Elliptic curve configurations on Fano surfaces

Xavier Roulleau

Elliptic curve configurations on Fano surfaces

Received: 5 May 2008 / Revised: 11 February 2009 Published online: 1 April 2009

Abstract. The elliptic curves on a surface of general type constitute an obstruction for the cotangent sheaf to be ample. In this paper, we give the classification of the configurations of the elliptic curves on the Fano surface of a smooth cubic threefold. That means that we give the number of such curves, their intersections and a plane model. This classification is linked to the classification of the automorphism groups of theses surfaces.

1. Introduction

Considering a varietyS, it is a natural question to study the ampleness of its cotan- gent sheaf. Recall that, by definition, a bundleE onSis ample if the tautological sheafOP(E^∗)(1)is an ample line bundle of the projective bundleP(E^∗)of one dimensional sub-spaces ofE^∗. Gieseker gives the following criteria of ampleness if in addition the bundleEis generated by its space of global sections:

Proposition 1.(Gieseker [8])The bundleEis ample if and only if for every curve C!→S, the bundleE⊗O^C has no quotient isomorphic to the trivial sheafO^C.

Hence, we consider varieties with the following assumptions:

Hypothesis 2.The variety S is a smooth complex surface of general type. The cotangent sheaf"_S of S is generated by its global sections and the irregularity q =dimH⁰("S)satisfiesq >3.

With the criteria of Gieseker in mind, a curveC !→ S is called non-ample if and only if the sheaf"S⊗O^Chas a quotient isomorphic to the trivial sheafO^C. These curves are the obstruction to the ampleness of"S. For example, a smooth curve of genus 1 onSis a non-ample curve.

LetTS="^∗_Sbe the tangent sheaf and letπ :P(TS)→Sbe the projection. As π_∗O_P(T_S)(1)="Sand the cotangent sheaf is generated by its global sections, we can define a map:

ψ:P(T_S)→P(H⁰("_S)^∗)=P^q−1

called the cotangent map ofS, such thatOP(T_S)(1)=ψ^∗(O_P^q−1(1)).

X. Roulleau: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan. e-mail: [email protected] Mathematics Subject Classification (2000):Primary 14J29; Secondary 14J45, 14J50, 14J70, 32G20

DOI: 10.1007/s00229-009-0264-5

This map is the object of the paper [10]. In the present paper, we study the cotangent map and the configurations of the non-ample curves on Fano surfaces.

These surfaces were discovered by Fano and interest in them has been stim- ulated by the works of Clemens and Griffiths [4] Tyurin [12], [13] in 1971. By definition, a Fano surface is the Hilbert scheme of lines of a smooth cubic threefold F !→P⁴. This scheme is a surfaceSthat verifies Hypothesis2and has irregularity q =5.

By [4, Tangent Bundle Theorem 12.37], the image of the cotangent mapψ : P(TS) → P(H⁰("S)^∗)of S is a hypersurface F^% of P(H⁰("S)^∗) & P⁴ that is isomorphic to the original cubic F. Moreover, when we identify F and F^%, the triple(P(TS), π, ψ )is the universal family of lines onF.

The main results presented here are:

Theorem 3.(A)The non-ample curves of a Fano surface S are the smooth curves of genus1.

(B)There are only10configurations of smooth curves of genus1on the Fano sur- faces. We know a plane model of each curve, the number of these curves and their intersection numbers.

(C)For each Fano surface S, we construct a particular sub-groupGSof its auto- morphism group. This construction goes in such a way that the knowledge of the groupGSgives the knowledge of the elliptic curve configurations on S and reciprocally, the knowledge of the elliptic curve configuration on S determines the groupGS.

The results are summarized in the Classification Theorem26. The numbern_Sof elliptic curves on a Fano surfaceSverifies 0≤n_S ≤30. The surfaces for which n_S>0 constitute a seven-dimensional family in the 10 dimensional moduli space of Fano surfaces.

The Picard numberρSof a Fano surfaceS satisfies 1≤ρS ≤25 and we can prove that it is 1 forSgeneric. In fact, the number of elliptic curves is linked with the Picard numberρS: we haveρS≥nSunlessnS=30, in which caseρS=25. The numbern_S is also the number of one-dimensional fibres of the cotangent mapψ.

This shows that the geometric properties ofψand the ampleness of the cotangent bundle vary non-trivially with the Fano surface.

We remark also that the results (A) and (B) of Theorem3are analogous to the classical statement on the canonical bundle of a minimal surface of general type which is ample if and only if the surface does not contain a(−2)-curve and the classification of canonical surfaces singularities.

We end this paper by an application of our study of Fano surfaces to construct cubic threefold whose intermediate Jacobian is isomorphic to a product of elliptic curves as an Abelian variety. The interest of this results is that in order to prove that the cubic threefolds are not rational, Clemens and Griffiths use the fact that their intermediate Jacobian cannot be isomorphic to a product of Jacobians of curves as a principally polarized Abelian variety.

Part of this paper was written at the Max-Plank Institute of Bonn, which is gratefully acknowledged.

