The Fano surface of the Fermat cubic threefold, the del Pezzo surface of degree 5 and a ball quotient

We study the Fano surface S of the Fermat cubic threefold. We prove that S is a degree 81 abelian cover of the degree 5 del Pezzo surface and that the complement of the union of 12 disjoint elliptic curves on S is a ball quotient. The lattice of this ball quotient is related to the Deligne-Mostow lattice number 1.


Introduction
Let us recall the following well known Theorem (see [8], Theorem 2) that relates an inequality between (log) Chern numbers with the theory of Ball quotients: Hirzebruch, Miyaoka, Sakai, Yau]. Let S be a smooth projective surface with ample canonical bundle K and let D be a reduced simple normal crossing divisor on S (may be 0). Suppose that K + D is nef and big. Then the following inequality: c 2 1 ≤ 3c 2 holds, wherec 2 1 ,c 2 are the logarithmic Chern numbers of S ′ = S − D defined by: c 2 1 = (K +D) 2 andc 2 = χ top (S ′ ). Furthermore, the equality occurs if and only if S ′ is a ball quotient i.e., if and only if we obtain S ′ by dividing the ball B 2 with respect to a discrete group Γ of automorphisms acting on B 2 properly discontinuously and with only isolated fixed points. If a smooth projective surface X contains a rational curve and its Chern numbers satisfies c 2 1 = 3c 2 > 0, then X is the projective plane.
Few examples of surfaces with Chern ratio c 2 1 c 2 equals 3 have been constructed algebraically i.e. by ramified covers of known surfaces. The first examples are owed independently to Inoue and Livné (for a reference, see [1]). Among these examples, there is a surface S, with Chern numbers: c 2 1 (S) = 3c 2 (S) = 3 2 5 2 , that is the blow-down of (−1)-curves of a certain cyclic cover of the Shioda modular surface of level 5. Hirzebruch [5] then constructed other examples. Starting with the degree 5 del Pezzo surface H 1 and for any n > 1, he constructed a cover η n : H n → H 1 of degree n 5 branched exactly over the ten (−1)-curves of H 1 , with order n, such that for n = 5: Theorem 2. [Hirzebruch]. The Chern numbers of H 5 satisfies: [7] established a link between the surface H 5 and the Inoue-Livné surface S: There is a étale map H 5 → S that is a quotient of H 5 by an automorphism group of order 25.
In the present paper, we give an example of a surface with log Chern numbers satisfyingc 2 1 = 3c 2 , and we obtain in part A) and B) of Theorem 4 below, results analogous to Theorem 2 and Proposition 3: The variety that parametrizes the lines on a smooth complex cubic threefold F ֒→ P 4 is a smooth surface of general type called the Fano surface of F [3]. Let S be the Fano surface of the Fermat cubic threefold: . It is the only Fano surface that contains 30 elliptic curves [11] ; the aim of this paper is to prove the following Theorem: There is an open subvariety S ′ ⊂ S, complement of 12 disjoint elliptic curves on S, such that S ′ is a ball quotient with log Chern numbersc 1 2 = 3c 2 = 3 4 . B) There is a étale map κ : H 3 → S that is a quotient of H 3 by an automorphism of order 3 and there is a degree 3 4 ramified cover η : S → H 1 branched with order 3 over the ten (−1)-curves of H 1 . C) The surface T = κ −1 S ′ ⊂ H 3 is a ball quotient. Let Λ be the lattice of T , i.e. the transformation group of the 2-dimensional unit ball B 2 such that Λ \ B 2 is isomorphic to T . The lattice Λ is the commutator group of the congruence group: where α is a primitive third root of unity, I is the identity matrix and H is Hermitian diagonal matrix with entries (1, 1, −1) defining the 2-dimensional unit ball B 2 .
We wish to remark that in order to prove the part B) and C), we use a result of Namba on ramified Abelian covers of varieties that, to the best of our knowledge, has never been used before this paper.
Let us explain how the Deligne-Mostow lattices occur. Let µ = (µ 1 , . . . , µ 5 ) be a 5-tuple of rational numbers with 0 < µ i < 1 and µ i = 2. Let d be the l.c.m. of the µ i and let n i be such that n i d = µ i . Let M be the moduli space of 5-tuples x = (x 1 , . . . , x 5 ) of distinct points on the projective line P 1 . For each point x of M , and 1 ≤ i < j ≤ 5, we consider the periods: These (multivalued) maps ω ij clearly factor to Q = M/Aut(P 1 ) (that is isomorphic to the complement of the 10 (−1)-curves of the degree 5 del Pezzo surface). They are called hypergeometric functions and they satisfy what is called the Appell differential equations system. It turns out that the ω ij span a 3 dimensional vector space W µ , and these yield a multivalued holomorphic map Q → P 2 = P(W * µ ). In fact, the image of that map lies in the (copy of a) unit Ball B 2 of C 2 ⊂ P 2 . The multivaluedness is measured by the monodromy representation π 1 (Q) → Aut(B 2 ) whose image is denoted by Γ µ . The main results of the fundamental papers of Deligne and Mostow [4] and Mostow [9] are to prove that the group Γ µ ⊂ P GL(W * µ ) is discontinuous and acts as a lattice on B 2 for only a finite number of 5-tuple µ, to compute these µ and to provide examples of non-arithmetic lattices acting on B 2 .
