 0, 0, 0, 2), s_{2} : (1, 2, 0, 2, 0). \end{matrix}

The subgroup of W_{Nar} which is generated by the transformations s_{2}, s_{3}, s_{4}, s_{5} is a copy of the Weyl group W (D_{4}) of the root lattice D_{4}. We want to consider the corresponding subgroup of O(5, 𝔽_{3}^{5}).

2.3 Remark. The subgroup “ W (D_{4})” of W_{Nar}, wich is generated by the substitutions s_{2}, s_{3}, s_{4}, s_{5}, corresponds to the subgroup of O^{+}(5, 𝔽_{3}^{5}), which stabilizes (1, 2, 2, 0, 2).

3. The ten dimensional representation

It is well known that the group W (E_{6}) admits a unique (up to isomorphism) ten dimensional representation. We denote by U the representation space of some concrete realization. In the next section we will give a concrete realization, which can be used to give proofs of the statements in this section. We will omit these proofs. Its better us to work with O(5, 𝔽_{3}) instead of W (E_{6}). Hence we extend this representation to a representation of O(5, 𝔽_{3}) such that the central involution − id acts by multiplication with −1.

There are two basic configurations of elements of V = 𝔽_{3}^{5}.

3.1 Definition. 1. A cross in V = 𝔽_{3}^{5} is a set {±a_{1}, …, ±a_{5}} of five pairs of vectors which are pairwise orthogonal and such that one of them is long and the other four are short (see [AF]).

2. A nine in V = 𝔽_{3}^{5} is a set of nine pairs ±a of short vectors whose common orthogonal complement contains an non-zero isotropic vector.

It is easy to verify that every long vector is contained in exactly three crosses. As a consequence there are 3 · 45 = 135 crosses.

It is also easy to verify that there are precisely nine short elements which are orthogonal to a given non-zero isotropic element. Hence the nines are in 1-1-correspondence to the pairs ±a of isotropic elements. We see that there are precisely 40 nines.

Finally we mention that W (E_{6}) acts transitively on the 135 crosses and also on the 40 nines.

3.2 Lemma and Definition. 1. Let C be cross. The space of all elements of the 10-dimensional representation space U, which change their sign under all five reflections along the elements of C is one-dimensional. We call such a space a cross-line.

2. Let N be a nine. The space of all elements of U which change their sign under all nine reflections along the elements of N is one-dimensional. We call such a space a nine-line

Different crosses (nines) lead to different one-dimensional subspaces of U. Hence we obtain 135 cross-lines in U and 40 nine-lines in U.

Sometimes it is useful to take representatives of these lines. A convenient way to do this is to exhibit one of them and to take its W (E_{6}) orbit:

§4. Modular forms 7

3.3 Lemma. 1. Let F ∈ U be one element which lies in a cross-line. The W (E_{6})-orbit of F consists of 2 · 135 = 270 elements. These 270 elements consist of 135 pairs ±G of elements which represent the 135 cross-lines.

2. Let F ∈ U be one element which lies in a nine-line. The W (E_{6})-orbit of F consists of 2 · 40 = 80 elements. These 80 elements consist of 40 pairs ±G of elements which represent the 40 cross-lines. The set of 270 (resp 80) elements of U which we obtain in this way is uniquely determined up to a common constant factor.

We describe some of the dependencies between the 270 elements. Let a be a fixed long element of V. Recall that there are three crosses which contain a and that in the 270 of an orbit there are six elements ±A, ±B, ±C. After suitable choice of the signs one has A + B = C:

3.4 Remark. Let a ∈ 𝔽^{5} be a long element. Let A, B be two representatives of two of the three cross lines belonging to a. We assume that A, B are in the same W (E_{6})-orbit. Then either A+B ore A−B is contained in the third cross line belonging to a.

4. Modular forms

Its is known that there exists up to isomorphism a unique 10-dimensional irreducible representation of W (E_{6}). From the explicit descriptions below one can see:

4.1 Remark. Let π : W (E_{6}) → GL(U) be an irreducible 10-dimensional representation. There exists a unique two dimensional subspace of U which is invariant under W (D_{4}), which denotes the subgroup generated by the reflections along e_{2}, e_{3}, e_{4}, e_{6}.

An important realization of this representation is given by modular forms. We denote by U(1, 4) ⊂ GL(5, ℂ) the unitary group of the Hermitean form z̄_{1}w_{1} − … − z̄_{5}w_{5}. We have to consider the subgroup U(1, 4)(E) the subgroup of all elements with entries in the ring of Eisenstein integers

E = ℤ[ω], ω = −\frac{1}{2} + i\frac{\sqrt{3}}{2}.

We denote by Γ the principle congruence subgroup of level \sqrt{−3}. Because of E/\sqrt{−3}E there is an exact sequence

1 ⟶ Γ ⟶ U(1, 4)(E) ⟶ O(5, 𝔽_{3}^{5}) ⟶ 1.

We consider the space [Γ, r] of modular forms of weight r, where r is divisible by 3 (otherwise this space will vanish). It is defined as the set of all holomorphic functions

f : B̃ ⟶ ℂ, B̃ = { z ∈ ℂ^{5}, (z, z) > 0 }.

which satisfy

a) f(tz) = t^{−r}f(z),

b) f(γz) = f(z) for γ ∈ Γ.

Let α ∈ E^{5} a non-zero isotropic element, (α, α) = 0. Then the limit

f(α) := lim_{z→α} f(z)

can be defined (see [Fr1]). (This is the value of f at the “cusp” α.) Let a ∈ 𝔽_{3}^{5} one of the 80 non-zero isotropic elements. It is easy to verify that there is a primitive a ∈ E^{5} which represents α. a is unique up to a third root of unity. We assumed that r is divisible by three. Hence f(a) := f(α) is well defined. Furthermore f(−a) = (−1)^{r}f(a). Hence the values of f at the cusps are given by 40 numbers. One knows [Fr1] that the ring of all modular forms is generated by [Γ, 3]. Moreover one knows th following basic facts:

1) The space [Γ, 3] is 10-dimensional.

2) The group O(5, 3) ≅ U(1, 4)(E)/Γ acts irreducibly on this space.

3) The elements of [Γ, 3] are determined by the 40 values at the cusps.

The following list is a system of representatives of the 40 non-zero pairs ±a of isotropic vectors.

(1,1,1,1,1)  (1,1,1,1,-1) (1,1,1,-1,1) (1,1,1,-1,-1) (1,1,0,0,0)  (1,1,-1,1,1)
(1,1,-1,1,-1) (1,1,-1,-1,1) (1,1,-1,-1,-1) (1,0,1,0,0)  (1,0,0,1,0)  (1,0,0,0,1)
(1,0,0,0,-1) (1,0,0,-1,0) (1,0,-1,0,0) (1,-1,1,1,1) (1,-1,1,1,-1) (1,-1,1,-1,1)
(1,-1,1,-1,-1) (1,-1,0,0,0) (1,-1,-1,1,1) (1,-1,-1,1,-1) (1,-1,-1,-1,1) (1,-1,-1,-1,-1)
(0,1,1,1,0) (0,1,1,0,1) (0,1,1,0,-1) (0,1,1,-1,0) (0,1,0,1,1) (0,1,0,1,-1)
(0,1,0,-1,1) (0,1,0,-1,-1) (0,1,-1,1,0) (0,1,-1,0,1) (0,1,-1,0,-1) (0,1,-1,-1,0)
(0,0,1,1,1)  (0,0,1,1,-1) (0,0,1,-1,1) (0,0,1,-1,-1)

As we mentioned, a modular F ∈ [Γ, 3] is determined by its forty values at the cusps, hence by a vectors (F (1), … F (40)), where F (i) denotes the value of F at the i-th element of the above list.