2. Properties of the Fano surfaces and of their cotangent map

2.1. Properties of the cotangent map

LetSbe a surface which verifies Hypothesis2. In the introduction, we defined the cotangent map

ψ:P(TS)→P(H⁰("S)^∗)=P^q−1

by the surjective morphismH⁰("S)⊗O_P(TS) →O_P(TS)(1). Here, we state general results about this morphism which will be used in the sequel; a complete treatment can be found in [10].

Let us recall thatπ :P(T_S)→ Sis the projection. Letsbe a point ofS. The restriction of the invertible sheafOP(T_S)(1)to the fibreπ⁻¹(s)&P¹is the degree 1 invertible sheaf and the image underψof that fibre is a line; we denote this line byLs !→P^q−1.

LetG(2,q)=G(2,H⁰("S)^∗)be the Grassmannian of projective lines inP^q−1. By the universal property of the Grassmannian, the cotangent map induces a map

G:S→G(2,q)

called the Gauss map ofS. The image byGof a pointsinSis the lineLs. LetAbe the Albanese varietyS; its tangent space at 0 isH⁰("_S)^∗.

Proposition 4.Let C!→S be a curve. The image underψofπ⁻¹(C)is a cone if and only if C is a non-ample curve.

An elliptic curve E !→ S is a non-ample curve. Let pE be the vertex of the coneψ (π⁻¹(E)). The underlying space of the point pEis the tangent space to the elliptic curveϑ (E)translated in0.

Proof. The first assertion is [10, lemme 2.1].

LetE be an elliptic curve on S. The natural quotient"_S⊗O^E → "_E is a trivial quotient, henceEis non-ample. The last assertion results from the definition

of the cotangent map. )*

Proposition 5.[10, cor. 2.12]Let C!→S be a non-ample curve on a surface with irregularity q > 4. Suppose that the vertex p of the cone T = ψ (π⁻¹(C))is a smooth point of the image ofψ. Then T is contained in the projective tangent space at p of F !→P^q−1and one of the two following possibilities occurs:

(a)C²<0and C is a smooth curve of genus1.

(b)C²=0and an integral multiple of C is a fiber of a fibration f :S→ B onto a curve of genus b with q−3≤b≤q−2.

LetC !→Sbe a curve and letK be a canonical divisor of S.

Proposition 6.[10, prop. 1.20]The degree of the cycleψ_∗π^∗C equals K C.

2.2. Main properties of Fano surfaces

We recall here some properties of Fano surfaces needed in the sequel. The main references are the works of Clemens and Griffiths [4] and of Tyurin [12].

Let F be a smooth cubic hypersurface ofP⁴and let S be its Fano surface of lines. This surface verifies Hypothesis2and has irregularityq =5. Moreover:

Theorem 7.[4, Tangent Bundle Theorem 12.37]The image of the cotangent map of S is a cubic hypersurface F^% !→ P(H⁰("S)^∗) &P⁴ isomorphic to the cubic F !→P⁴. Under the identification of H⁰("S)^∗and H⁰(P⁴,O_P⁴(1)), and F^%and F, the triple(P(TS), π, ψ )is the universal family of lines of F.

The Chern numbers of a Fano surface verify :c²₁=45,c₂=27. The cotangent map has degree 6: there are 6 lines through a generic point ofF.

Lemma 8.[4, Thm. 12.37], [2, Cor. Parag. 4]The Gauss map and the Albanese map are embeddings.

Letsbe a point ofSand letCs be the closure of pointst +=sinSsuch that the lineLt intersects the lineLs.

Proposition 9.[4, Parag. 10]The incidence divisor C_sis ample, has self-intersec- tion C_s²=5and arithmetical genus11. The divisor3Cs is numerically equivalent to a canonical divisor.

2.3. Properties of a non-ample curve on a Fano surface

LetS be a Fano surface and let F !→ P⁴be the image of its cotangent mapψ.

For a pointpofF, we denote byTF,p!→P⁴the projective tangent hyperplane to F at p. ForE !→ S a non-ample curve, we denote bypE the vertex of the cone ψ (π⁻¹(E)).

Proposition 10.A curve E on S is non-ample if and only if it is an elliptic curve.

Let E !→ S be elliptic curve. The coneψ (π⁻¹(E)) is the section of F by the hyperplane T_F,pE, furthermore : E²= −3, C_sE =1.

Proof. Let E !→ S be a non-ample curve. By Proposition5, the intersection of F byT_F,pE contains the coneψ (π⁻¹(E)). As a smooth cubic threefold does not contain a plane, the hyperplane section of F by T_F,pE is irreducible and is the coneψ (π⁻¹(E)). This cone has degree 3. As the Gauss map is an embedding, the restriction ofψonπ⁻¹(E)is birational onto its image, thusψ_∗π^∗E=ψ (π⁻¹(E)) and Proposition6implies that

K E =degψ_∗π^∗E =3, whereK is a canonical divisor ofS.