2. The Fano surface of the Fermat cubic as a cover of H 1 .
Let X be the quotient of S by the group G and let η : S → X be the quotient map.
Proposition 6. The surface X is (isomorphic to) the del Pezzo surface H 1 , and the cover η is branched with index 3 over the ten (−1)-curves of X.
Proof. Let us outline the proof of Proposition 6: using classical results on the quotient of a surface by a group action, we show that X is a smooth surface, then we compute its Chern numbers, and prove that the blowing down of four (−1)curves on X is the plane, and that allows us to conclude that X is the degree 5 del Pezzo surface H 1 .
In order to prove that the surface X is smooth, we need to recall two lemmas: Let s be a point of S. Let us denote by T S,s the tangent space of S at s, by L s ֒→ F the line on F ֒→ P 4 = P(C 5 ) corresponding to s and by P s ⊂ C 5 the subjacent plane to the line L s .
parametrizes the lines on a cone in the cubic F . Their configuration is as follows : The elements of the group G have order 3, for each of them it is easy to compute it closed set of fixed points. Let I be the set of points s in S such that s is an isolated fixed point of an element of G : I is the set of the 135 intersection points of the 30 elliptic curves. Let i, j, s, t be indices such that {i, j} ∩ {s, t} = ∅ and let s be the intersection point of E 1 ij and E 1 st . The orbit of s by G is the set of the 9 intersection points of the curves E β ij and E γ st , β, γ ∈ µ 3 . Let be s ∈ I, the group: G s = {g ∈ G/s is a isolated fixed point of g} is isomorphic to µ 2 3 and, by Proposition 7, its representation on the space T S,s is isomorphic to the representation: .(x, y) = (α 1 x, α 2 y) ∈ C 2 on C 2 . That implies by [2], that the image of s is a smooth point of X, thus X is smooth.
The ramification index of η : S → X at the points of I is 9 and the ramification index of η on the curve E β ij is 3. Let us denote by K V the canonical divisor of a surface V . Let be Σ = i,j,β E β ij ; the ramification divisor of η : S → X is 2Σ and K S = η * K X + 2Σ.
By [3], Lemma 8.1 and Proposition 10.21, we know moreover that Σ = 2K S , hence 3 4 (K X ) 2 = (η * K X ) 2 = (−3K S ) 2 = 9.45 and (K X ) 2 = 5. The stabilizer in G of an elliptic curve E β ij ֒→ S contains 27 elements and the group that fixes each point of E β ij has 3 elements. Let η β ij : E β ij → X ij be the restriction of η to E β ij . The curve X ij is smooth because it is the quotient of a smooth curve by an automorphism group. The map η β ij is a degree 9 ramified cover over 3 points with ramification index 3, hence: 3 and χ top (X ij ) = 2 : X ij is a smooth rational curve. As: , we deduce that the 10 curves X ij have the following configuration: otherwise.
Let I ′ = η(I) and let Σ ′ = X ij . By additive property of the Euler characteristic, we have: Moreover, χ top (Σ ′ ) = 5 (for an example of such computation see [12]) and we obtain χ top (X) = 7. We can blow-down four disjoint (−1)-curves among the ten curves X ij and we obtain a surface with Chern numbers: c 2 1 = 3c 2 = 9 but this surface contains 6 rational curves. Hence, by Theorem 1, it is the plane : X is the blow-up of the plane at four points. These points are in general position because of the intersection numbers of the X ij , therefore X is the degree 5 del Pezzo surface H 1 and the X ij are its ten (−1)-curves.
We proved that the quotient map η : S → X = H 1 is an Abelian cover branched over the ten (−1)-curves of X with ramification index 3, and that completes the proof of Proposition 6.
Let us now prove that: Proposition 9. There exists an étale map κ : H 3 → S of degree 3.
Proof. To prove Proposition 9, we begin to recall Namba's results on Abelian covers of algebraic varieties.
Let D 1 , . . . , D s be irreducible hypersurfaces of a smooth projective variety M and let e 1 , . . . , e s be positive integers. A covering π : Y → M is said to branch (resp. to branch at most) at D = e 1 D 1 + · · · + e s D s if the branch locus is (resp. is contained in) ∪D i and the ramification index over D i is e i (resp. divides e i ). An Abelian covering π : Y → M which branches at D is said to be maximal if for every Abelian covering π 1 : Y 1 → M which branches at most at D, there is a map κ : Y → Y 1 such that π = π 1 • κ. Let : We say that F 1 , F 2 ∈ Div 0 (M, D) are linearly equivalent F 1 ∼ F 2 if F 1 − F 2 is integral and is a principal divisor. The following result is due to Namba: Theorem 10. (Namba [10], Thm. 2.3.18.). There is a bijective map of the set of (isomorphism classes of ) Abelian coverings π : Y → M branched at most at D onto the set of finite subgroups G of Div 0 (M, D)/ ∼. Let G π be the transformation group of the cover π : Y → M . The bijective map satisfies: (1) G π ≃ G(π) (2) let π 1 : Y 1 → M and π 2 : Y 2 → M be Abelian covers branched at most at D.