We take from [AF] the following data: There is a basis

F_{0}, … F_{9} ∈ [Γ, 3]

such that the values at the cusps of F_{i} are given by the rows of the following matrix:

3 0 0 0 0 0 0 0 0 1 1 1-2-2-2 0 1 1 0-1 1 0 0-2-1-1-1-1-1-1-1 2-1-1 2 2-2-1-1 1
0 3 0 0 0 0 0 0 0 1 1-2 1-2-2 0 1 1 0-1 1 0-3 1-1-1-1-1-1-1 2-1-1 2-1 2-2-1 2-2
0 0 3 0 0 0 0 0 0 1-2 1-2 1-2 0 1 1 0-1 1-3 0 1-1-1-1-1-1 2-1-1 2-1 2-1-2 2-1-2
0 0 0 3 0 0 0 0 0 1-2-2 1 1-2 0 1 1 0-1-2 0 0 1-1-1-1-1 2-1-1-1 2 2-1-1 1-1-1-2
0 0 0 0 3 0 0 0 0 1 1 1 1 1 1 0-2-2 0-1-2 0 0-2-1-1-1-1-1-1-1-1-1-1-1-1 1-1-1 1
0 0 0 0 0 3 0 0 0-2 1 1-2-2 1 0 1 1-3-1 1 0 0 1-1-1 2 2-1-1-1 2-1-1-1-1-2 2 2 1
0 0 0 0 0 0 3 0 0-2 1-2 1-2 1 0 1-2 0-1 1 0 0 1-1 2-1 2-1-1 2-1-1-1-1-1 1-1 2 1
0 0 0 0 0 0 0 3 0-2-2 1-2 1 1 0-2 1 0-1 1 0 0 1 2-1 2-1-1 2-1-1-1-1-1-1 1 2-1 1
0 0 0 0 0 0 0 0 3-2-2-2 1 1 1 0-2-2 3-1-2 3 3-2 2 2-1-1 2-1-1-1-1-1-1-1 4-1-1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3-3-3 3 0-3 3 3-3 0 0 0 0 0 0 0 0 0 0 0 0 3-3-3 3

We consider the 10-dimensional space, which is generated by the rows of this matrix. The action of O(3, 𝔽_{3}^{5}) on this space is very easy: The group O(5, 3) permutes the 80 non-zero isotropics of 𝔽_{3}^{5}. In the same way it permutes the 80 values at the cusps. Because the 80 values are determined by the 40 representatives of the pairs ±a this gives an action at the chosen 40 values. (The values are permuted up to certain signs.)

We need the two dimensional subspace described in 4.1. By 2.3 the group W (D_{4}) corresponds to the stabilizer of ±(1, 1, 1, 0, 1).

4.2 Lemma. The two dimensional subspace of [Γ, 3], which is invariant under W (D_{4}) (in the sense of 2.3) contains the two modular forms

h := F_{0} − F_{1} + F_{4} − F_{5} − F_{9}, g := F_{0} − F_{1} − F_{2} + F_{3} − F_{5} + F_{7} − F_{9}.

The forms h, g, h − g are contained in cross lines.

1. The reflection along (1, 2, 0, 2, 0) interchanges h and g.

2. The reflection along (0, 0, 0, 2, 0) has the effect (h, g) ↦ (h − g, −g).

We also have to consider the field of modular functions K(Γ) which consists of quotients of modular forms of the same weight and which can be considered as the rational functions on M.

4.3 Definition. A cross ratio in K(Γ) is the quotient f = A/B of two elements F, G ∈ [Γ, 3] with the following properties:

1. They belong to the same W (E_{6})-orbit.

2. They are contained in different cross-lines which belong to the same long element of 𝔽_{3}^{5}.

3. A − B (and not A + B) is contained in a cross-line. The function h/g (4.2) is a cross ratio.

Using 3.3 we see that there exist precisely 6 · 45 = 270 cross ratios. Note that f^{−1} is a cross ratio if f is one. We also mention that the cross ratios constitute one orbit under W (E_{6}).

The variety M = \overline{B_{4}/Γ} contains certain divisors, which are called mirrors [AF]: A long (short) mirror is the fixed point set of a reflection along a long (short) element of V = 𝔽_{3}^{5}. So we have 45 long and 36 short mirrors. It is known that mirrors are irreducible subvarieties of codimension 1.

4.4 Proposition. 1. Let C be a cross and F ∈ [Γ, 3] a non-zero element of the corresponding cross line. Then F vanishes on the five mirrors belonging to the cross. (The multiplicities are one for the long mirror and three for the four short mirrors, if they are counted on B_{4}.) There are no other zeros.

2. Let N be a nine and F ∈ [Γ, 3] a non-zero element of the corresponding nineline. Then F vanishes on the 9 mirrors belonging to the nine. (The multiplicities are three, if they are counted on B_{4}.) There are no other zeros.

The first part of this proposition is contained in [AF]. That the forms vanish on the described mirrors is trivial. The basic result is that there are no other zeros. The proof uses the theory of Borcherds products. The details for the first case are given in [AF]. The proof of the second case goes along the same lines. We skip it.

We want to mention that a modular form F ∈ [Γ, 3] which belongs to a nine must be the third power of a modular form of weight one. But this form of weight one must have a character! Actually these forms of weight one are also in the literature and we want to give a few comments:

Borcherd’s theory treats modular forms for the group O(2, n). From the fact that the real part of Hermitean form is a symmetric form on obtains an embedding U(1, 4) ⊂ O(2, 8). This is the way how Borcherd’s theory can be applied to U(1, 4). But one can also use the imaginary part of a a Hermitean form. This is an alternating form and in this way on gets an embedding U(1, 4) ⊂ Sp(10, ℝ). Now on can use Siegel modular forms of weight 1/2 (theta constants with suitable characteristics) and restrict them to get modular forms for U(1, 4). It turns out that on gets modular forms of weight one on B_{4}. This program has been carrried through by Matsumoto and Terasoma [MT] and it is possible to identify their forms with our cube roots of “nines”. Actually Matsumoto and Terasoma identify their (restrictions of) thea series with Coble polynomials. In the next section we will identify our nines with Coble polynomials. In the way on can read off the explicit expressions. The identities

Borcherd’s additive lifts = Borcherd’s products = theta series

yield non-trivial identities of complicated expressions.

5. Van Geemen’s 10-dimensional space

Van Geemen’s work is related to the work of Naruki [Na]. The starting point is the field ℂ(λ, ρ, μ, ν). We already explained that W (E_{6}) ≅ W_{6} acts on this field as group of field automorphisms. We use the notation

f^{s}(λ, ρ, μ, ν) := f(s^{−1}(λ, ρ, μ, ν)) (s ∈ W_{Nar}).

Van Geemen modifies this representation a little bit. the result is a ℂ-linear representation of W (E_{6}). The essential ingredient is the function

C_{1} = \frac{(λ^{2}μνρ^{2} − 1)^{3}}{(λ − 1)(λρ − 1)(λνρ − 1)(λμνρ^{2} − 1)(λμνρ − 1)(λμρ − 1)}.

§5. Van Geemen’s 10-dimensional space 11

We also need

C_{6}(λ, ρ, μ, ν) := C_{1}(ν, μ, λ, ρ).

The modified action of W_{Nar} will be denoted by s(f) It is defined by

s_{1}(f) := −\frac{f^{s_{1}}}{C_{1}} , s_{2}(f) = f^{s_{2}}, …, s_{5}(f) = f^{s_{5}}, s_{6}(f) = −\frac{f^{s_{6}}}{C_{6}} .

It can be checked that this defines a linear representation. A basic result of van Geemen is:

5.1 Proposition. Let

f_{1} = \frac{ρ(λ − 1)(μ − 1)(ν − 1)(λμνρ^{2} − 1)}{λμνρ^{2}}.

The W_{Nar}-orbit of this function with respect to the modified representation s(f) generates a 10-dimensional space vG. The group W_{Nar} acts irreducibly on this space.

From the uniqueness of this representation we obtain the existence of an isomorphism

[Γ, 3] ⟶^{∼} vG,

which is invariant with respect to the actions of O(5, 𝔽_{3}^{5}) ≅ W_{Nar} (using the explicit isomorphism 2.2 of both groups). This isomorphism is unique up to a constant factor. For the description of this isomorphism we need:

5.2 Lemma. The two dimensional subspace of vG, which is invariant under the group W (D_{4}) (generated by s_{2}, s_{3}, s_{4}, s_{5}) is generated by the two functions

\begin{aligned}f_{1} & = \frac{(λρ − 1)(μρ − 1)(νρ − 1)(λμνρ − 1)}{λμνρ^{2}}, \\ g_{1} & = \frac{(ρ − 1)(λμρ − 1)(λνρ − 1)(μνρ − 1)}{λμνρ^{2}}. \end{aligned}

1. The transformation s_{2} interchanges both.

2. The transformation s_{4} has the effect (f_{1}, g_{1}) ↦ (f_{1} − g_{1}, −g_{1}). The quotient f_{1}/g_{1} is a cross ratio.

