Fano Surfaces with 12 or 30 Elliptic Curves

Fano Surfaces with 12 or 30 Elliptic Curves

X av i e r Rou l l e au

Introduction

A Fano surface is a surface of general type that parameterizes the lines of a smooth cubic threefold. First studied by Fano and then by many others—including Bombieri and Swinnerton-Dyer [4], Gherardelli [7], Tyurin [14; 15], Clemens and Griffiths [5], and Collino [6]—these surfaces carry many remarkable properties.

In our previous paper [12], we classified Fano surfaces according to the config- urations of their elliptic curves. The aim of the present paper is to give various applications of this study when the Fano surface contains 12 or 30 elliptic curves.

The main result of the first part of this paper is as follows.

Proposition1.

(i) The Picard numberρ_Sof a Fano surfaceSsatisfies1≤ρ_S≤25and is1for Sgeneric.

(ii) A Fano surface that contains12elliptic curves is a triple ramified cover of the blow-up of 9points of an abelian surface.

(iii) The Néron–Severi group of such a surface has rank12, 13or25=h^1,1(S).

(iv) ForSgeneric among Fano surfaces with12elliptic curves, the Néron–Severi group has rank12and is rationally generated by its12elliptic curves.

(v) An infinite number of Fano surfaces with12elliptic curves have maximal Picard number25=h^1,1(S).

Recall that among the K3 surfaces, the Kummer surfaces are recognized as those K3 having 16 disjoint(−2)-curves. They are the double cover of the blow-up over the 2-torsion points of an abelian surface (see [11]). Our theorem is the analogue for Fano surfaces that contains 12 elliptic curves among Fano surfaces.

In the second part, we study the Fano surfaceS of the Fermat cubic threefold F →P⁴:

F = {x₁³+x₂³+x₃³+x³₄+x₅³ =0}.

Letµ3be the group of third roots of unity and letα∈µ3be a primitive root. For sa point ofS, we denote byLsthe line onF corresponding to the pointsand we denote byC_s the incidence divisor that parameterizes the lines inF that cut the lineL_s.

Received August 3, 2009. Revision received February 5, 2010.

313

Theorem2. The surfaceSis the unique Fano surface that contains30smooth curves of genus1.These curves are numbered

E_ij^β, 1≤i < j ≤5, β∈µ3, in such a way that, for two such curvesE_ij^γ andE_st^β,

E_ij^βE_st^γ =







1 if {i,j} ∩ {s,t} = ∅,

−3 if E_ij^β=E_st^γ, 0 else.

The Néron–Severi groupNS(S)ofShas rank25=dimH¹(S,S)and discrim- inant3¹⁸.These30elliptic curves generate an index-3sublattice of NS(S)and, with the class of an incidence divisorCs (s∈S),they generate the Néron–Severi group.

Given a smooth curve of low genus and with a sufficiently large automorphism group, it is sometimes possible to calculate the period matrix of its Jacobian [3].

In this paper, we calculate also the period lattice of the Albanese variety of the 2-dimensional varietyS.This computation is used to determine the Néron–Severi group ofS.We determine also the fibrations ofSonto an elliptic curve as well as the intersection numbers between the fibers of these fibrations, and we discuss some of the more interesting fibrations.

Acknowledgments. A part of this paper written during the author’s stay at the Max Planck Institute of Bonn and at the University of Tokyo, under JSPS fel- lowship. The author wishes to thank Bert van Geemen for a useful discussion on Theorem 11.

1. Preliminaries on Fano Surfaces

1.1. Tangent Bundle Theorem

LetF → P⁴ be a smooth cubic threefold and letSbe its Fano surface of lines.

We consider the following diagram:

U ^ψ //

π

F ^// _P⁴

S

whereU is the universal family of lines andπ,ψare the projections.

Theorem3 (Tangent Bundle Theorem [5]). There is an isomorphism π_∗ψ^∗O(1)_S,

where_Sis the cotangent sheaf. By this isomorphism, we can identify the spaces H⁰(F,O(1))andH⁰(S,_S),the varietiesP⁴andP(H⁰(S,_S)^∗),and the vari- etiesU andP(T_S)forT_S=^∗_S.

We always work with the identifications of this theorem. In particular, the line L_s → F corresponding to a pointsinSis the projectivized tangent space toS ats.

1.2. Properties of Fano Surfaces with an Elliptic Curve

Let us denote byCa cone on the cubicF.The following lemmas come from [12].

Lemma4. The coneCis a hyperplane section ofF.The curveEparameterizing the lines onCis naturally embedded intoSand is an elliptic curve. Conversely, if E →Sis an elliptic curve onS,the surfaceψ(π⁻¹(E))is a cone.