By Proposition5, the curve E satisfies E² ≤ 0. The number 2pa(E)−2 = E²+K E = E²+3 must be divisible by 2, henceE² +=0. Proposition5imply that E is an elliptic curve, thus: E² = −3. Since K is numerically equivalent to

3Cs, we obtain:ECs =1. )*

The following proposition is [4, Parag. 8, 10]:

Proposition 11.A Fano surface contains at most30smooth curves of genus1.

2.4. The automorphism groups of the cubic and of the Fano surface

Let us denote by Aut(X)the automorphisms group of a varietyX. LetSbe a Fano surface and letFbe the image of the cotangent map ofS.

An elementh∈Aut(F)preserves the lines and by the Tangent Bundle Theorem 7, it acts onSby an element denoted byρ(h).

Let beτ ∈ Aut(S). The automorphismτ acts on the space H⁰("_S)^∗ by an automorphism denoted byτ^∗ ∈G L(H⁰("_S)^∗). We remark also thatτ induce an automorphism of the Albanese variety ofS: the morphismτ^∗is the differential of that automorphism.

We denote by!τ ∈ PG L(H⁰("S)^∗)the projectivisation ofτ^∗.

The following proposition is an immediate consequence of the definitions and the Tangent Bundle Theorem7; we skip its proof because of its length.

Proposition 12.The morphismρ :Aut(F)→Aut(S)is an isomorphism and its inverse is the morphismAut(S)→Aut(F);τ →!τ.

In particular, for a pointsofSandτ ∈Aut(S), we have :Lτs =!τ (Ls).

3. Configurations of the elliptic curves

3.1. Configurations of2or3elliptic curves

For two sub-varietiesV₁,V₂ofP⁴, we denote by-V₁,V₂.their linear hull.

LetE !→Sbe an elliptic curve on the Fano surfaceSand let pEbe the vertex of the coneψ (π⁻¹(E)): this cone is the intersection ofFandTF,p_E.

Letsbe a point ofSoutside the curveE: the lineLscorresponding to the point sis not inside the coneψ (π⁻¹(E))andXs := -Ls,pE.is a plane. This plane cuts the cubicFin three lines:

(1) the lineLs,

(2) the lineLγ_Es(on the cone) through the vertex pEand the intersection point of Lsand the hyperplaneTF,p_E,

(3) the residual lineLσ_Essuch that:

XsF=Ls +L_γEs+L_σEs.

As an Albanese morphism ofSis an embedding, the surfaceSdoes not contain a rational curve ; furthermoreE has genus>0. Then [6, Cor. 1.44] implies that the rational mapsσE :S →SandγE :S → Eare everywhere defined. As the plane Xs (sinS\E) is equal toXσ_Es, the morphismσ_E²is the identity onS\E, thusσE

is an involutive automorphism.

Lets,t be two points of the curveE !→S. The line Ls cuts the lineLt at the vertex of the coneψ (π⁻¹(E)): thus the pointslies on the incidence divisorCtand there exists a residual divisorR_t such that:

C_t =E+R_t.

Theorem 13. (a)Let t be a point of E!→S. The divisor R_t is the fibre at t ofγ_E and has arithmetical genus7. The morphismsγ_Eandσ_E satisfyγ_Eσ_E =γ_E. (b)The automorphism −σ_E^∗ ∈ G L(H^o("_S)^∗)is a complex order 2 reflection.

The eigenspace ofσ_E^∗ with eigenvalue1is the underlying space of the vertex pE ∈P⁴.

(c)Let E^%!→S be an elliptic curve, distinct from E. Then0≤E E^%≤1.

(d)The automorphismsσE andσ_E% verify:

(σEσE%)^{3−E E%} =I dS. (e)If E E^%=1, then the fibrationγ_Econtracts E^%.

(f)If E E^%=0, then there exists a third elliptic curve E^%%such that σ_E(E^%)=σ_E%(E)=E^%%.

Moreover the curves E^%and E^%%are sections ofγ_E.

Let us prove Theorem13. Lettbe a point ofE!→S. Letsbe a generic point of R_t =C_t−E. The lineL_scuts the lineL_t !→ψ (π⁻¹(E))in a point different from p_Eand by definition:γ_Es=t, thussis a point ofγ_E⁻¹(t). This proves thatR_tis a component ofγ_E⁻¹(t). Conversely, we haveR²_t =(Ct−E)²=5−2+(−3)=0, by [1, Chap. III, Zariski’s Lemma 8.2], that implies thatR_t is the fibre attofγE.

A canonical divisorKis numerically equivalent to 3Ct, henceK Rt =K(Ct− E)=12 andRt has arithmetical genus¹²⁺⁰₂ +1.

The plane Xs(sinS\E) is equal to the planeXσ_Es, henceγEs=γEσEs.

LetE^%!→Sbe an elliptic curve. AsRt is a fibre, we have:

RtE^%=(Ct −E)E^%=1−E E^%≥0.

If E = E^%, we see thatγE has degree 4 on E. Suppose now thatE += E^%, then E E^%≥0 and hence 0≤ E E^%≤1. We proved (a) and (c).