One applies this Theorem to the ten (−1)-curves X ij of X = H 1 with e i = 3. The group Div 0 (H 1 , D)/ ∼ is isomorphic to (Z/3Z) 5 (for brevity, we skip the proof, but for an example of such a computation, see the proof of Lemma 13). As η 3 : H 3 → H 1 is a degree 3 5 Abelian cover branched over the (−1)-curves with index 3, the group G(η 3 ) is equal to Div 0 (H 1 , D)/ ∼. Thus by Theorem 10, there exists κ : H 3 → S such that η 3 = η • κ. As the maps η and η 3 are branched with order 3 over the ten (−1)-curves of H 1 , the map κ is étale. That completes the proof of Proposition 9.
3. The Fano surface of the Fermat cubic as a ball quotient.
By [3], Theorem 7.8, (9.14) and (10.11), the Fano surface S of the Fermat cubic is smooth with invariants c 2 1 = 45 and c 2 = 27. Let S ′ ⊂ S be the complement of the union D of 12 disjoints elliptic curves on S (there are 5 such sets of 12 elliptic curves, we can take by example the 12 curves E β 1i , 2 ≤ i ≤ 5, β 3 = 1). Let c 2 1 , c 2 be the logarithmic Chern numbers of S ′ .
Let H be the Hermitian diagonal matrix with entries (1, 1, −1) defining the 2dimensional unit ball B 2 into P 2 . Let α be a third primitive root of unity and let Γ be the congruence group: where I is the identity matrix. As κ is étale and S ′ is a ball quotient, the surface T = κ −1 S ′ ⊂ H 3 is a ball quotient. Let Λ be the transformation group of the 2-dimensional unit ball B 2 such that Λ \ B 2 ≃ T . We have: Theorem 12. The group Λ is the commutator group of Γ.
Proof. In order to compute Λ, we combine ideas in [14], where Yamazaki and Yoshida computed the lattice of the Ball quotient surface H 5 , and we use Namba's Theorem 10.
Let ℓ 1 , . . . , ℓ 6 ∈ H 0 (P 2 , O(1)) be the linear forms defining the 6 lines on the plane going through 4 points in general position. Let H ′ 3 be the normal algebraic surface determined by the field It is an Abelian cover π : H ′ 3 → P 2 of degree 3 5 of the plane branched with order 3 over the 6 lines {ℓ i = 0} and the surface H 3 is the fibered product of π : H ′ 3 → P 2 and the blow-up map τ : H 1 → P 2 (see [13] (1.3)). The situation is as follows: We apply Namba's Theorem 10 to the 6 lines of the complete quadrilateral on the plane, with weights e i = 3.
Proof. Let L i be the line {ℓ i = 0} and let L be a generic line. The group: it is a sub-group of: where the rational divisor E = aL + a i 3 L i in Div(P 2 , D) is equivalent to 0 if and only if the a i , 1 ≤ i ≤ 6 are divisible by 3 and c 1 (E) = a + 1 3 a i = 0 (here we use that linear and numerical equivalences are equal on the plane). The map : is well defined and is an isomorphism. The group Div 0 (P 2 , D)/ ∼ is isomorphic to: {(a,ā 1 , . . . ,ā 6 ) ∈ Z × (Z/3Z) 6 /a = 0 and ā i = 0} and is therefore isomorphic to (Z/3Z) 5 .
Let b : P 2 → N be the function such that b(p) = 1 outside the complete quadrilateral, b(p) = 3 on the complete quadrilateral minus the 4 triple points p 1 , . . . , p 4 , and b(p) = ∞ on these 4 points. The pair (P 2 , b) is an orbifold (for the theory of orbifold we refer to [15], Chap. 5). By [6], Chap. 5, the universal cover of that orbifold is B 2 with the transformation group Γ. Therefore, a cover Z → P 2 with branching index 3 over the complete quadrilateral corresponds to a normal sub-group K of Γ and Γ/K is isomorphic to the group of transformation of the covering Z → P 2 .
If moreover, the cover Z → P 2 is Abelian, the group K contains the commutator group [Γ, Γ], thus B 2 /[Γ, Γ] is the maximal Abelian cover of (P 2 , b). We have seen that the cover π : H ′ 3 → P 2 of degree 3 5 is maximal among Abelian covers of (P 2 , b), thus the lattice of the ball quotient T is the commutator group [Γ, Γ].
Moreover, we remark that : Proof. This is the fact that the universal cover of the orbifold (P 2 , b) of the proof of Theorem 12 is B 2 with the transformation group Γ.