LetE →Sbe an elliptic curve.

Lemma5. We can canonically associate toE → San involutionσ_E:S → S and a fibrationγ_E:S→E.

Let us recall the construction ofσE andγE. For a generic points ofS, the line L_s cuts the hyperplane sectionψ(π⁻¹(E))into a pointp_s. The line betweenp_s and the vertex of the coneψ(π⁻¹(E))lies inside this cone and is represented by a pointγ_EsinE.The linesL_sandL_γEslie on a plane; that plane cuts the cubic into a third residual line, denoted byL_σEs.

Letsbe a point ofEand letC_sbe the incident divisor parameterizing the lines inF that cutL_s.

Lemma6. The fiberγ_E^∗ssatisfiesCs =γ_E^∗s+E.We haveC_s² =5,CsE=1, andE²= −3.

LetAbe the Albanese variety of the Fano surfaceS;its tangent space isH⁰(S,S)^∗. We denote byϑ: S → Athe Albanese map. Sinceϑ is an embedding [5], we considerSas a subvariety ofA.To the morphismsσEandγEcorrespond an invo- lutionE:A→AofAand a morphism _E:A→Esuch thatϑσE=Eϑ and _Eϑ =ϑγE.

Lemma7 [12, Lemma 29]. The differentialsd_Eandd _Eof_E and _E are endomorphisms ofH⁰(S,_S)^∗.They satisfy

I+dE+d E=0,

whereI is the identity. The eigenspace of the eigenvalue1of the involutiondE

is the tangent spaceTEof the curveE →A(translated in0).

Let us denote byf the projectivization ofd_E∈GL(H⁰(S,_S)^∗): it is an auto- morphism ofP⁴ =P(H⁰(S,_S)^∗).Letp_Ebe the point ofP⁴ corresponding to the 1-dimensional spaceTE⊂H⁰(S,S)^∗.

Lemma8. The involutionf preserves the cubic threefoldF → P⁴. The point p_E is the vertex of the coneψ(π⁻¹(E)). The hyperplaneP(Ker(d _E))andp_E constitute the closed set of fixed points off.

Conversely, letfbe an involution of P⁴acting onFand fixing an isolated point and a hyperplane. The isolated fixed point is the vertex of a cone onF.

We will use Lemma 7 as in the next example.

Example9. Letx1,...,x5 be homogenous coordinates of P⁴. Then the point (1 : 0 :· · ·: 0)is the vertex of a cone on the cubic threefold

F = {x₁²x2+G(x2,...,x5)=0}

(whereGis a cubic form such thatF is smooth). LetE →Sbe the elliptic curve parameterizing the lines of that cone. By Lemma 7, we see that the involutiondE

satisfies

d_E:(x1,x2,...,x5)→(x1,−x2,...,−x5), and we deduce thatd Eis defined by

d E:(x1,x2,...,x5)→(−2x1, 0,..., 0).

1.3. Theta Polarization

LetSbe a Fano surface, letAbe its Albanese variety, and letϑ: S →Abe the Albanese map. By [5, Thm. 13.4], the image %ofS ×S under the morphism (s1,s2)→ϑ(s1)−ϑ(s2)is a principal polarization ofA. Letτ be an automor- phism ofSand let τbe the automorphism ofAsuch thatϑ τ = τϑ.Let (s1,s2)be a point ofS×S;thenτ(ϑ(s1)−ϑ(s2))=ϑ(τ(s1))−ϑ(τ(s2)).This leads to the following statement.

Lemma10. The automorphismτpreserves the polarizationτ^∗%=%.

For a varietyX, we denote byH²(X,Z)_f the groupH²(X,Z)modulo torsion.

We denote by NS(X)=H^1,1(X)∩H²(X,Z)_fits Néron–Severi group and byρ_X its Picard number. For a divisorDinX, we denote its Chern class byc1(D).

Theorem11.

(a) IfDandDare two divisors ofA,then ϑ^∗(D)ϑ^∗(D)=

A

1 3!

3

c1(%)∧c1(D)∧c1(D).

(b) The following sequence is exact:

0→NS(A)−→^ϑ∗ NS(S)−→Z/2Z−→0.

(c) The Néron–Severi group ofSis generated byϑ^∗NS(A)and by the class of an incidence divisorCs(s∈S).The class of ϑ^∗(%)is equal to2Cs.

(d) We haveρ_A=ρ_S≤25=dimH¹(S,_S)andρ_S=1forSgeneric.