Comparing this lemma with lemma 4.2, we obtain

5.3 Lemma. There is an equivariant isomorphism

[Γ, 3] ⟶^{∼} vG

with the property

F_{0} − F_{1} + F_{4} − F_{5} − F_{9} ⟼ \frac{(λρ − 1)(μρ − 1)(νρ − 1)(λμνρ − 1)}{λμνρ^{2}}.

Repeated application of reflections gives the complete list for all cross lines and one can deduce the expressions of the modular forms F_{0}, …, F_{9} as elements of vG. In the following list each polynomial has to be divided by 3λμνρ^{2}.

F_{0}: 3 − λνρ + νμ^{2}λρ^{2} − λρ − 2μρ − νρ − 2λμνρ + λ^{2}μνρ^{2} + ν^{2}μλρ^{2} − ν^{2}μ^{2}ρ^{3}λ + μ^{2}ρ^{3}λ^{2}ν^{2} − 2μ^{2}ρ^{4}λ^{2}ν^{2} + 2νμ^{2}λρ^{3} − μ^{2}λ^{2}ρ^{3}ν + λρ^{3}ν^{2}μ + λ^{2}ρ^{3}μν

F_{1}: −λνρ − 2νμ^{2}λρ^{2} + 2λρ + μρ − νρ + λμνρ − 2λ^{2}μνρ^{2} + ν^{2}μλρ^{2} − ν^{2}μ^{2}ρ^{3}λ − 3μλρ^{2} + μ^{2}ρ^{3}λ^{2}ν^{2} − 2μ^{2}ρ^{4}λ^{2}ν^{2} + 2νμ^{2}λρ^{3} + 2μ^{2}λ^{2}ρ^{3}ν + λρ^{3}ν^{2}μ + λ^{2}ρ^{3}μν

F_{2}: 3ρ − λνρ + νμ^{2}λρ^{2} − λρ − 2μρ − νρ + λμνρ + λ^{2}μνρ^{2} + ν^{2}μλρ^{2} − ν^{2}μ^{2}ρ^{3}λ − 2μ^{2}ρ^{3}λ^{2}ν^{2} + μ^{2}ρ^{4}λ^{2}ν^{2} + 2νμ^{2}λρ^{3} − 3λμνρ^{3} − μ^{2}λ^{2}ρ^{3}ν + λρ^{3}ν^{2}μ + λ^{2}ρ^{3}μν

F_{3}: νμ^{2}λρ^{2}+λμνρ−νμ^{2}λρ^{3}−λνρ+2λρ+μρ−νρ+λ^{2}μνρ^{2}+ν^{2}μλρ^{2}−ν^{2}μ^{2}ρ^{3}λ− 2μ^{2}ρ^{3}λ^{2}ν^{2} + μ^{2}ρ^{4}λ^{2}ν^{2} + 2μ^{2}λ^{2}ρ^{3}ν + λρ^{3}ν^{2}μ − 2λ^{2}ρ^{3}μν − 3λμρ

F_{4}: −νρ+μ^{2}ρ^{4}λ^{2}ν^{2} −3λ^{2}ν^{2}ρ^{3}μ−ν^{2}μ^{2}ρ^{3}λ+ν^{2}μλρ^{2} −νμ^{2}λρ^{3} +2λνρ+νμ^{2}λρ^{2} − 2λμνρ − λρ + μρ + λ^{2}μνρ^{2} + μ^{2}ρ^{3}λ^{2}ν^{2} − μ^{2}λ^{2}ρ^{3}ν + λρ^{3}ν^{2}μ + λ^{2}ρ^{3}μν

F_{5}: −3μνρ^{2} − λνρ − 2νμ^{2}λρ^{2} − λρ + μρ + 2νρ + λμνρ + λ^{2}μνρ^{2} − 2ν^{2}μλρ^{2} + 2ν^{2}μ^{2}ρ^{3}λ+μ^{2}ρ^{3}λ^{2}ν^{2} − 2μ^{2}ρ^{4}λ^{2}ν^{2} +2νμ^{2}λρ^{3} −μ^{2}λ^{2}ρ^{3}ν +λρ^{3}ν^{2}μ+λ^{2}ρ^{3}μν

F_{6}: 2λνρ − 2νμ^{2}λρ^{2} − λρ + μρ − νρ + λμνρ − 2λ^{2}μνρ^{2} − 2ν^{2}μλρ^{2} + 2ν^{2}μ^{2}ρ^{3}λ + μ^{2}ρ^{3}λ^{2}ν^{2} − 2μ^{2}ρ^{4}λ^{2}ν^{2} − νμ^{2}λρ^{3} + 2μ^{2}λ^{2}ρ^{3}ν + λρ^{3}ν^{2}μ + λ^{2}ρ^{3}μν

F_{7}: νμ^{2}λρ^{2} +λμνρ−νμ^{2}λρ^{3} −λνρ−3νμρ−λρ+μρ+2νρ+λ^{2}μνρ^{2} +ν^{2}μλρ^{2} + 2ν^{2}μ^{2}ρ^{3}λ − 2μ^{2}ρ^{3}λ^{2}ν^{2} + μ^{2}ρ^{4}λ^{2}ν^{2} − μ^{2}λ^{2}ρ^{3}ν − 2λρ^{3}ν^{2}μ + λ^{2}ρ^{3}μν

F_{8}: 2λνρ + νμ^{2}λρ^{2} − λρ + μρ − νρ − 2λμνρ + λ^{2}μνρ^{2} + ν^{2}μλρ^{2} + 2ν^{2}μ^{2}ρ^{3}λ − 2μ^{2}ρ^{3}λ^{2}ν^{2} +μ^{2}ρ^{4}λ^{2}ν^{2} −4νμ^{2}λρ^{3} +3λμνρ^{3} +2μ^{2}λ^{2}ρ^{3}ν −2λρ^{3}ν^{2}μ−2λ^{2}ρ^{3}μν

F_{9}: 3νμ^{2}λρ^{2} + 3λμνρ^{3} − 3λμνρ − 3νλρ^{2} − 3νμ^{2}λρ^{3} + 3λνρ

The modular forms F_{0}, …, F_{9} satisfy certain cubic relations. One of them is

\begin{matrix}(F_{1} − F_{2}) & (F_{0} − F_{3} − F_{5} + F_{8} − F_{9})(F_{4} + F_{6} + F_{7}) \\ = (F_{1} − F_{2} − F_{6} + F_{7})(F_{0} − F_{3})(F_{4} + F_{5} + F_{8} − F_{9}). \end{matrix}

By means of the above table for the F_{i} one verifies that the corresponding rational functions in vG satisfy the same relation. In [Fr2] it is shown that the O(5, 𝔽_{3})-orbit of this relation generates the whole ideal of relations and moreover that the F_{i} generate the full ring of modular forms. This can be expressed in a formal way as follows. We denote by vG[n] the vector space of all rational functions which can be expressed as polynomials of degree n in the elements of vG, especially vG[0] = ℂ, vG[1] = vG. We consider the graded algebra which is defined by

A[vG] := ⨁_{n=0}^{∞} vG[n].

We obtain a surjective homomorphism

A(Γ) ⟶ A[vG].

A dimension argument shows that this is an isomorphism.

5.4 Theorem. The isomorphism 5.3 extends to an isomorphism of rings

A(Γ) ⟶^{∼} A[vG].

This isomorphism is equivariant with respect to O^{+}(5, 𝔽_{3}) ≅ W_{Nar}.

The isomorphism 6.7 induces an isomorphism of the field K(Γ) of modular functions (=quotients of modular forms of equal weight) and the field of rational functions in λ, μ, ν, ρ,

K(Γ) ⟶ ℂ(λ, μ, ν, ρ).

This isomorphism can be considered as a birational map

M ⟶ Č.

On clearly expects that this is the biholomorphic biholomorphic map which comes from the fact that both spaces represent the standard GIT-model of marked cubic surfaces. We refer to [AF] for some details. But this needs a proof. At the present state we only know that there is an automorphism of ℂ(λ, μ, ν, ρ) which intertwines the two isomorphisms K(Γ) → ℂ(λ, μ, ν, ρ). We know that this automorphism is W_{Nar}-equivariant and we also know that this automorphism permutes the cross ratios. Here one has to use the basic result of [AF] that cross ratios in the sense of 4.3 are the same as the classical cross ratios of Cayley (theorem 7.2 in [AF]). The claimed equality of the two isomorphisms hence follows from

5.5 Lemma. Let σ be an automorphism of the function field ℂ(λ, μ, ν, ρ) which commutes with all elements from W_{Nar} and which has the further property that the image of a cross ratio is a constant multiple of may be another cross ratio. Then σ is the identity.