If E E^% =1, then RtE =0 and E^% is contained in a fibre ofγE, hence (e). If E E^%=0, thenRtE^%=1, thusE^%is a section ofγE.

Let us prove the property (b). Up to a variable change, we can suppose that the vertexpE of the coneψ (π⁻¹(E))is the point(1:0:0:0:0). It is easy to prove that, up to a variable change, an equation ofFis:

F= {x₁²x₂+G(x₂,x₃,x₄,x₅)=0}.

The automorphismh : x → (−x₁ :x₂ : x₃ : x₄ : x₅)acts onF. The geometric interpretation ofhis given as follows: ifq is a generic point ofF, then the points p,q,h(q)lie on a line. Letsbe a generic point ofS. It is then easy to check that the linesLs,h(Ls)and the point pEspan a plane : this is thus the planeXsand we see thath(Ls)=L_σEs, thus :h ="σE (see the notations of Theorem12).

The automorphismσ"E =hfix the pointpEand the hyperplane{x₁=0}!→P⁴. The fixed locus ofσEis thus the union ofEand the 27 points corresponding to the lines in the intersection of F and the hyperplane{x₁ =0}(as we can verify this intersection is a smooth cubic surface).

By Proposition12, the automorphismσ_E^∗ ∈G L(H⁰("_S)^∗)is equal to:

g:x→(−x₁,x₂,x₃,x₄,x₅)

or to−g. Letϑ : S → Abe an Albanese morphism. Asσ_E is the identity onE, the automorphismσ_E^∗ ∈ G L(H⁰("_S)^∗)(which is the differential of the action of σ_EonA) is the identity on the tangent space of the curveϑ (E)translated at 0. By Proposition4, this space isC(1,0,0,0,0)⊂H⁰("_S)^∗. Thusσ_E^∗ = −gand−σ_E^∗ is a complex reflection of order 2. We proved b).

Let E^% += E a second smooth curve of genus 1 on S. Let us prove that (σ_Eσ_E%)^{3−E E%} =I d_S.

Case E E^%=1.Suppose thatE E^%=1. A generic hyperplane sectionY of the cone ψ (π⁻¹(E^%))parameterizes the lines of this cone and is a plane model ofE^%: we will identifyYandE^%. Lettbe the intersection point ofEandE^%: with this neutral element, the curveE^%is an elliptic curve.

Letsbe a point ofE^%different fromt, then by definition ofσE andγE: X_E,sF =L_s+L_γEs+L_σEs

whereXE,s := -Ls,pE.. Since the lineLs cuts the line Lt at the vertexpE%, we haveXE,s = -Ls,Lt.. Thus the pointtis one of the three pointss, γEs, σEs. As s+=tand the automorphismσEpreservesE, the pointσEsis not an element ofE, henceγ_Es=t. ThusE^%is a component ofγ_E⁻¹(t).

The plane XE,s contains the lines Ls, Lt that are in the hyperplane section ψ (π⁻¹(E^%))ofF, thus the third lineLσ_Es is also in the coneψ (π⁻¹(E^%))and the pointσEsis onE^%. Because of the relation

X_E,sF=L_s+L_γEs+L_σEs,

the three pointss,t =γEs, σEsare on a line in the plane modelY, henceσEs= −s for all pointssofE^%.

Remark 14.Since σ_E(s) = −s on E^%, the endomorphismσ_E^∗ ∈ G L(H⁰("_S)^∗) is the morphism of multiplication by−1 on the tangent space to the curveϑ (E^%) (translated to 0). Thus :σ"Ep_E% = p_E% and p_E% is the intersection point of the line Lt and the hyperplane of fixed points ofσ"E. This implies that the points pE and p_E% are the only vertices of cones on the lineLt.

Letsbe a generic point ofS, then

"

σE%XE,s =σ"E%-pE,Ls. =#

pE,Lσ_E%s$

=XE,σ_E%s, hence:

"

σ_E%XE,sF =L_σE%s+L_γEσE%s+L_σEσE%s. ButX_E,sF =L_s+L_γEs+L_σEs hence:

"

σ_E%XE,sF =L_σE%s+L_σE%γEs+L_σE%σEs.

Sinceγ_Eσ_E%s andσ_E%γ_Es are points of E, we see thatσ_Eσ_E%s = σ_E%σ_Es, thus (σ_Eσ_E%)²=I d_S.

Case E E^%=0.Suppose now thatE E^%=0. Letsbe a point ofE^%. By definition X_E,s = -p_E,L_s.and:

XE,sF=Ls+Lγ_Es +Lσ_Es.

Suppose thatσEsis a point ofE^%, then planeXE,s cuts the coneψ (π⁻¹(E^%))into two lines :Ls andL_σEs. As this cone has degree 3, the intersection of XE,s and ψ (π⁻¹(E^%))contains the third lineL_γEs. Sinceψ (π⁻¹(E^%))is a cone, that implies thatγEsis a point ofE^%: this is a impossible becauseE E^%=0. ThusσEsis not a point ofE^%.