Proof. The morphismϑ is an embedding, and the homological class ofϑ(S)is equal to_3!¹%³[2, Prop. 7]. This proves part (a).

Since%is a polarization, the bilinear symmetric form Q_%: H²(A,C)×H²(A,C)→C

defined by

Q_%(η1,η2)=

A

1 3!

3

c1(%)∧η1∧η2

is nondegenerate (Hodge–Riemann bilinear relations; [8, Chap. 0, Sec. 7]). This implies that the morphism

ϑ^∗: H²(A,C)→H²(S,C)

is injective, and sinceSandAhave the same second Betti number (see [7, (2)]), it follows that the homomorphism

ϑ∗:H2(S,Z)f →H2(A,Z) is injective. By [6, 2.3.5.1] we have the exact sequence

H2(S,Z)_f −→^ϑ∗ H2(A,Z)→Z/2Z→0;

thus

0→H2(S,Z)f −→ϑ∗ H2(A,Z)→Z/2Z→0 is exact. By duality, this yields:

0→H²(A,Z)−→^ϑ∗ H²(S,Z)_f →Z/2Z→0.

Because the spacesH^1,1(A)andH^1,1(S)have dimension 25 (see [5]), we have ϑ^∗(H^1,1(A))=H^1,1(S).This implies that the sequence

0→NS(A)−→^ϑ∗ NS(S)→Z/2Z

is exact. This sequence is exact on the right also becauseϑ^∗% = 2C_s by [5, Lemma 11.27](%is a principal polarization and is not divisible by 2, so the class ofC_s andϑ^∗NS(A)generate NS(S)).This proves part (b).

By [6], any Jacobian of an hyperelliptic curve of genus 5 is a limit of the Al- banese varieties of Fano surfaces endowed with their principal polarization. By [9], the endomorphism ring of a Jacobian of a generic hyperelliptic curve is iso- morphic toZ. If the generic Albanese variety of Fano surface were not simple, then also its limit would be nonsimple. This is a contradiction, soρA =1 for a generic Fano surface.

2. Fano Surfaces with 12 Elliptic Curves Letλ∈Candλ³=1;then the cubic threefold

F_λ= {x₁³+x₂³+x³₃−3λx1x2x3+x₄³+x₅³=0}→P⁴ is smooth. Lete1,...,e5be the dual basis ofx1,...,x5.The 12 points

C(e4−βe5),C(ei−βej)∈P⁴, 1≤i < j ≤3, β³ =1

are vertices of cones in Fλ. Let Sλ be the Fano surface of Fλ. We denote by E_ij^β →Sλ the elliptic curve that parameterizes the lines of the cone of vertex

C(e_i−βe_j)(see Lemma 4). Let(y1: y2 : y3)be projective coordinates of the plane. LetE_λ→P²be the elliptic curve

E_λ= {y₁³+y³₂+y³₃−3λy1y2y3=0}

with neutral element(1 : −1 : 0). By [12], the 9 curvesE_ij^β (1 ≤ i < j ≤ 3, β³ =1)are disjoint and isomorphic toE0;the 3 curvesE^β₄₅(β³ =1)are disjoint and isomorphic toE_λ, and

E₄₅^βE_ij^γ =1 for all 1≤i < j ≤3, β³=γ³=1.

Conversely, letSbe a Fano surfaces with 12 elliptic curves. Then we have the fol- lowing result.

Lemma12 [12, 3.3]. There is a set of 3disjoint elliptic curves onSisomorphic toE_µ(for someµ∈C)such that the9remaining elliptic curves are disjoint and such that the surfaceSis isomorphic toS_µ.

LetY be the surfaceY =Eλ×Eλ.LetT1andT2be the elliptic curves T1= {x+2y =0/(x,y)∈E_λ×E_λ},

T2 = {2x+y=0/(x,y)∈E_λ×E_λ}

onYand let1 →Ybe the diagonal. Any 2 of the 3 curvesT1,T2,1meet transver- sally at the 9 points of 3-torsion of1.We denote byZthe blow-up ofY at these 9 points.

Proposition13. The Fano surfaceS_λis a triple cyclic cover ofZbranched along the proper transform of1+T1+T2inZ.

Proof. Letα∈µ3be a primitive root. The order-3 automorphism f:x →(αx1:αx2 :αx3:x4:x5)

acts onFλ.The automorphismfacts on the Fano surface of lines ofFλby an auto- morphism denoted byτ.Since we know the action off, we can check immediately that the fixed locus ofτ is the smooth divisorE₄₅¹ +E₄₅^α +E₄₅^α2.The quotient of Sλbyτ is a smooth surface Zwith Chern numbers c₁² = −9 andc2 =9, and the degree-3 quotient mapη:Sλ→Zis ramified overE¹₄₅+E₄₅^α +E₄₅^α2.

For an elliptic curveE →S, we denote byγE:S→Ethe associated fibration ( Lemma 5). By Lemma 6, the morphism

g=(γE₄₅^α,γE₄₅^α2):Sλ→Y

has degree 3=(Cs−E₄₅^α)(Cs−E₄₅^α2).Let beE=E₄₅^β (forβ³=1).

Letsbe a generic point ofS.By definition (see Lemma 4), the lineL_scuts the lineL_γEs. BecauseL_γEs is stable byf, the linef(L_s)= L_τs cuts also the line L_γEs;thus, by definition ofγ_E, we haveγ_Eτs=γ_Es.This proves thatγ_Eτ =γ_E andgτ =g. Hence, by the property of the quotient map, there is a birational morphism

h:Z→Y such thatg=hη.

Lett be the intersection point ofE₁₂¹ andE¹₄₅and letϑ: Sλ→ Aλbe the Al- banese map such thatϑ(t)=0.It is an embedding and we considerSλas a sub- variety ofAλ.The tangent space to the curveE₄₅^β →Aλ(translated to 0)isVβ = C(βe4−β²e5).The tangent space ofE₄₅^α ×E₄₅^α2 isVα⊕V_α².With the help of Lemma 7 and Example 9, it is easily checked that the images undergof the curves E₄₅¹ ,E₄₅^α, andE₄₅^α2are respectively1,T1, andT2.

Moreover, the morphismghas degree 1 on these 3 elliptic curves and contracts the 9 elliptic curvesE_ij^β (1 ≤i < j ≤ 3,β³ =1).This implies that the image undergofE₄₅¹ +E₄₅^α +E₄₅^α2 isT1+T2+1and thatZis isomorphic toZ.

LetDbe the proper transform of1+T2+T2inZ. By Proposition 13 and [1, Chap. I, Para. 17 & 18], the divisorDis divisible by 3 in NS(Z).

SinceYis an Abelian surface, there exist 3⁴invertible sheavesLonZsuch that L^⊗3=OZ(D).LetS(L)→Zbe the degree-3 cyclic cover ofZthat is branched overDand associated to suchL.

Corollary14. The surfaceS(L)contains12elliptic curves, E₄₅^β,E_ij^γ, 1≤i < j≤3, β³=γ³=1, that have the same configuration as forS_λ;moreover, the divisor

K=

β³=1

2E₄₅^β +E₁₂^β +E₁₃^β +E₂₃^β is a canonical divisor ofS(L).

Among these81invertible sheaves, if there exists a uniqueLsuch thatS(L)is a Fano surface, then that surface is isomorphic toS_λ.

Proof. See [1, Chap. I, Para. 17 & 18]. For the uniqueness of the invertible sheafL, suppose thatS(L)andS(L)are Fano surfaces. By construction, they contain 12 elliptic curves, 3 of which are isomorphic toEλand cut the 9 others. Thus, by Lemma 12,S(L)andS(L)are isomorphic toSλand thereforeL=L.

Remark15. The remaining 80 surfacesS(L)are thus “fake” Fano surfaces and are on different components of the moduli space of surfaces withc₁² = 45 and c2=27.

Letαbe a third primitive root of unity. Let us now study the Néron–Severi group ofS.

Proposition16. (1)Suppose thatE_λhas no complex multiplication. The Néron–

Severi group of S_λhas rank12. The sublattice generated by the elliptic curves and the class of an incidence divisorC_s has rank12and discriminant6¹⁰.

(2)If E_λ has complex multiplication by a field different fromQ(α),then the Néron–Severi group ofS_λhas rank13.

(3)IfE_λhas complex multiplication byQ(α),then the Néron–Severi group of S_λhas rank25.

Proof. We can easily compute the Picard number of the Abelian varietyE₀³×E_λ². By [12], the Albanese varietyAofSλis isogenous toE₀³×E_λ².Hence their Néron–

Severi groups have the same rank and, according to the cases (1), (2), and (3), this rank is 12, 13, or 25. Now Theorem 11 implies that the Picard number ofSλis 12, 13, or 25, respectively.

3. The Fano Surface of the Fermat Cubic 3.1. Elliptic Curve Configuration of the Fano Surface

of the Fermat Cubic

LetSbe the Fano surface of the Fermat cubicF →P⁴=P(H⁰(S,S)^∗): x₁³+x₂³+x₃³+x₄³+x₅³=0.

