3. Modular forms

the moduli space marked cubic surfaces

Eberhard Freitag

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 B4/Γ of the quotient of the 4-ball by a certain Picard modular group, namely the principal congruence subgroup Γ of level√

−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 B4/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. HenceMappears as subvariety ofP⁹(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 → Mwhich identifiesC with the blow-up ofMat its 40 singular points (the cusps). HenceM ∼= ˇC 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 ofC (which is a rational variety). This means that the mapC → 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 P²(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 ofMinto P⁸⁰⁻¹(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.

We will give the explicit isomorphism between the space generated by the Coble polynomials an 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!

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 coordinatesλ, ρ, µ, ν. This work is essential for the explicit formulae in this paper. We will not include proofs and therefore van Geemens paper stays behind the scene.

2 Marked cubic surfaces 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.

2. The ten dimensional representation

It is well known that the group W(E₆) admits a unique (up to isomorphism) ten dimensional representation. We denote by U the representation space of some concrete realization. It is better for us to work with O(5,F3) instead ofW(E6). This group contains a copy ofW(E6) as subgroup of index two. The group O(5,F3) here is defined by means of the quadratic formq(x) =x²₁−x²₂−· · ·−x²₅. An elementx is called shortif q(x) = −1 and longif q(x) = 1. We extend the representation of W(E6) to a representation of O(5,F3) such that the central involution −id acts by multiplication with−1.

There are two basic configurations of elements ofV =F⁵3.

2.1 Definition. 1. A crossin V =F⁵3 is a set{±a₁, . . . ,±a₅} 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. AnineinV =F⁵3is a set of nine pairs±aof 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 thatW(E6) acts transitively on the 135 crosses and also on the 40 nines.

2.2 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 ofC is one-dimensional. We call such a space across-line.

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

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

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 itsW(E₆) orbit:

2.3 Lemma. 1. Let F ∈ U be one element which lies in a cross-line. The W(E6)-orbit of F consists of 2·135 = 270elements. 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. TheW(E6)-orbit ofF consists of 2·40 = 80 elements. These 80 elements consist of 40 pairs ±Gof 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 commonconstant factor.

We describe some of the dependencies between the 270 elements. Let abe a fixed long element of V. Recall that there are three crosses which containa 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:

2.4 Remark. Let a∈ F⁵ 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(E6)-orbit. Then eitherA+B oreA−B is contained in the third cross line belonging to a.

3. Modular forms

An important realization of the ten-dimensional representation is given by modular forms. We denote by U(1,4)⊂GL(5,C) the unitary group of the Hermitean form ¯z1w1−. . .−z¯5w5. We have to consider the subgroup U(1,4)(E) of all elements with entries in the ring of Eisenstein integers

E=Z[ω], ω=−1 2+ i

√3 2 . We denote by Γ the principle congruence subgroup of level√

−3. Because ofE/√

−3E ∼=F3there is an exact sequence

1−→Γ−→U(1,4)(E)−→O(5,F⁵3)−→1.

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

f : ˜B −→C, B˜={

z∈C⁵, (z, z)>0} . which satisfy

a)f(tz) =t^−rf(z), b)f(γz) =f(z) forγ∈Γ.

One knows [Fr1] that the ring of all modular forms is generated by [Γ,3]. Moreover one knows the following basic facts:

1) The space [Γ,3] is 10-dimensional.

2) The group O(5,F3)∼= U(1,4)(E)/Γ acts irreducibly on this space.

In [Fr2] an explicit basisY₀, . . . , Y₉ of this space has been constructed. We use this basis here and denote it by

F₀, . . . F₉∈[Γ,3].

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

3.1 Definition. Across ratioinK(Γ)is the quotientf =A/B of two elementsF, G∈[Γ,3]with the following properties:

1. They belong to the sameW(E6)-orbit.

2. They are contained in different cross-lines which belong to the same long element of F⁵3. 3. A−B (and not A+B) is contained in a cross-line.

The function

F₀−F₁+F₄−F₅−F₉, g:=F0−F1−F2+F3−F5+F7−F9. is a cross ratio.

Using 2.3 we see that there exist precisely 6·45 = 270 cross ratios. Note thatf⁻¹ is a cross ratio if f is one. We also mention that the cross ratios constitute one orbit underW(E6).