The automorphismσE fixesE. Hence the pointσEsis not a point of E. This proves that the surfaceScontains a third smooth curve E^%% =σE(E^%)of genus 1 and thatE E^%%=E^%E^%%=0.

For a pointsofE^%, the planeX_E,s = -p_E,L_s.contains the lineL_γEs and the pointp_E% ∈ L_s, hence:

XE,s = -pE,Ls. =#

pE%,Lγ_Es$

=XE%,γEs. But we have

XE,sF=Ls+Lγ_Es +Lσ_Es

and

X_E%,γ_EsF=L_γEs+L_γE%γEs+L_σE%γEs.

Since the pointss, γEs andσEsare respectively points of E^%,E andE^%%, we see that for all pointssofE^%:

σ_E%γ_Es=σ_Es.

Hence the restriction of σE% to E is a morphism from E to E^%% andσE%(E) = σ_E(E^%)=E^%%, this proves f).

Letsbe a point of E^%. The linesL_s,L_γEs andL_σEs contain respectively the vertices p_E%,p_E, and p_E%%. Assvaries inE^%, the planeX_E,s varies and the linear hull*of the pointsp_E, p_E%andp_E%%cannot be a plane: it is a line.

Remark 15.The line *lies outside the cubic F, otherwise the curves E and E^% would have a common point. The points p_E, p_E% and p_E%% are the intersection points of*and the cubicF.

Let s be a point of S. The morphism !σ_E% verifies :!σ_E%(L_s) = L_σE%s, and furthermore:

!σE%XE,σ_E%s =!σE%#

pE,Lσ_E%s$

= -pE%%,Ls. =XE%%,s. (3.1) We haveXE,σ_E%sF =L_σE%s+L_γEσE%s+L_σEσE%s. Hence

!σ_E%X_E,σE%sF =L_σ2

E%s+L_σE%γEσE%s+L_σE%σEσE%s,

but by3.1:

!σ_E%X_E,σE%sF=X_E%%,sF.

SinceXE%%,sF =Ls+Lγ_E%%s+Lσ_E%%s, we see thatσE%σEσE% =σE%%. So the group generated byσ_E, σ_E%, σ_E%%is isomorphic to+₃and(σ_Eσ_E%)³=I d_S.

This ends the proof of Theorem13. )*

Let us now study the configuration of three elliptic curves.

Proposition 16.Let E₁,E₂and E be three elliptic curves on S such that E₁E = E₂E =1. Then: E1E₂=0and the curve E₃=σE₁(E₂)verifies E₃E=1.

Proof. Suppose E₁E₂ = 1. This implies that the line through p_E1 and p_E2 lies on the coneψ (π⁻¹(E))and hence goes throughpE. But by Remark14, the three pointspE, pE₁ andpE₂ cannot be on a line. HenceE₁E₂=0.

Since E E₁ = 1, we have σE₁(E) = E and E₃E =σE₁(E₂)σE₁(E) = E₂

E = 1. )*

3.2. The graph of the configuration of the vertices of cones

LetEbe the set of elliptic curves onS. Let us consider the following graphG: the set of vertices ofGisEand an edge linksE∈EtoE^%∈Eif and only ifE E^%=0.

LetGS be the sub-group of Aut(S)generated by the automorphismsσE, E ∈ E. We have the three following relations between its generators:

(a) for allE ∈E,σEverifiesσ_E² =I dS,

(b) an edge linksEandE^%if and only if(σEσ_E%)³=I dS, (c) otherwise(σEσE%)²=I dS.

The following corollary is a consequence of Proposition16.

Corollary 17.Let E₁,E₂and E₃be three elements ofE. At least one edge links two of the three vertices E₁,E₂,E₃of the graphG.

There is no sub-group ofGSgenerated by some elementsσ_E,E ∈ Eand iso- morphic to(Z/2Z)³.

Proof. If there are no edges between the verticesE₁andE₃and between the ver- tices E₂and E₃, then E₁E₃ = E₂E₃ = 1 and the Proposition 16implies that E₁E₂ =0. Thus an edge links the verticesE₁andE₂. The second assertion is a

reformulation of the first. )*

We remark that if the graphGhasmconnected components, then the group GSis the direct product ofmsub-groups. The Corollary17can be reformulated as follows:

Corollary 18.The graph G is connected or Ghas two connected components G1,G2such that two different vertices of a componentGⁱare linked by an edge.

LetE^%be a sub-set ofE. LetG^%be the graph whose set of vertices isE^%and such that an edge links two elements ofE^%if and only if these vertices are linked by an edge inG. Suppose that the three relations (a), (b) and (c) above are the only ones between the elementsσ_E, E ∈ E^%. The group generated by the automorphisms σE, E ∈E^%is then a Weyl group, which we denote byG^%_S.

Corollary 19.If the graphG^%is connected and has n vertices, then1≤n≤4and the groupG^%_Sis the permutation group of the set of n+1elements.