Lete1,...,e5∈H⁰(S,S)^∗be the dual basis of basis ofx1,...,x5. Letµ3be the group of third roots of unity, let 1≤i < j ≤5, and letβ∈µ3.The point

p^β_ij=C(e_i−βe_j)∈P⁴

is the vertex of a cone on the cubicF.We denote byE_ij^β→S the elliptic curve that parameterizes the lines on that cone. The complex reflection groupG(3, 3, 5) (in the basise1,...,e5 of H⁰(_S)^∗)is the group generated by the permutation matrices and the matrix with diagonal elementsα,α,α,α,α²(whereα∈µ3is a primitive root).

We recall the following.

Proposition17. The Fano surface of the Fermat cubic possesses30 smooth curves of genus1numbered

E_ij^β, 1≤i < j ≤5, β∈µ3.

(i) Each smooth genus-1curve of the Fano surface is isomorphic to the Fermat plane cubicE:= {x³+y³+z³=0}.

(ii) LetE_ij^γ andE_st^β be two smooth curves of genus1.Then

E_ij^βE^γ_st=







1 if {i,j} ∩ {s,t} = ∅,

−3 if E_ij^β =E^γ_st, 0 else.

(iii) LetE →Sbe a smooth curve of genus1.The fibrationγ_Ehas20sections and contracts9elliptic curves.

(iv) The automorphism group of S is isomorphic to the complex reflection G(3, 3, 5).

Proof. See [12]. For part (iv) we use the fact that an automorphism ofF must pre- serve the configuration of the 30 vertices of cones and the 100 lines that contain 3 such vertices.

3.2. The Albanese Variety of the Fano Surface of the Fermat Cubic LetSbe the Fano surface of the Fermat cubicF.Our main aim is to compute the full Néron–Severi group ofS, which will be done in Section 3.3. We first need to study the Albanese varietyAofS.

3.2.1. Construction of Fibrations

In order to know the period lattice of the Albanese variety ofS, we construct mor- phisms of the Fano surface onto an elliptic curve and then study their properties.

Letϑ:S→Abe a fixed Albanese map. It is an embedding, and we considerS as a subvariety ofA.Recall that ifτ is an automorphism ofS, we denote byτ∈ Aut(A)the unique automorphism such thatτϑ =ϑτ.

By [12], the reflection groupG(3, 3, 5)is the analytic representation of the auto- morphismsτ,τ ∈Aut(S).The ringZ[G(3, 3, 5)] ⊂End(H⁰(_S)^∗)is then the analytic representation of a subring of endomorphisms of the Abelian varietyA.

Let us denote by7^∗_Athe rank-5 sub-Z[α]-module ofH⁰(_S)generated by the forms

x_i−βx_j (i < j,β∈µ3).

Let8be an element of7^∗_A. The endomorphism ofH⁰(S)^∗ = H⁰(S,S)^∗ de- fined by x → 8(x)(e1−e2)is an element of Z[G(3, 3, 5)]. Let us denote by

8: A→Ethe corresponding morphism of Abelian varieties, whereE→Ais the elliptic curve with tangent spaceC(e1−e2).We denote byγ₈: S → Ethe morphism ₈ϑ.

For 1≤i < j≤5 andβ∈µ3, the space

C(e_i−βe_j)⊂H⁰(_S)^∗

is the tangent space to the elliptic curveE_ij^β→Atranslated to 0 ( Lemma 7).

LetH1(A,Z)⊂H⁰(S)^∗be the period lattice ofA. The elliptic curveEhas complex multiplication by the principal ideal domainZ[α].There exists ac∈C^∗ such that

H1(A,Z)∩C(e1−e2)=Z[α]c(e1−e2).

Up to the basis change ofe1,...,e5byce1,...,ce5, we may suppose thatc=1.

SinceG(3, 3, 5)acts transitively on the 30 spacesC(ei−βej), we have H1(A,Z)∩C(ei−βej)=Z[α](ei−βej).

We define the Hermitian product of two forms8,8∈7^∗_Aby 8,8:=^k=

5 k=1

8(e_k)8(e_k) and the norm of8by8 =

8,8.LetC_sbe an incidence divisor.

Theorem18. Let8be a nonzero element of7^∗_Aand letF₈be a fiber of γ₈. (a) The intersection number ofF₈andE_ij^β→Sis equal to

E_ij^βF8= |8(ei−βej)|². (b) We haveF₈C_s =28²,and the fiberF₈has genus

g(F₈)=1+38².

(c) Let 8and 8be two linearly independent elements of7^∗_A ⊂ H⁰(_S). The morphismτ₈,8 =(γ₈,γ₈):S→E×Ehas degree equal toF₈F₈,and

F8F8 = 8²8²− 8,88,8.

Remark19. The known intersection numbersF₈E_ij^βandF₈C_senable us to write the numerical equivalence class of the fiberF₈in theZ-basis given in Theorem 29 (to follow).

Proof of Theorem 18. Proving part (a) involves a trick. For 1≤ i < j ≤ 5 and β∈µ3, we can interpret the intersection numberE_ij^βF8geometrically as the de- gree of the restriction ofγ8toE_ij^β→S. It is also the degree of the restriction of

8toE_ij^β → A. Since this restriction is the multiplication map by8(ei −βej), the degree of the morphism ₈onE_ij^β is equal to|8(e_i −βe_j)|². ThusF₈E_ij^β =

|8(e_i−βe_j)|².

Let us now study the genus ofF₈.

Lemma20. The fiber ofF8has genus1+38²andCsF8=28²(wheresis a point ofSandCsis the incidence divisor).

Proof. Letbe the sum of the 30 elliptic curves onS.We have F8=

i,j,β

F8E_ij^β =

i,j,β

|8(ei−βej)|²=128².

BecauseF₈is a fiber, we haveF₈² =0.Since is twice a canonical divisor [5], we deduce thatF₈has genus 1+¹₂(0+¹₂F₈)=1+38².

The divisor 3C_sis numerically equivalent to a canonical divisor [5]. Therefore C_sF₈=28².

We identify the Chern class of a divisor of the Abelian varietyAwith an alternat- ing form on the tangent spaceH⁰(S)^∗ofA(cf. [3, Thm. 2.12]). Let%be the principal polarization defined in Section 1.3.

Lemma21. The Chern class of %is equal to a i

√3 5 j=1

dxj∧dx¯j, whereais a scalar andi²= −1.

Proof. LetH be the matrix (in the basise1,...,e5)of the Hermitian form asso- ciated toc1(%) (see [3, Lemma 2.17]). The automorphismτinduced by τ ∈ Aut(S)preserves the polarization%( Lemma 10). This implies that, for allM = (mjk)1≤j,k≤5∈G(3, 3, 5), we have

tMHM¯ =H,

whereM¯ is the matrixM¯ =(m¯jk)1≤j,k≤5.This proves that H = 2

√3aI5,

whereI5is the identity matrix anda∈C.Therefore,c1(%)=a^√i₃ ⁵_j=1dxj∧dx¯j. BecauseH1(A,Z)∩C(e1−e2)=Z[α](e1−e2), the Néron–Severi group of the elliptic curveEis theZ-module generated by

η= i

√3dz∧dz¯, wherezis the coordinate on the spaceC(e1−e2).

Let8=a1x1+ · · · +a5x5be an element of7^∗_A.The pull-back of the formη by the morphism ₈: A→Eis

8∗η= i

√3d8∧d8.¯

The form ₈^∗ηis the Chern class of the divisor ₈^∗0 andγ₈^∗η=ϑ^∗ ₈^∗ηis the Chern class of the divisorF8.

Lemma22. Let8and8be two elements of 7^∗_A.Then F8F8= 8²8²− 8,88,8 andc1(%)= ^√i₃ ⁱ⁼_i=1⁵dx_i∧dx¯_i.

Proof. By Theorem 11,ϑ^∗c1(%)is the Chern class of the divisor 2Cs(s∈S)and 2CsF8=ϑ^∗c1(%)ϑ^∗ ₈^∗η=

A

1 3!

4

c1(%)∧ ₈^∗η;

as a result, we have 2C_sF₈=

i

√3 ₅

A

a_jdx_j

∧

¯ a_jdx¯_j

∧4a⁴

1≤k≤5 j=k

(dx_j∧dx¯_j)

and

2C_sF₈= 4

a k=5 k=1

a_ka¯_k 1

5!

A

5

c1(%).

Since%is a principal polarization, we have _5!¹

A

₅

c1(%)=1;hence 2C_sF₈ =

4a8².We have seen in Lemma 20 thatC_sF₈=28².Thus we deduce thata=1.

By Theorem 11, for8=a1x1+ · · · +a5x5and8=b1x1+ · · · +b5x5∈7^∗_A, F₈F₈ =

A

1 3!

3

c1(%)∧ ₈^∗η∧ ₈^∗η.

Since 1 3!

i

√3 2

d8∧d8¯∧d8∧d8¯∧³ c1(%)

=

k=j

a_ka¯_kb_jb¯_j −a_ka¯_jb_jb¯_k 1

5!

5

c1(%), the result follows.

Let8and8be two linearly independent elements of7^∗_A.The degree of the mor- phismτ₈,8 =(γ₈,γ₈)is equal toF₈F₈ becauseτ₈^∗_,8(E× {0})=F₈ ∈NS(S), τ₈^∗_,8({0} ×E)=F8∈NS(S), and the intersection number of the divisors{0} ×E andE× {0}is equal to 1.

This completes the proof of Theorem 18.

3.2.2. Period Lattice ofA.

We compute here the period lattice of the Albanese varietyAin the basise1,...,e5. Theorem23.

(1) The latticeH1(A,Z)is equal to:

Z[α](e1−e5)+Z[α](e2−e5)+Z[α](e3−e5)+Z[α](e4−e5) +1+α

1−αZ[3α](α²e1+α²e2+αe3+αe4+e5).

(2) The varietyAis isomorphic toE⁴×E,whereE=C/Z[α]andE=C/Z[3α].

(3) The image of the morphismϑ^∗: NS(A)→NS(S)is the sublattice of rank25 and discriminant2²3¹⁸generated by the divisors

F_xi−β²_xj =Cs−E_ij^β, 1≤i < j ≤5, β∈µ3,

i<j

E¹_ij. Proof. The groupG(3, 3, 5)acts onH1(A,Z), and

H1(A,Z)∩C(e_i−βe_j)=Z[α](e_i−βe_j);

henceH1(A,Z)contains the lattice 70=

i≤j,β∈µ3

Z[α](e_i−βe_j).

For 1≤ i < j ≤5 andβ∈µ3, the differential of _xi−βxj is the morphismx → (x_i−βx_j)(e1−e2).Thus,

λ_i−βλ_j∈Z[α] ∀λ=(λ1,...,λ5)∈H1(A,Z).

Let us define

7= {x =(x1,...,x)∈C⁵/x_i−βx_j∈Z[α], 1≤i < j ≤5, β∈µ3}.

This lattice7containsH1(A,Z)and is equal to Z[α]e1⊕ · · · ⊕Z[α]e4⊕ 1

α−1Z[α]w,

wherew =e1+ · · · +e5. Letφ:7 → 7/70be the quotient map. The group 7/70is isomorphic to(Z/3Z)²and contains 6 subgroups. The reciprocal images of these groups are the lattices

70=φ⁻¹(0), 7_α² =70+ α² α−1Zw, 71=70+ 1

α−1Zw, 7α−1=70+Zw, 7_α=70+ α

α−1Zw, 7=70+ 1

α−1Z[α]w.

These are the 6 lattices7that verify70⊂7⊂7, so the latticeH1(A,Z)must be equal to one of them.

Letωbe the alternating formω= ^√i₃ ^k=_k=1⁵dx_k∧dx¯_k(see Lemma 22). We have 1

α−1w, α

α−1w∈7.

However,

ω 1

α−1w, α α−1w

= −5 3 is not an integer and so7is different fromH1(A,Z).

The Pfaffian ofc1(%)relative toH1(A,Z)is equal to 1 because%is a principal polarization. The Pfaffian ofωrelative to the lattice70is equal to 9; hence70is different fromH1(A,Z).

We have 71−α =

Z[α]e_i, and the principally polarized Abelian variety (C⁵/71−α,ω)is isomorphic to a product of Jacobians. Since(A,c1(%))cannot be isomorphic to a product of Jacobians ([5], 0.12), it follows thatH1(A,Z)=71−α.

The lattice7_αⁱ is equal to

Z[α](e1−e5)+Z[α](e2−e5)+Z[α](e3−e5)+Z[α](e4−e5) + αⁱ

1−αZ[3α](α²e1+α²e2+αe3+αe4+e5).

The lattices71and7αdepend on a choice ofαsuch thatα²+α+1=0, hence the latticeH1(A,Z)is equal to7_α².

Let

u1=e1−e2, u2 =e2−e3, u3=e3−e4, u4=e4−e5, u5= α²

1−α(α²e1+α²e2+αe3+αe4+e5).

The Hermitian formH= ^√2₃I5 in the basisu1,...,u5defines a principal polar- ization ofA.Let End^s(A)be the group of symmetrical morphisms for the Rosati

involution associated toH.An endomorphism ofAcan be represented by a size-5 matrixMin the basisu1,...,u5.The symmetrical endomorphisms satisfy^tMH= HM;¯ that is,^tA= ¯A.Since we already knowH1(A,Z), we can easily compute a basisBof End^s(A).