Proof. Because σ commutes with W_{Nar} the W (D_{4}) invariant part has been mapped into itself. From this one easily derives that a cross ratio has been mapped to a constant multiple of the same cross ratio. The variables λ, …, ρ can be written as products of cross ratios. Hence σ acts on the variables be multiplication with scalars. Looking at the explicit expressions of the cross ratios as rational functions on sees that the four scalars must be 1. ⊔ ⊓

We also mention that we have no a rigorous proof for a result which is less more contained in [vG]:

5.6 Remark. The identity on ℂ(λ, μ, ν, ρ) induces a biholomorphic map

proj(ℂ[vG]) ⟶^{∼} Č

between van Geemen’s model and Naruki’s Č.

It is easy now to express the Naruki-coordinates as modular functions: Applying group elements to the formula in 5.3 and taking quotients, we can obtain all the 270 cross ratios and the corresponding modular functions. As Naruki observed, every of the four coordinates λ, …, ρ can be written as the product of two cross ratios, [Na], (5.5). After some calculation one gets:

5.7 Theorem. The for modular functions

\begin{aligned}λ & = −\frac{(F_{0} + F_{4} + F_{7})(F_{1} − F_{3} + F_{9})}{(−F_{0} + F_{2} + F_{5} − F_{7})(F_{1} + F_{5} + F_{7} + F_{9})} \\ μ & = \frac{(F_{1} − F_{3} + F_{5} − F_{6} − F_{7} + F_{8})(F_{0} − F_{6})}{(F_{1} − F_{4} + F_{5} − F_{6} + F_{9})(F_{0} − F_{2} − F_{6} + F_{8} − F_{9})} \\ ν & = −\frac{(F_{0} + F_{7} + F_{8} − F_{9})(−F_{4} + F_{5})}{(−F_{0} + F_{1} + F_{9})(F_{1} + F_{2} − F_{3} + F_{5} + F_{8})} \\ ρ & = \frac{(F_{1} + F_{5} + F_{7} + F_{9})(F_{2} + F_{4} + F_{5})}{(F_{3} + F_{5} + F_{7})(F_{0} + F_{4} + F_{7})} \end{aligned}

generate the field of modular functions. Under the isomorphism induced by 5.6 they correspond to the Naruki coordinates

It is worth while to mention that we now have explicit formulae to construct an equation of a cubic surface with given peroids z ∈ B_{4}. For given z one computes the values of the modular forms F_{i}(z) and then λ, …, ρ using 5.7. A cubic surface with periods z then is defined by the cubic polynomial in the variables X, Y, W, Z (see [Na] (5.1)):

\begin{matrix}ρW [λX^{2} + μY^{2} + νZ^{2} + (ϱ − 1)^{2}(λμνρ − 1)^{2}W^{2} + (μν + 1)Y Z + (λν + 1)XZ \\ +(λμ + 1)XY − (ρ − 1)((λμνρ − 1)W \{(λ + 1)X + (μ + 1)Y + (ν + 1)Z\}] + XY Z. \end{matrix}

§6. Coble-Yoshida polynomials 15

6. Coble-Yoshida polynomials

The Coble polynomials are contained in the fundamental paper [Co] of Coble which is rather hard to read nowadays. These polynomials have been recovered by M. Yoshida [Yo] and we will use his approach:

The starting point is that any non-singular cubic surface is isomorphic to the blow-up of a projective plane P^{2}(ℂ) at six points, which must be in general position (no three of them are collinear and the six points are not on a conic). The six points are assumed to be ordered. Then the associated cubic surface inherits a natural marking. The group GL(3) acts on P^{2}(ℂ) and hence on the set of ordered 6-tuples of points (in general position). A system of representatives is given by

[1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1], [1, x_{1}, x_{3}], [1, x_{2}, x_{4}].

These six points are in general position, if a certain concrete discriminant polynomial in the x_{1}, …, x_{4} vanishes [Yo]. The open dense subset M ⊂ ℂ^{4} which is obtained in this way is a model of the moduli space of non-singular marked cubic surfaces. As we know, the group W (E_{6}) must act on M. Yoshida gives generators of this representation of W (E_{6}), namely the following 6 transformations which we denote here by the letter t (and not s as in Yoshidas paper, because we used already the letter s for Naruki’s transformations).

\begin{matrix}t_{1} : & (x_{1}, x_{2}, x_{3}, x_{4}) ⟼ (\frac{1}{x_{1}} , \frac{1}{x_{2}} , \frac{x_{3}}{x_{2}} , \frac{x_{4}}{x_{2}} ) \\ t_{2} : & (x_{1}, x_{2}, x_{3}, x_{4}) ⟼ (x_{3}, x_{4}, x_{1}, x_{2}) \\ t_{3} : & (x_{1}, x_{2}, x_{3}, x_{4}) ⟼ (\frac{x_{1} − x_{3}}{1 − x_{3}} , \frac{x_{2} − x_{4}}{1 − x_{4}} , \frac{x_{3}}{x_{3} − 1}, \frac{x_{4}}{x_{4} − 1}) \\ t_{4} : & (x_{1}, x_{2}, x_{3}, x_{4}) ⟼ (\frac{1}{x_{1}} , \frac{x_{2}}{x_{1}} , \frac{1}{x_{3}} , \frac{x_{4}}{x_{3}} ) \\ t_{5} : & (x_{1}, x_{2}, x_{3}, x_{4}) ⟼ (x_{1}, x_{1}, x_{4}, x_{3}) \\ t_{6} : & (x_{1}, x_{2}, x_{3}, x_{4}) ⟼ (\frac{1}{x_{1}} , \frac{1}{x_{2}} , \frac{1}{x_{3}} , \frac{1}{x_{4}} ) \end{matrix}

We denote the group, which is generated by this six relations by W_{Yos}. The Coxeter graph is

t_{1} — t_{2} — t_{3} — t_{4} — t_{5}
|
t_{6}

Yoshida defines a certain linear representation of W_{Yos} on the space of all rational functions in x_{1}, …, x_{4}:

6.1 Remark. There is a linear representation of W_{Yos} on the vector space of all rational functions in x_{1}, …, x_{4}, which is given by

t(f)(x) := −c(t)f(t^{−1}(x)) (t ∈ W_{Yos})

where c(t) is a factor which defined on the generators t_{1}, …, t_{6} as

(x_{1}x_{2})^{3}, 1, (1 − x_{3})^{3}(1 − x_{4})^{4}, (x_{1}x_{2})^{3}, 1, (x_{1}x_{2}x_{3}x_{4})^{2}.

One has to verify that this definition is compatible with the defining relations. ⊔ ⊓

6.2 Proposition. There are 80 polynomials in the variables x_{1}, …, x_{4} which are characterized by the following properties:

1. The are transitively permuted under the action 6.1.

2. One of them is

(x_{2} − x_{1})(x_{3} − x_{1})(x_{4} − x_{2})(x_{4} − x_{3}).

3. They generate a 10-dimensional space CY with irreducible action of W_{Yos}.

4. If f is one of the 80 polynomials then also −f, hence we have essentially only 40 polynomials.

It is easy to compute the full list of all these polynomials. We will not reproduce it here (see [Yo]). These polynomials have already been detected 1917 by Coble [Co] 331–372. We will call these polynomials Coble-Yoshida polynomials.

We have to identify W_{Yos} with the Weyl group W (E_{6}) and its realizations introduced in section 2. We use the concrete isomorphisms:

6.3 Remark. There is a isomorphism W (E_{6}) ≅ W_{Yos} given on generators by

R(e_{j}) ↔ t_{j} (1 ≤ j ≤ 6).

The induced isomorphism W_{Nar} ≅ W_{Yos} is

s_{1} ↔ t_{1}, s_{3} ↔ t_{2}, s_{4} ↔ t_{3}, s_{5} ↔ t_{4}, s_{6} ↔ t_{5}, s_{1} ↔ t_{6}.

There is an induced isomorphism W_{Yos} ≅ O^{+}(5, 𝔽_{3}) (use 2.2).