Proof. By corollary17and the classification of the Weyl groups [9], the graphG^% must be one of the graphs An,1 ≤n ≤4. The Weyl groupW(An)associated to

Anis the permutation group ofn+1 elements. )*

3.3. Restrictions on the complex reflection groups

Let us denote byGSthe complex reflection group generated by the order 2 reflec- tions−σ_E^∗ ∈ G L(H⁰("S)^∗), E !→ S elliptic curve. For the basic properties of reflection groups see [5,11] or [7], here we recall some them:

Definition 20.A reflection in a spaceV is a linear transformation ofV of finite order with fixed point set an hyperplane ofV. A reflection group is a finite group generated by reflections.

LetG₁andG₂be two reflection groups acting on spaces V₁andV₂, we say (improperly) thatG₁is a reflection sub-group of G₂, if there exists an injective morphismG₁ → G₂ of complex reflection groups. In this case, we denote the elements ofG₁andG₂by the same letters.

The list of the 37 types of irreducible reflection groups was compiled by Shep- hard and Todd [11].

Let us fix some notation. Let m,n > 0 be integers and let p be an integer dividingm. We denote by+n ⊂ G Ln(C)the group of permutation matrices and byA(m,p,n)⊂G Ln(C)the group of diagonal matricesDof ordermsuch that det(D)^mp =1. The type 2 reflection groups are the groupsG(m,p,n)generated byA(m,p,n)and+n.

The type 3 groups constitute the family[ ]ⁿforn ∈N\{0,1}where the[ ]ⁿis the group of morphismsC→C; x→ξx,ξⁿ=1.

Theorem 21.An irreducible sub-group of GSgenerated by reflections of order2 is isomorphic to one of the following groups:

{1},[ ]²,G(3,3,n),G(1,1,n)=+_n,2≤n≤5.

Proof. By the Remarks14and15, a projective line ofP⁴contains at most 3 verti- ces of a cone, hence a 2 dimensional reflection sub-group ofGScontains at most 3 reflections of order 2. The groups of type 4 to 22 are 2 dimensional irreducible groups. They either have no or at least 6 reflections of order 2 [5, Table p. 395] : none of these groups can be a reflection sub-group ofGS.

The groups numbered 25,29,31,32,33,34 possess either no or at least 40 reflections of order 2 [5, p. 412]. Since reflections of order 2 ofG_Sare in bijection with elliptic curves on S and a Fano surface contains at most 30 elliptic curves (Proposition11), none of these groups is a sub-group ofG_S.

The groups 23,28,30,35,36,37 are real reflection groups [11, p. 299] which have been excluded by the Corollary19.

The group 26 has a reflection sub-group isomorphic to the group number 4 already excluded [11, p. 302].

The groups 24 and 27 have two reflections R₁ and R₂ of order 2 such that (R₁R₂)⁴=1 [11, p. 299] and hence they cannot be sub-groups ofGS.

It thus remains to study the reflection groups of types 1, 2 and 3.

LetM be an irreducible sub-group ofGSgenerated by order 2 reflections.

Type3. It is immediate that there existsn > 1 such that[ ]ⁿ & M if and only if n=2.

Type2and1. The groupG(m,p,n)(where pdividesm ∈ N^∗andn >1) is an n dimensional irreducible reflection group if and only ifm >1 and(m,p,n)+=

(2,2,2). The representationG(1,1,n)=+nof the permutation group breaks up into a 1 dimensional trivial representation and ann−1 dimensional irreducible representationW(A_n−1)called the standard representation. The groups:

W(A_n), n∈N^∗

constitute the number 1 reflection type in the Shephard-Todd classification.

Letm, p,n be integers such thatm > 0, pdividesm andn > 1. Suppose thatMis the groupG(m,p,n). Theorem13implies that a group generated by two reflections of order 2 ofG_Sis the diedral group of order 4 or 6. Form>1, the group G(m,p,n)contains a diedral sub-group of order 2mgenerated by two reflections of order 2. Thus the integermis an element of{1,2,3}; moreovern ≤5.

The groupG(2,2,2)is not irreducible and is thus excluded. The groupsG(2,1,n) withn≥3 and the groupsG(2,2,n)withn≥4 contain sub-groups isomorphic to (Z/2Z)³generated by reflections of order 2. By Corollary18, such groups cannot be reflection sub-groups ofGS.

The reflection group+₄is isomorphic to the groupG(2,2,3)plus the trivial representation. SoG(2,2,3)is implicitly in the list of Theorem21.

The groupG(3,1,n)cannot be isomorphic toMbecause its order 2 reflections generate the strict sub-groupG(3,3,n)⊂G(3,1,n).

By Corollary19, if W(An)is a reflection sub-group ofGS, thenn ≤ 4. The group+nis isomorphic to the reflection groupW(A_n−1)plus the trivial represen- tation.