The varietyM=B4/Γ contains certain divisors, which are called mirrors in [AF]. A long (short) mirror is the fixed point set of a reflection along a long (short) element ofV =F⁵3. So we have 45 long and 36 short mirrors. It is known that mirrors are irreducible subvarieties of codimension 1.

3.2 Proposition. 1. Let C be a cross andF ∈[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 onB4.) There are no other zeros.

2. LetN be a nine andF∈[Γ,3]a non-zero element of the corresponding nine-line. ThenF vanishes on the 9 mirrors belonging to the nine. (The multiplicities are three, if they are counted onB4.) 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.

It is easy now to express the Naruki-coordinatesλ, µ, ν, ρfrom [Na] as modular functions: After some calculation one gets:

4 Marked cubic surfaces

3.3 Theorem. The cross ratio’s in the sense of 3.1 agree with the classical ones (described for example in [Na]). The for modular functions

λ=− (F₀+F₄+F₇)(F₁−F₃+F₉) (−F0+F2+F5−F7)(F1+F5+F7+F9) µ= (F1−F3+F5−F6−F7+F8)(F0−F6)

(F1−F4+F5−F6+F9)(F0−F2−F6+F8−F9) ν=− (F0+F7+F8−F9)(−F4+F5)

(−F₀+F₁+F₉)(F₁+F₂−F₃+F₅+F₈) ρ= (F₁+F₅+F₇+F₉)(F₂+F₄+F₅)

(F3+F5+F7)(F0+F4+F7)

agree with Naruki’s coordinates. They generate the field of modular functions.

4. 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 planeP²(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 P²(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, x₁, x₃], [1, x₂, x₄].

These six points are in general position, if a certain concrete discriminant polynomial in thex₁, . . . , x₄ vanishes [Yo]. The open dense subsetM ⊂C⁴which is obtained in this way is a model of the moduli space of non-singular marked cubic surfaces. We use the action of the groupW(E₆) on this model as described in [Yo].

4.1 Proposition. There are 80 polynomials in the variablesx₁, . . . , x₄ which are characterized by the following properties:

1. The are transitively permuted under the action ofW(E6).

2. One of them is

(x2−x1)(x3−x1)(x4−x2)(x4−x3).

3. They generate a 10-dimensional space with irreducible action ofW(E6).

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

4.2 Proposition. Up to a constant factor there is a unique isomorphism between [Γ,3] and the space generated by the Coble-Yoshoda polynomials, such that the modular form F9 corresponds the Coble-Yoshida polynomial

3(−x4+x3)(−1 +x1)(x2−1)(x4x1−x3x2).

Our formulae allow to give explicitly the birational transformation between six points in a plane given byx1, . . . , x4and the coefficientsλ, . . . , ρof the corresponding cubic surface in its Cayley normal form:

4.3 Remark. The transformation of the six point parameters x1, . . . , x4 and Cayley’s λ, . . . , ρ is given by

x₁= (λνµρ−1)(−1 +λρ) (λνρ−1)(λµρ−1) x₂= µ(−1 +λρ)

(λµρ−1)

x₃= (ρ−1)(νµρ−1) (−1 +ρµ)(νρ−1) x₄= µ(ρ−1)

(−1 +ρµ) λ=x2(−1 +x4)(−x4+x3)(−1 +x1)

(x3−1)(x1−x2)(x2−1)x4

µ= −x4x2(−x1+x1x4−x4+x2−x3x2+x3) (x₁x₄−x₁x₂x₄−x₁x₃x₄+x₁x₂x₃+x₂x₃x₄−x₃x₂) ν =−(−1 +x4)(x2−1)(−x3x2+x1x4)

(x2−x4)(x1−x2)(−x4+x3) ρ= −(x1−x2)(x3−1)(x2−x4)

(−1 +x₄)x₂(−x₁+x₁x₄−x₄+x₂−x₃x₂+x₃)

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 Francaise326, 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 do- mains,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)

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[Co] Coble, A. B.: Point sets and allied cremona groups (Part III), Trans. AMS18, 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(E6)-equivariant projective embedding of the moduli space of cubic surfaces, eprint math.AG/0002102.