Our gaol is to describe an equivariant isomorphism [Γ, 3] → Yo. Such an isomorphism is unique up to a constant factor. It will turn out that the Coble-Yoshida polynomials are contained in nine-lines in the sense of 3.2. Therefore we need an explicit description for the nine-lines. The description of the ninelines is extremely easy if we use the realization (O^{+}(5, 𝔽_{3}), [Γ, 3]) of the ten dimensional representation. Recall that a modular form F is determined by the values F (α) at the non-zero isotropic elements α ∈ 𝔽_{3}^{5}.

The following observation is due to D. Allcock:

6.4 Lemma. Let α ∈ 𝔽_{3}^{5} be a non-zero isotropic vector. Assume that another three short vectors α_{1}, α_{2}, α_{3} have been chosen such that α, α_{1}, α_{2}, α_{2} are pairwise orthogonal. Then there exists a unique modular form F ∈ [Γ, 3] whose values at the non-zero isotropic elements β are given by

F (β) := \{ \begin{matrix}0 & if (β, α) ≠ 0 \\ (β, α_{1})(β, α_{2})(β, α_{3}) & else. \end{matrix}

(Here we use the embedding 𝔽_{3} ↪ ℤ, which maps 0_{𝔽_{3}}, 1_{𝔽_{3}} and −1_{𝔽_{3}} to the integers 0, 1, −1.) This modular form is contained in the cross line corresponding to α.

It is sufficient to verify this for one α, which can be done by a direct computation. As an example we consider α = (1, 2, 0, 0, 0). We take

α_{1} = (0, 0, 1, 0, 0), α_{2} = (0, 0, 0, 1, 0), α_{3} = (0, 0, 0, 0, 1)

Comparing with the list above we see:

6.5 Propsition. The modular form F_{9} is contained in the nine line which corresponds to the non-zero isotropic vector (1, 2, 0, 0, 0).

We determine the corresponding line in the Yoshida representation. In the table 2.1 three of the 6 short vectors are orthogonal to (1, 2, 0, 0, 0). The belong to e_{1}, e_{3}, e_{5} and e_{6}. The associated reflections change the sign of F_{9}. The corresponding reflections in W_{Yos} are t_{1}, t_{3}, t_{5}, t_{6}. hence the Coble-Yoshida polynomial which corresponds to F_{9} must change its sign under these 4 transformations. A computation shows that up to a constant factor only one polynomial with this property exist, namely (−x_{4}+x_{3})(−1+x_{1})(x_{2}−1)(x_{4}x_{1}− x_{3}x_{2}). This gives

6.6 Proposition. Up to a constant factor there is a unique isomorphism [Γ, 3] ≅ Yo, such that the modular form F_{9} corresponds the Coble-Yoshida polynomial

3(−x_{4} + x_{3})(−1 + x_{1})(x_{2} − 1)(x_{4}x_{1} − x_{3}x_{2}).

Now it is easy to derive the expressions of the basic modular forms:

F_{0}: −2x_{1}x_{2}x_{3}x_{4} − x_{1}x_{3}x_{4}^{2} + x_{1}x_{2}x_{4}^{2} + x_{2}^{2}x_{3}x_{4} − 2x_{3}^{2}x_{2}^{2} + x_{3}^{2}x_{2} + x_{3}x_{2}^{2} + x_{4}x_{3}^{2}x_{2} + x_{1}x_{2}x_{3}x_{4}^{2} − x_{3}^{2}x_{2}x_{1}x_{4} + x_{1}^{2}x_{2}x_{3}x_{4} + x_{4}^{2}x_{1}^{2} + x_{1}x_{2}^{2}x_{4}x_{3} − 2x_{1}^{2}x_{2}x_{4}^{2} + x_{1}x_{3}^{2}x_{2} + x_{1}x_{3}x_{4} − x_{1}x_{2}x_{3} − 2x_{2}x_{3}x_{4} + x_{1}x_{2}x_{4} − x_{1}^{2}x_{2}x_{3} + 2x_{2}x_{4}x_{1}^{2} − x_{4}x_{1}^{2} + x_{1}x_{2}^{2}x_{3} − 2x_{4}x_{1}x_{2}^{2}

F_{1}: 4x_{1}x_{2}x_{3}x_{4} − x_{1}x_{3}x_{4}^{2} − 2x_{1}x_{2}x_{4}^{2} + x_{2}^{2}x_{3}x_{4} − 2x_{3}^{2}x_{2}^{2} + x_{3}^{2}x_{2} + x_{3}x_{2}^{2} + x_{4}x_{3}^{2}x_{2} + x_{1}x_{2}x_{3}x_{4}^{2} −x_{3}^{2}x_{2}x_{1}x_{4} −2x_{1}^{2}x_{2}x_{3}x_{4} +x_{4}^{2}x_{1}^{2} −2x_{1}x_{2}^{2}x_{4}x_{3} +x_{1}^{2}x_{2}x_{4}^{2} +3x_{1}x_{2}^{2}x_{3}^{2} − 2x_{1}x_{3}^{2}x_{2} +x_{1}x_{3}x_{4} −x_{1}x_{2}x_{3} −2x_{2}x_{3}x_{4} +x_{1}x_{2}x_{4} +2x_{1}^{2}x_{2}x_{3} −x_{2}x_{4}x_{1}^{2} −x_{4}x_{1}^{2} − 2x_{1}x_{2}^{2}x_{3} + x_{4}x_{1}x_{2}^{2}

F_{2}: −2x_{1}x_{2}x_{3}x_{4} +2x_{1}x_{3}x_{4}^{2} +x_{1}x_{2}x_{4}^{2} +x_{2}^{2}x_{3}x_{4} −2x_{3}^{2}x_{2}^{2} +x_{3}^{2}x_{2} +x_{3}x_{2}^{2} +x_{4}x_{3}^{2}x_{2} − 2x_{1}x_{2}x_{3}x_{4}^{2}+2x_{3}^{2}x_{2}x_{1}x_{4}−2x_{1}^{2}x_{2}x_{3}x_{4}−2x_{4}^{2}x_{1}^{2}+x_{1}x_{2}^{2}x_{4}x_{3}+x_{1}^{2}x_{2}x_{4}^{2}+3x_{1}^{2}x_{3}x_{4}+ x_{1}x_{3}^{2}x_{2}+x_{1}x_{3}x_{4}−x_{1}x_{2}x_{3}−2x_{2}x_{3}x_{4}+x_{1}x_{2}x_{4}−x_{1}^{2}x_{2}x_{3}−3x_{1}x_{3}^{2}x_{4}+2x_{2}x_{4}x_{1}^{2}− x_{4}x_{1}^{2} + x_{1}x_{2}^{2}x_{3} − 2x_{4}x_{1}x_{2}^{2}

F_{3}: 4x_{1}x_{2}x_{3}x_{4} − x_{1}x_{3}x_{4}^{2} − 2x_{1}x_{2}x_{4}^{2} − 2x_{2}^{2}x_{3}x_{4} + x_{3}^{2}x_{2}^{2} + x_{3}^{2}x_{2} + x_{3}x_{2}^{2} − 2x_{4}x_{3}^{2}x_{2} − 2x_{1}x_{2}x_{3}x_{4}^{2}+3x_{2}x_{3}x_{4}^{2}+2x_{3}^{2}x_{2}x_{1}x_{4}−2x_{1}^{2}x_{2}x_{3}x_{4}+x_{4}^{2}x_{1}^{2}+x_{1}x_{2}^{2}x_{4}x_{3}+x_{1}^{2}x_{2}x_{4}^{2}− 2x_{1}x_{3}^{2}x_{2} +x_{1}x_{3}x_{4} −x_{1}x_{2}x_{3} −2x_{2}x_{3}x_{4} +x_{1}x_{2}x_{4} +2x_{1}^{2}x_{2}x_{3} −x_{2}x_{4}x_{1}^{2} −x_{4}x_{1}^{2} − 2x_{1}x_{2}^{2}x_{3} + x_{4}x_{1}x_{2}^{2}

F_{4}: −2x_{1}x_{2}x_{3}x_{4} −x_{1}x_{3}x_{4}^{2} +x_{1}x_{2}x_{4}^{2} −2x_{2}^{2}x_{3}x_{4} +x_{3}^{2}x_{2}^{2} −2x_{3}^{2}x_{2} +x_{3}x_{2}^{2} +x_{4}x_{3}^{2}x_{2} + x_{1}x_{2}x_{3}x_{4}^{2} − x_{3}^{2}x_{2}x_{1}x_{4} + x_{1}^{2}x_{2}x_{3}x_{4} + x_{4}^{2}x_{1}^{2} + x_{1}x_{2}^{2}x_{4}x_{3} − 2x_{1}^{2}x_{2}x_{4}^{2} + x_{1}x_{3}^{2}x_{2} + x_{1}x_{3}x_{4}+2x_{1}x_{2}x_{3}+x_{2}x_{3}x_{4}−2x_{1}x_{2}x_{4}−x_{1}^{2}x_{2}x_{3}+2x_{2}x_{4}x_{1}^{2}−x_{4}x_{1}^{2}−2x_{1}x_{2}^{2}x_{3}+ x_{4}x_{1}x_{2}^{2}

F_{5}: 4x_{1}x_{2}x_{3}x_{4} − x_{1}x_{3}x_{4}^{2} − 2x_{1}x_{2}x_{4}^{2} + x_{2}^{2}x_{3}x_{4} − 2x_{3}^{2}x_{2}^{2} + x_{3}^{2}x_{2} + x_{3}x_{2}^{2} + x_{4}x_{3}^{2}x_{2} + x_{1}x_{2}x_{3}x_{4}^{2} + 3x_{4}^{2}x_{1} − x_{3}^{2}x_{2}x_{1}x_{4} + x_{1}^{2}x_{2}x_{3}x_{4} − 2x_{4}^{2}x_{1}^{2} − 2x_{1}x_{2}^{2}x_{4}x_{3} + x_{1}^{2}x_{2}x_{4}^{2} + x_{1}x_{3}^{2}x_{2}−2x_{1}x_{3}x_{4}−x_{1}x_{2}x_{3}−2x_{2}x_{3}x_{4}−2x_{1}x_{2}x_{4}−x_{1}^{2}x_{2}x_{3}−x_{2}x_{4}x_{1}^{2}+2x_{4}x_{1}^{2}+ x_{1}x_{2}^{2}x_{3} + x_{4}x_{1}x_{2}^{2}

F_{6}: −2x_{1}x_{2}x_{3}x_{4} −x_{1}x_{3}x_{4}^{2} −2x_{1}x_{2}x_{4}^{2} +x_{2}^{2}x_{3}x_{4} +x_{3}^{2}x_{2}^{2} −2x_{3}^{2}x_{2} −2x_{3}x_{2}^{2} +x_{4}x_{3}^{2}x_{2} + x_{1}x_{2}x_{3}x_{4}^{2} − x_{3}^{2}x_{2}x_{1}x_{4} + x_{1}^{2}x_{2}x_{3}x_{4} + x_{4}^{2}x_{1}^{2} − 2x_{1}x_{2}^{2}x_{4}x_{3} + x_{1}^{2}x_{2}x_{4}^{2} + x_{1}x_{3}^{2}x_{2} + x_{1}x_{3}x_{4} + 2x_{1}x_{2}x_{3} + x_{2}x_{3}x_{4} + x_{1}x_{2}x_{4} − x_{1}^{2}x_{2}x_{3} − x_{2}x_{4}x_{1}^{2} − x_{4}x_{1}^{2} + x_{1}x_{2}^{2}x_{3} + x_{4}x_{1}x_{2}^{2}

F_{7}: 4x_{1}x_{2}x_{3}x_{4} +2x_{1}x_{3}x_{4}^{2} +x_{1}x_{2}x_{4}^{2} −2x_{2}^{2}x_{3}x_{4} −2x_{3}^{2}x_{2}^{2} +x_{3}^{2}x_{2} +x_{3}x_{2}^{2} − 2x_{4}x_{3}^{2}x_{2} + 3x_{4}x_{3}^{2}x_{2}^{2}−2x_{1}x_{2}x_{3}x_{4}^{2}−x_{3}^{2}x_{2}x_{1}x_{4}+x_{1}^{2}x_{2}x_{3}x_{4}−2x_{4}^{2}x_{1}^{2}−2x_{1}x_{2}^{2}x_{4}x_{3}+x_{1}^{2}x_{2}x_{4}^{2}+ x_{1}x_{3}^{2}x_{2} −2x_{1}x_{3}x_{4} −x_{1}x_{2}x_{3} +x_{2}x_{3}x_{4} −2x_{1}x_{2}x_{4} −x_{1}^{2}x_{2}x_{3} −x_{2}x_{4}x_{1}^{2} +2x_{4}x_{1}^{2} + x_{1}x_{2}^{2}x_{3} + x_{4}x_{1}x_{2}^{2}

F_{8}: −8x_{1}x_{2}x_{3}x_{4} + 2x_{1}x_{3}x_{4}^{2} + x_{1}x_{2}x_{4}^{2} − 2x_{2}^{2}x_{3}x_{4} + 4x_{3}^{2}x_{2}^{2} − 2x_{3}^{2}x_{2} − 2x_{3}x_{2}^{2} − 2x_{4}x_{3}^{2}x_{2} + x_{1}x_{2}x_{3}x_{4}^{2} − 3x_{4}^{2}x_{1} + 2x_{3}^{2}x_{2}x_{1}x_{4} + 4x_{1}^{2}x_{2}x_{3}x_{4} − 3x_{4}^{2}x_{1}^{2}x_{3} + 4x_{4}^{2}x_{1}^{2} + x_{1}x_{2}^{2}x_{4}x_{3} − 2x_{1}^{2}x_{2}x_{4}^{2} − 3x_{1}x_{2}^{2}x_{3}^{2} + x_{1}x_{3}^{2}x_{2} + x_{1}x_{3}x_{4} + 2x_{1}x_{2}x_{3} + 4x_{2}x_{3}x_{4} + x_{1}x_{2}x_{4} − x_{1}^{2}x_{2}x_{3} − x_{2}x_{4}x_{1}^{2} − x_{4}x_{1}^{2} + x_{1}x_{2}^{2}x_{3} + x_{4}x_{1}x_{2}^{2}

F_{9}: 3x_{1}x_{2}x_{4}^{2} − 3x_{2}^{2}x_{3}x_{4} − 3x_{4}^{2}x_{1} + 3x_{2}x_{3}x_{4} − 3x_{1}^{2}x_{2}x_{4}^{2} + 3x_{1}x_{2}^{2}x_{4}x_{3} + 3x_{4}^{2}x_{1}^{2} − 6x_{1}x_{2}x_{3}x_{4} + 3x_{3}^{2}x_{2}^{2} + 3x_{1}x_{3}x_{4} − 3x_{3}^{2}x_{2} + 3x_{1}^{2}x_{2}x_{3}x_{4} − 3x_{1}x_{2}^{2}x_{3}^{2} − 3x_{1}^{2}x_{3}x_{4} + 3x_{1}x_{3}^{2}x_{2}

Similarly to A[vG] we define the algebra

A[Yo] := ⨁_{n=0}^{∞} Yo[n].

§7. Table of the Coble polynomials 19

Here Yo[n] denotes the vector space of all rational functions which can be expressed as polynomials of degree n in the elements of Yo, especially Yo[0] = ℂ, Yo[1] = Yo. One checks that the images of the cubic relations in the modular forms F_{0}, …, F_{9} are also relations in A[Yo]. We obtain a surjective homomorphism

A(Γ) ⟶ A[Yo].

Again a dimension argument shows that this is an isomorphism.

6.7 Theorem. The isomorphism 6.6 extends to an isomorphism of rings

A(Γ) ⟶^{∼} A[vG].

This isomorphism is equivariant with respect to O^{+}(5, 𝔽_{3}) ≅ W_{Yos}.

Corollary. The Coble-Yoshida algebra A[vG] is a normal algebra. The cubic relations described in [Yo] are defining relations.

Our formulae allow to give explicitly the birational transformation between six points in a plane given by x_{1}, …, x_{4} and the coefficients λ, …, ρ of the corresponding cubic surface in its Cayley normal form:

6.8 Remark. The transformation of the six point parameters x_{1}, …, x_{4} and Cayley’s λ, …, ρ is given by

\begin{aligned}x_{1} & = \frac{(λνμρ − 1)(−1 + λρ)}{(λνρ − 1)(λμρ − 1)} \\ x_{2} & = \frac{μ(−1 + λρ)}{(λμρ − 1)} \end{aligned} \begin{aligned}x_{3} & = \frac{(ρ − 1)(νμρ − 1)}{(−1 + ρμ)(νρ − 1)} \\ x_{4} & = \frac{μ(ρ − 1)}{(−1 + ρμ)} \end{aligned}

\begin{aligned}λ & = \frac{x_{2}(−1 + x_{4})(−x_{4} + x_{3})(−1 + x_{1})}{(x_{3} − 1)(x_{1} − x_{2})(x_{2} − 1)x_{4}} \\ μ & = \frac{−x_{4}x_{2}(−x_{1} + x_{1}x_{4} − x_{4} + x_{2} − x_{3}x_{2} + x_{3})}{(x_{1}x_{4} − x_{1}x_{2}x_{4} − x_{1}x_{3}x_{4} + x_{1}x_{2}x_{3} + x_{2}x_{3}x_{4} − x_{3}x_{2})} \\ ν & = \frac{−(−1 + x_{4})(x_{2} − 1)(−x_{3}x_{2} + x_{1}x_{4})}{(x_{2} − x_{4})(x_{1} − x_{2})(−x_{4} + x_{3})} \\ ρ & = \frac{−(x_{1} − x_{2})(x_{3} − 1)(x_{2} − x_{4})}{(−1 + x_{4})x_{2}(−x_{1} + x_{1}x_{4} − x_{4} + x_{2} − x_{3}x_{2} + x_{3})} \end{aligned}

7. Table of the Coble polynomials

In the following table we give the expressions of 40 Coble polynomials. They are parametrized by isotropic vectors α ∈ 𝔽_{3}^{5}. Recall that they correspond to 9 pairs ±a of short vectors and that these correspond to 9 pairs ±r of roots of the E_{6}-lattice. (This is the encoding which Yoshida uses in [Yo].) We express the Coble-polynomials in the 6-point coordinates x_{1}, …, x_{4}, in the modular forms F_{0}, …, F_{9} and in the Cayley parameters λ, …, ρ.

(1, −1, 0, 0, 0)

(−x_{4} + x_{3})(−1 + x_{1})(x_{2} − 1)(x_{4}x_{1} − x_{3}x_{2}) F_{9} −λνρ(ρ − 1)(−1 + μ)(μρ − 1)

(0, −1, 1, 1, 0)

(−x_{2} + x_{1})(x_{3} − 1)(x_{4} − 1)(x_{4}x_{1} − x_{3}x_{2}) F_{7} − F_{8} + F_{9} −νρ(λρ − 1)(−1 + μ)(λμρ − 1)

(0, −1, 0, 1, 1)

(x_{4} − 1)(−1 + x_{1})(−x_{2} + x_{1})(−x_{2} + x_{4})x_{3} F_{3} − F_{8} + F_{9} −λρ(νρ − 1)(−1 + μ)(μνρ − 1)

(0 − 1, −1, 1, 0)

(−x_{2} + x_{1})(−x_{1} + x_{3})(−x_{2} + x_{4})(x_{4} − x_{3}) F_{2} − F_{3} −(λμνρ^{2} − 1)(λ − 1)ρ(−1 + μ)

(0, −1, 1, −1, 0)

−(x_{4}x_{1} − x_{3}x_{2})(−x_{3}x_{2} + x_{2} − x_{4} + x_{4}x_{1} − x_{1} + x_{3}) F_{5} − F_{6} −νρ(λ − 1)(μρ − 1)(λμρ − 1)

(−1, −1, 1, −1, 1)

−(x_{2} − 1)(x_{3} − 1)(x_{4} − x_{3})(−x_{2} + x_{4})x_{1} F_{2} − F_{4} + F_{9} −ρ(λνρ − 1)(−1 + μ)(λμνρ − 1)

(0, −1, 0, −1, 1)

−x_{3}x_{2}(−x_{3}x_{2} + x_{2} − x_{4} + x_{4}x_{1} − x_{1} + x_{3})(−1 + x_{1}) F_{1} − F_{6} −λρ(ν − 1)(μρ − 1)(μνρ − 1)

(0, −1, 0, 1, −1)

(x_{3} − 1)(x_{2} − 1)(−x_{2} + x_{1})(−x_{1} + x_{3})x_{4} F_{2} − F_{7} −(λμνρ^{2} − 1)(ν − 1)ρ(−1 + μ)

(0, −1, −1, −1, 0)

−(x_{4} − x_{3})x_{2}x_{1}(−x_{3}x_{2} + x_{2} − x_{4} + x_{4}x_{1} − x_{1} + x_{3})

F_{0} − F_{1} −(λμνρ − 1)(λρ − 1)(μρ − 1),

(−1, −1, −1, −1, 1)

−(x_{2} − 1)(−x_{1} + x_{3})(x_{4}x_{1} − x_{3}x_{2})(x_{4} − 1) −F_{4} + F_{7} −νρ(λμνρ^{2} − 1)(λ − 1)(−1 + μ)

(1, −1, 1, 1, −1)

−(−x_{2} + x_{4})(−x_{3}x_{2} + x_{2} − x_{4} + x_{4}x_{1} − x_{1} + x_{3})x_{1}x_{3} F_{2} + F_{5} + F_{8} −ρ(λ − 1)(μνρ − 1)(λμνρ − 1)

(−1, 0, 1, 0, 0)

−(x_{4}x_{1} − x_{3}x_{2})(−x_{4}x_{1} + x_{1}x_{3}x_{4} − x_{1}x_{2}x_{3} + x_{3}x_{2} − x_{2}x_{3}x_{4} + x_{1}x_{2}x_{4}) F_{5} − F_{6} − F_{7} + F_{8} −μνρ(λ − 1)(ρ − 1)(λρ − 1)

(−1, −1, 1, 1, 1)

(−x_{1} + x_{3})x_{2}(x_{4} − x_{3})(x_{2} − 1) F_{0} − F_{4} −(λνρ − 1)(μρ − 1)(λμνρ^{2} − 1)

(−1, −1, 1, −1, −1)

−(−1 + x_{1})(x_{4} − 1)(x_{4} − x_{3})(−x_{1} + x_{3})x_{2} F_{3} − F_{4} −(λμνρ^{2} − 1)(ν − 1)(−1 + μ)λρ

(1, −1, −1, 1, 1)

−x_{1}x_{4}(−x_{3}x_{2} + x_{2} − x_{4} + x_{4}x_{1} − x_{1} + x_{3})(x_{3} − 1) F_{1} + F_{2} + F_{8} −ρ(ν − 1)(λμρ − 1)(λμνρ − 1)

(0, −1, 0, −1 − 1)

−x_{4}x_{1}(−x_{3}x_{2} + x_{2} − x_{4} + x_{4}x_{1} − x_{1} + x_{3})(x_{2} − 1) F_{0} − F_{5} −(λμνρ − 1)(νρ − 1)(μρ − 1)

(−1, 0, 0, 0, 1)

−(−x_{4}x_{1} + x_{1}x_{3}x_{4} − x_{1}x_{2}x_{3} + x_{3}x_{2} − x_{2}x_{3}x_{4} + x_{1}x_{2}x_{4})(−1 + x_{1})x_{4} F_{1} − F_{3} − F_{6} + F_{8} −λμρ(ν − 1)(ρ − 1)(νρ − 1)

(−1, −1, 0, 0, 0)

(−x_{2} + x_{1})x_{4}x_{3}(−x_{3}x_{2} + x_{2} − x_{4} + x_{4}x_{1} − x_{1} + x_{3}) F_{0} − F_{1} − F_{2} + F_{3} − F_{5} + F_{6} + F_{7} − F_{8} −(μνρ − 1)(ρ − 1)(λμρ − 1)

(1, −1, −1, 1, −1)

−(x_{4} − 1)x_{2}(−x_{3}x_{2} + x_{2} − x_{4} + x_{4}x_{1} − x_{1} + x_{3})x_{3} F_{0} + F_{3} + F_{7} −(λμρ − 1)(λνρ − 1)(μνρ − 1)

22 Marked cubic surfaces (−1, 0, −1, 0, 0)

−(x_{4} − x_{3})(−x_{4}x_{1} + x_{1}x_{3}x_{4} − x_{1}x_{2}x_{3} + x_{3}x_{2} − x_{2}x_{3}x_{4} + x_{1}x_{2}x_{4}) F_{0} − F_{1} − F_{2} + F_{3} − F_{9} −(λνρ − 1)(λμρ − 1)(ρ − 1)

(−1, −1, −1, 1, 1)

(x_{2} − 1)x_{1}(x_{4}x_{1} − x_{3}x_{2})(x_{3} − 1) −F_{4} + F_{5} + F_{9} −νρ(λμνρ − 1)(λρ − 1)(μρ − 1)

(−1, −1 − 1, −1, −1)

−(−1 + x_{1})(−x_{2} + x_{4})(x_{4}x_{1} − x_{3}x_{2})(x_{3} − 1) −F_{4} + F_{8} −λμνρ^{3}(λ − 1)(−1 + μ)(ν − 1)

(1, −1, 1, −1, −1)

−x_{2}(−x_{2} + x_{1})(−x_{1} + x_{3})x_{3}(x_{4} − 1) F_{0} + F_{6} + F_{7} −(λρ − 1)(μνρ − 1)(λμνρ^{2} − 1)

(1, −1, 1, 1, 1)

−(−x_{1} + x_{3})(−x_{3}x_{2} + x_{2} − x_{4} + x_{4}x_{1} − x_{1} + x_{3})x_{2}x_{4} F_{3} + F_{6} + F_{7} −(λμνρ^{2} − 1)(ν − 1)(λ − 1)μρ

(0, 0, 1, −1, −1)

−(−x_{2} + x_{4})(−x_{4}x_{1} + x_{1}x_{3}x_{4} − x_{1}x_{2}x_{3} + x_{3}x_{2} − x_{2}x_{3}x_{4} + x_{1}x_{2}x_{4}) F_{3} + F_{4} + F_{5} −μρ(λ − 1)(νρ − 1)(λνρ − 1)

(1, −1, −1, −1, 1)

−(−x_{1} + x_{3})x_{4}(−x_{2} + x_{1})(x_{4} − 1) F_{0} + F_{3} + F_{6} −(νρ − 1)(λμρ − 1)(λμνρ^{2} − 1)

(−1, −1, 1, 1, −1)

(−x_{2} + x_{4})x_{1}(x_{4} − x_{3})(−1 + x_{1}) F_{1} − F_{4} + F_{9} −(λμνρ − 1)(νρ − 1)ρλ(μρ − 1)

(0, 1, 1, 0, 1)

(−x_{1} + x_{3})(x_{4} − x_{3})(x_{2} − 1)x_{1}x_{4} F_{0} − F_{2} −(λμνρ − 1)(ρ − 1)(λμνρ^{2} − 1)

(0, 0, −1, −1, 1)

−(−x_{4}x_{1} + x_{1}x_{3}x_{4} − x_{1}x_{2}x_{3} + x_{3}x_{2} − x_{2}x_{3}x_{4} + x_{1}x_{2}x_{4})(x_{3} − 1)x_{2} F_{1} + F_{4} + F_{7} −μρ(ν − 1)(λρ − 1)(λνρ − 1)

(−1, 0, 0, 0, −1)

−(−x_{4}x_{1} + x_{1}x_{3}x_{4} − x_{1}x_{2}x_{3} + x_{3}x_{2} − x_{2}x_{3}x_{4} + x_{1}x_{2}x_{4})(x_{2} − 1)x_{3} F_{0} − F_{2} − F_{5} + F_{7} − F_{9}

−(λνρ − 1)(μνρ − 1)(ρ − 1)

(1, 0, 0, 1, 0)

(−x_{2} + x_{1})(−x_{4}x_{1} + x_{1}x_{3}x_{4} − x_{1}x_{2}x_{3} + x_{3}x_{2} − x_{2}x_{3}x_{4} + x_{1}x_{2}x_{4}) F_{0} − F_{1} − F_{5} + F_{6} − F_{9} −(νρ − 1)(μρ − 1)(λρ − 1)

(1, −1, −1, −1, −1)

−(−x_{2} + x_{4})(−x_{2} + x_{1})x_{3}(x_{3} − 1) F_{1} + F_{2} + F_{5} + F_{9} −ρ(λμρ − 1)(λνρ − 1)(μνρ − 1)

(0, 0, −1, −1, −1)

−(x_{4} − 1)(−x_{4}x_{1} + x_{1}x_{3}x_{4} − x_{1}x_{2}x_{3} + x_{3}x_{2} − x_{2}x_{3}x_{4} + x_{1}x_{2}x_{4})x_{1} F_{0} + F_{4} + F_{8} − F_{9} −(λρ − 1)(λμνρ − 1)(νρ − 1)

(0, −1, 1, 0, 1)

(x_{4} − 1)x_{3}(x_{4}x_{1} − x_{3}x_{2})(−1 + x_{1}) F_{6} − F_{8} + F_{9} −λνρ(μνρ − 1)(ρ − 1)(λμρ − 1)

(−1, −1, −1, 1, −1)

(−1 + x_{1})x_{2}(x_{4}x_{1} − x_{3}x_{2})(x_{4} − 1) −F_{4} + F_{6} −λμνρ(λρ − 1)(μρ − 1)(νρ − 1)

(0, 1, −1, 0, 1)

(x_{2} − 1)(x_{4}x_{1} − x_{3}x_{2})(x_{3} − 1)x_{4} F_{5} − F_{7} −μνρ(λνρ − 1)(λμρ − 1)(ρ − 1)

(1, −1, 1, −1, 1)

−x_{1}(−x_{2} + x_{1})(−x_{2} + x_{4})x_{4}(x_{3} − 1) F_{1} + F_{5} + F_{8} −(λμνρ − 1)(νρ − 1)μρ(λρ − 1)

(0, 0, 1, −1, 1)

−(x_{1} + x_{3})(−x_{4}x_{1} + x_{1}x_{3}x_{4} − x_{1}x_{2}x_{3} + x_{3}x_{2} − x_{2}x_{3}x_{4} + x_{1}x_{2}x_{4}) F_{2} + F_{4} + F_{6} −ρ(λμνρ^{2} − 1)(ν − 1)(λ − 1)

(0, 1, 1, 0, −1)

(−x_{2} + x_{4})(x_{4} − x_{3})(−1 + x_{1})x_{2}x_{3} F_{1} − F_{3} −λμρ(λνρ − 1)(μνρ − 1)(ρ − 1)

(1, 0, 0, −1, 0)

−(−x_{4}x_{1} + x_{1}x_{3}x_{4} − x_{1}x_{2}x_{3} + x_{3}x_{2} − x_{2}x_{3}x_{4} + x_{1}x_{2}x_{4})·

· (−x_{3}x_{2} + x_{2} − x_{4} + x_{4}x_{1} − x_{1} + x_{3})

F_{2} − F_{3} − F_{7} + F_{8} −ρ(ν − 1)(−1 + μ)(λ − 1)

Literature

[ACT1] Allcock, D. Carlson, J. and Toledo, D.: A complex hyperbolic structure for the moduli of cubic surfaces, Comptes Rendus de l’Aacademie Scientifique Francaise 326, ser I, 49–54 (1988) (alg geom /970916)

[ACT2] Allcock, D. Carlson, J., and Toledo, D.: The complex hyperbolic geometry of the moduli space of cubic Surfaces, preprint

[BB] Baily, W.L., and Borel,A.: Compactification of arithmetic quotients of bounded symmetric domains, Annals of Math. 84, No 3, 442–528 (1966)

[Bo1] Borcherds, R.: Automorphic forms with singularities on Grassmannians, Invent. math. 132, 491–562 (1998) http://www.dpmms.cam.ac.uk/home/

emu/reb/.my-home-page.html

[Bo2] Borcherds, R.: The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J. 97, No 219–233 (1999) http://www.dpmms.cam.ac.uk/home/

emu/reb/.my-home-page.html

[Co] Coble, A. B.: Point sets and allied cremona groups (Part III), Trans. AMS 18, 331–372, (1917).

[Fr1] Freitag, E.: Modular forms related to cubic surfaces, preprint 1999

[Fr2] Freitag, E.: A graded algebra related to cubic surfaces, to appear in the Kyushu Journal of Mathematics

[vG] van Geemen, B.: A linear system on Naruki’s moduli space of marked cubic surfaces, preprint (2001)

[Na] Naruki, I.: Cross ratio variety as moduli space of cubic surfaces, Proc. London Math. Soc. (3) 45, 1–30 (1982)

[Yo] Yoshida, M.: A W (E_{6})-equivariant projective embedding of the moduli space of cubic surfaces, eprint math.AG/0002102.