Hence, we proved thatMis isomorphic to one of the groups{1},[ ]²,G(3,3,n),

G(1,1,n)=+_n,2≤n≤5. )*

3.4. Classification of groups and Fano surfaces

Let us classify the Fano surfaces according to the configuration of their elliptic curves. Let us recall the notations : to each elliptic curve E !→ S corresponds

an automorphismσ_E ofS and the groupG_Sis generated by the elements−σ_E^∗ ∈ G L(H⁰("_S)^∗), where the automorphismsσ_E^∗are defined in paragraph2.4. We will use the following remark:

Remark 22.LetFeqbe an equation of the imageFof the cotangent map of a Fano surfaceS. There exists a morphism:

χ:GS→C^∗ such thatF_eq◦N =χ (N)F_eqfor allN∈G_S.

Thus we are looking for cubic formsFeq, reflection groupsGand morphisms χ:G→C^∗such thatFeq◦N =χ (N)Feqfor allN ∈Gand such that{Feq =0} is smooth.

We need some notations and preliminary materials.

The order ofG(m,p,n)is equal to^m_pmⁿ⁻¹n!and the number of its order 2 reflec- tions ismⁿ⁽ⁿ₂⁻¹⁾. The groupG(m,m,n)acts on the polynomial space ofCⁿ. The algebra of invariant polynomials is generated by the polynomials:

i=n

%

i=1

x_i^mk, k∈ {1, . . . ,n−1}

and byx₁x₂. . .xn(see [11]).

LetSbe a Fano surface such that the groupG(m,m,n)(m∈ {1,3},2≤n ≤5) is a reflection sub-group ofGS. Let bem ∈ {1,3}andn > 1. We easily check that the only non-trivial morphism fromG(m,m,n)toC^∗is the determinant. We call a polynomialP an anti-invariant ofG(m,m,n)if P◦N =(detN)P for all N ∈G(m,m,n). We verify that:

Lemma 23.The only reflection groups G(m,m,n)with m∈ {1,3}and n≥2that possess an anti-invariant polynomial of degree≤3are G(1,1,2), G(3,3,2)and G(1,1,3).

Notation 24.Forλ³+=1, we denote byE_λthe smooth plane cubic:x³+y³+z³− 3λx yz=0.

We denote by Athe Albanese variety ofS, byϑ : S → Aan Albanese mor- phism, bye₁, . . . ,e₅ the dual basis of the basis x₁, . . . ,x₅ ∈ H^o("_S). If v ∈ H⁰("S)^∗is a non zero vector,Cvis the vector space generated byvor the point of P⁴=P(H⁰("S)^∗)corresponding to this space, we specify as need be. We denote byµ₃the group of third roots of unity.

Recall (see Theorem13and its proof) that there is a one to one correspondence between:

(a) the elliptic curvesEonS.

(b) the order 2 reflectionsR= −σ_E^∗ ofGS. (c) the conesψ (π⁻¹(E))onF.

(d) the vertices pE of the conesψ (π⁻¹(E)) !→F.

When we consider an order 2 reflectionRcorresponding to an elliptic curveE!→S, the vertex p_E of the coneψ (π⁻¹(E))is the point ofP⁴corresponding to the ei- genspace with eigenvalue−1 ofR∈G L(H⁰("_S)^∗)=G L₅(C).

The coneψ (π⁻¹(E))is the intersection of the cubicFand the tangent space at pE. A plane model of the curveEis the intersection of this cone with the hyperplane of fixed points ofσ"E(where"σE is the projectivization ofR).

In order to know the intersection number between two elliptic curvesE₁and E₂corresponding to the reflectionsR₁andR₂, it suffice to compute the order (=2 or 3) of the elementR₁R₂and to use the formula:

(R₁R₂)^3−E1E2 =I d.

It can also be verified by hand if the line between the vertices pE₁ and pE₂ is or not inside the cubic, accordingly:E₁E₂=1 or 0.

In the sequel, we proceed to the classification of the elliptic curve configurations on Fano surfaces according to the dimension of the studied irreducible reflection sub-group ofGS; this group must be in the list of Theorem21.

• 1 dimensional sub-group ofGS.

We proved in paragraph3.1, that whenGScontains the group[ ]²generated by a reflection R, the cubic can be written as follows:

F= {x₁²x₂+G(x₂, . . . ,x₅)=0}

whereG is a cubic form and R acts byx → (−x₁,x₂,x₃,x₄,x₅). The tangent space at p=(1:0:0:0:0)isTF,p= {x₂=0}. This pointpis the vertex of a cone inF. This cone is the intersection ofTF,pandF. Let us denote by E !→ S the elliptic curve on the Fano surface that parameterizes the lines on the cone.

The hyperplane{x₁ = 0}is an invariant of the cubic because it is the fixed locus ofσ"_E. A plane model ofEis obtained by the intersection ofFand the plane {x₁=x₂=0}.

We see in that way, that given a pair(T,E)whereT is a smooth cubic surface (say given byT = {G(x₂, . . . ,x₅)=0}) andEis a smooth hyperplane section of T (sayE =T ∩ {x₂=0}), we can associate the pair(S,E^%)whereSis the Fano surface of{x₁²x₂+G(x₂, . . . ,x₅)=0}andE^%is an elliptic curve on it isomorphic toE. Reciprocally, given the pair(S,E^%), we can recover the cubicFand the pair (T,E), up to isomorphism. As the moduli space of isomorphism class of such pairs (T,E)is 7 dimensional, the moduli of Fano surfaces that contains an elliptic curve is a 7 dimensional sub-space of the 10 dimensional moduli of Fano surfaces.