By [3, Prop. 5.2.1] and [3, Rem. 5.2.2], the map φ_H: End^s(A)→NS(A)

M→ m(·^tMH¯·)

is an isomorphism of groups. We obtain the base of the Néron–Severi group ofA by taking the image byφ_Hof the baseB.Then we get the image of the morphism ϑ^∗: NS(A)→NS(S).

Remark24. By Theorem 23, the varietyAis biregular toE⁴×E.Yet by [5, (0.12)], this cannot be an isomorphism of principally polarized Abelian varieties.

3.2.3. Study of Some Fibrations, Remarks

LetXbe a smooth surface,C a smooth curve, andγ: X → C a fibration with connected fibers. A point ofXis called acritical point of γ if it is a zero of the differential

dγ:T_X→γ^∗T_C.

A fiber ofγ is singular at a point if and only if this point is a critical point [1, Chap. III, Sec. 8].

Let us suppose thatCis an elliptic curve. The critical points ofγ are then the zeros of the formγ^∗ω∈H⁰(X,_X), whereωis a generator of the trivial sheaf_C. Let us assume that the cotangent sheaf ofX is generated by global sections.

There are morphisms

P(TX) ^ψ //

π

P(H⁰(X,X)^∗)

X

where π is the natural projection and the morphism ψ, called the cotangent map[13], is defined byπ_∗ψ^∗O(1)=_X.

A pointx ofXis a critical point ofγ if and only if the lineLx =ψ(π⁻¹(x)) lies in the hyperplane

{γ^∗ω=0}→P(H⁰(X,_X)^∗).

The initial motivation for studying Fano surfaces is that, for those surfaces, the cotangent map is known: it is the projection map of the universal family of lines.

The example of Fano surfaces illustrates that the knowledge of the image of the cotangent map is powerful for the study of a surface.

LetSbe the Fano surface of the Fermat cubicF.

Notation25. For 1≤i < j ≤5, we defineBij=Bji = _β∈µ3E_ij^β.

Let 1≤i≤5 andj < r < s < tbe such that{i,j,r,s,t} = {1, 2, 3, 4, 5}.We define8_i =(1−α)x_i∈7^∗_A.

Corollary26. The fibrationγ8i:S →Eis stable and has connected fibers of genus10,and its only singular fibers are

Bjr+Bst, Bjs+Brt, Bjt+Brs.

The27intersection points of the curvesE_jr^β andE^τ_st(β,γ∈µ3)constitute the set of critical points of this fibration.

Proof. By Theorem 18(a), a fiber ofγ8ihas genus 1+3|1−α|²=10.

Letβ ∈ µ3,h,k ∈ {j,r,s,t}, andh < k. The form8i is zero on the space C(eh−βek);henceE_hk^β is contracted to a point and is a component of the fiber ofγ8i.

The divisorD1=Bjr+Bst is connected, satisfies(Bjr+Bst)² =0, and has genus 10. Its irreducible components are contracted byγ8i. Hence, it is a fiber andγ8i has connected fibers. Likewise, the divisorsD2 =Bjs +Brt andD3 = Bjt+Brsare fibers ofγ8i.

The 27 lines inside the intersection of the Fermat cubic and the hyperplane {8i =0}correspond to the 27 intersection points of the curvesE_hk^β andE^γ_lmsuch thath < k,l < m, and{i,h,k,l,m} = {1, 2, 3, 4, 5}.These 27 critical points lie in the fibersD1,D2,D3, which are thus the only singular fibers ofγ_8i.

The singularities ofD1,D2, andD3are double ordinary, and the surface pos- sesses no rational curve. Therefore, the fibration is stable.

Let(a1,...,a5)∈µ⁵₃be such thata1... a5=1, and let 8=(1−α)(a1x1+ · · · +a5x5)∈7^∗_A. Corollary27. The divisor

D=

1≤i<j≤5

E_ij^ai/aj

is a singular fiber of the Stein factorization of γ₈.

Proof. The connected divisorDsatisfiesD²=0 and has genus 16. Moreover, by Theorem 18, the irreducible components ofDare contracted byγ8.

Letw∈H⁰(S)^∗bew=e1+ · · · +e5.Then H¹(A,Z)∩Cw= α²

1−αZ[3α]w.

The morphismx → (a1x1+ · · · +a5x5)w∈ End(H⁰(_S)^∗)is an element of Z[G(3, 3, 5)]. It is the differential of a morphism ₈: A → E, where E = C/_1−α^α2 Z[3α]

w.

The morphism ₈has a factorization by ₈and a degree-3 isogeny betweenE andE.The divisorDis a connected fiber of the morphismϑ ₈, the Stein fac- torization ofγ₈.