• 2 dimensional reflection sub-groups ofGS.

– The anti-invariant polynomials of the reflection groupG(3,3,2)yield sin- gular cubics. The invariants of the groupG(3,3,2)are generated by the polynomials:

x₁³+x₂³,x₁x₂, x₃,x₄,x₅. Up to a change of variables, the cubicF is:

F = {x₁³+x₂³+3x1x₂l(x₃,x₄,x₅)+x₃³+x₄³+x³₅−3λx3x₄x₅=0} wherelis a linear form andλ∈C.

The 3 pointsC(e₁−βe₂),β ∈ µ₃(see Notations24) are vertices of cones.

The Fano surfaceS contains three disjoint elliptic curvesE₁₂^β, β ∈ µ₃that have the following model:

{x₃³+x₄³+x₅³−3λx₃x₄x₅+l³=0}.

Note that we also have studied the reflection groupG(1,1,3)=+₃because its rep- resentation decomposes into the trivial one plus the standard representationW(A₂) which is equal toG(3,3,2).

• 3 dimensional reflection sub-groups ofGS. – The invariants ofG(3,3,3)are generated by:

x₁³+x₂³+x₃³,x₁x₂x₃, x₄,x₅.

Up to a variables change, there exist coordinates there existsλ∈Csuch that:

F = {x₁³+x₂³+x₃³−3λx1x₂x₃+x₄³+x₅³=0}.

The groupG(3,3,3)×G(3,3,2)is a reflection sub-group of G_S. The 9 points C(ei −βej),1 ≤i < j ≤ 3, β ∈ µ₃are vertices of cones. The corresponding elliptic curves:

E_{i j}^β,1≤i< j ≤3, β ∈µ₃

on the Fano surfaceSare isomorphic to the curveE₀and are disjoint.

The 3 pointsC(e₄−βe₅), β ∈ µ₃are vertices of cones. The corresponding 3 elliptic curvesE₄₅^β, β ∈µ₃are disjoint and isomorphic to the curveE_λ. For all 1≤i < j ≤3, (β, γ )∈µ²₃, we have:

E_{i j}^βE^γ₄₅=1.

The Fano surface contains 12 elliptic curves.

– The invariants of degree less than or equal to 3 of+₄are generated by:

x₅^k, x₁^k+x^k₂+x₃^k+x₄^k, k∈ {0,1,2,3}.

Let F be a smooth cubic defined by an element of the 6 dimensional space of invariant cubics. For 1≤i < j ≤4, the pointC(e_i −e_j)ofP⁴is the vertex of a cone onF; let us denote by:

E_{i j} !→S

the corresponding elliptic curve on the Fano surfaceS. We have:

Ei jEst =





1 if{i,j} ∩{s,t} = ∅

−3 ifEi j =Est

0 otherwise.

The surfaceScontains 6 elliptic curves.

• 4 dimensional reflection sub-groups ofG_S.

– Suppose that+₅is a reflection sub-group ofG_S. The invariants of degree less than or equal to 3 of+₄are generated by:

x₁^k+x₂^k+x₃^k+x₄^k+x₅^k, k∈ {0,1,2,3}.

There existλ, µ∈Csuch that the image of the cotangent map ofSis:

F =





i=5

%

i=1

x_i³+λ )_i=5

%

i=1

xi

* )_i=5

%

i=1

x_i²

* +µ

)_i=5

%

i=1

xi

*3

=0



.

For 1≤i < j ≤5, the point

pi j =C(ei−ej)

is the vertex of a cone and we denote byEi j !→Sthe corresponding elliptic curve.

We have:

Ei jEst =





1 if{i,j} ∩{s,t} = ∅

−3 ifEi j =Est

0 otherwise.

The dual graph of this configuration of 10 elliptic curves is the trivalent Petersen graph.

• 5 dimensional reflection sub-group ofGS. – The Fermat cubic:

F = {x₁³+x₂³+x₃³+x₄³+x₅³=0}

is, up to isomorphism, the only cubic stable underG(3,3,4)andG(3,3,5). The pointsC(e_i−βe_j),1≤i < j ≤5, β ∈µ₃are vertices of cones onF. Its Fano surface S is the unique Fano surface that contains 30 smooth curves of genus 1.

These curves are numbered:

E^β_{i j},1≤i < j ≤5, β ∈µ₃. LetE_{i j}^γ andE_st^β be two such curves, then:

E_{i j}^βE^γ_st =





1 if{i,j} ∩{s,t} = ∅

−3 ifE_{i j}^β =E^γ_st 0 otherwise.

A remarkable property of this surface is that its Néron-Severi group N S(S)has rank 25=dimH¹(S, "S)andN S(S)⊗Qis generated by the 30 elliptic curves.

Now, we study the case for which the reflection groupGS is not irreducible.

Corollary18proves that: