Geometry of stable ruled surface over an elliptic curve

We consider the stable ruled surface $S_1$ over an elliptic curve. There is a unique foliation on $S_1$ transverse to the fibration. The minimal self-intersection sections also define a 2-web. We prove that the 4-web defined by the fibration, the foliation and the 2-web is locally parallelizable.


Preliminaries
2.1 Some properties on an elliptic curve where t ∈ C \ {0, 1} be an elliptic curve. Throughout this article, we use the following background of an elliptic curve.
Proposition 2.1. The set of points of C forms an abelian group, with p ∞ as the 0 element and with addition characterized for any couple of points p = (x 1 , y 1 ), q = (x 2 , y 2 ) in C by: is an automorphism of C which fixes the points of order 2: it is the standard involution of the curve C.
If we denote Jac(C), the jacobian of C, we have: Lemma 2.3. There exists a bijection between C and its jacobian defined by this following map: From now on, we will use the isomorphism between the additive group structure (C, p ∞ ) and the group structure on C induced by its jacobian.

Ruled surface over an elliptic curve
Let C be a smooth curve on C.
Definition 2.4. A ruled surface over C is a holomorphic map of two dimensional complex variety S onto C π : S → C which makes S a P 1 -fibration over C.
Exemple 2.5. The fiber bundle associated to a vector bundle of rank 2 over C is a ruled surface. We denote it, P(E).
Conversely, we have the following theorem proved by Tsen in [5] : Theorem 2.6. Let π : S → C be a ruled surface over C: 1. there exists a vector bundle E of rank 2 over C such that S = P(E); 2. there exists a section, i.e a map σ : C → S such that π • σ = id; 3. P(E) ∼ = P(E ) if and only if there is a holomorphic line bundle L over C such that E ∼ = E ⊗ L.
Definition 2.7. A ruled surface P(E) is decomposable if it has two disjoint sections.
The following lemma whose proof is in ( [7], page 16) shows the relationship between the ruled surface S = P(E) and the vector bundle E. Lemma 2.8. There exists a one-to-one correspondance between the line subbundles of E and the sections of S. Futhermore, if σ L is the section related to the line subbundle L then: Notation 2.9. We recall that the notation σ L .σ L means the self-intersection of the section σ L .
Consider κ = min {σ.σ, σ : C → S /π • σ = id}. This number only depends on the ruled surface S = P(E). Indeed, it does not change when we replace E by E ⊗ L for a line bundle L on C.
Definition 2.11. The ruled surface P(E) is stable if κ > 0. Definition 2.12. A minimal section of S is a section σ : C → S such that, the self-intersection is minimal. That is to say, σ.σ = κ.
Using lemma 2.8, we notice that a minimal section corresponds to a line subbundle of E with maximal degree. Thus, the invariant κ can be written as: Now, we are interested in indecomposable ruled surfaces over an elliptic curve.
There are unique nontrivial extensions of invertible sheaves: Recall the following Atiyah's theorem as proved in ([1], Th.6.1): Theorem 2.13. Up to isomorphism, the unique indecomposable ruled surfaces over C are S 0 = P(E 0 ) and S 1 = P(E 1 ).
Remark 2.14. Equivalency, any indecomposable vector bundle E of rank 2 on C takes the form E = E i ⊗ L, for i = 0, 1 and L a line bundle.
As our aim in this paper is the study of the ruled surface S 1 , we will show firstly some important properties of E 1 .
Lemma 2.15. The degree of the maximal line subbundles of E 1 is zero.
Proof. Let L be a subbundle of E 1 and consider the quotient M := E 1 /L. By the following exact sequence 0 / / L / / E 1 / / M / / 0 and the fact that E 1 is indecomposable, we have H 1 (M −1 ⊗ L) = 0. Thus, due to Serre's duality, we have 2 deg(L) deg(E 1 ) and then deg(L) 0 because deg(E 1 ) = 1. Since the trivial line bundle O C is a line subbundle over E 1 , we have the result.
Remark 2.16. By this lemma, we can deduce that the ruled surface S 1 is stable. More precisely, up to isomorphism, it is the unique stable ruled surface over an elliptic curve.
If we consider max E 1 = {L → E 1 , deg L = 0} the set of line subbundles of E 1 having a maximal degree, we have: Lemma 2.17. The jacobian of C sets the parameters of the set max E 1 . More precisely, the map is a bijection.
To prove this lemma, we have to use a key lemma of Maruyama in ([7], page 8): Lemma 2.18. Let E be a vector bundle of rank 2 over a curve. If L 1 and L 2 are distinct maximal line subbundles of E such that L 1 and L 2 are isomorphic, then E = L 1 ⊕ L 1 . Now, we can prove lemma 2.17: Proof.
• Let L 1 and L 2 be two elements in max E 1 such that L 1 ∼ = L 2 . We have two possibilities, either L 1 = L 2 or they are both distinct. According to the lemma 2.18, the last case cannot occur because E 1 is not decomposable. Thus, the map M is injective.
• Let L ∈ Jac (C) be distinct from the trivial line bundle. If we apply the functor Hom (L, −−) to the exact sequence Remark 2.19. The minimal sections of S 1 have self-intersection equal to 1 and they are parametrised by the jacobian which is isomorphic to C. Hence for every point ∈ C, we denote σ the minimal section corresponding via lemma 2.17 to the subbundle isomorphic to Lemma 2.20. Let σ and σ be two minimal sections of S 1 . If we consider D their intersection divisor, we have: where π : S 1 → C is the projection map.
Proof. Intuitively, the divisor D is defined by the points above at which the line bundles related to σ and σ coincide. More precisely, π (D) is a effective divisor equivalent to divisor det( . Since the degree of D is equal to 1, we obtain the result. Remark 2.21. Let Q be a point of S 1 belonging to the fiber π −1 (p). If the minimal section σ passes through the point Q, then the unique other minimal section passing through the same point Q is the section σ −p− . They might be the same for some Q. By this theorem, the Atiyah's bundle E 1 is not flat because deg E 1 = 1. However what can we say about its associated ruled surface ? The answer of this question is given by Frank Loray and David Marin in [6]. Consider C as a torus C/Z + τ Z and let : Z + τ Z → PSL 2 (C) be the representation of the fundamental group of C defined by (1) = −z and (τ ) = 1 z .
Up to conjugacy in PSL 2 (C), it is the unique representation onto the 4-order The orbits of the elements of C × P 1 modulo the action given by the representation and the universal cover of C form a ruled surface over C, denoted by E. It is obvious to see, the horizontal foliation of C × P 1 lifts to a regular foliation R transverse to the fibration of E. The foliated surface (E, R) is Thus, we can deduce that every leaf of foliation R is a cover of C. It is not difficult to show that the intersection of any fiber of E and the leaf R [x 0 ,z 0 ] is given by the set Using the finitude of the representation, we have every leaf of the foliation is a cover of finite degree. Moreover, the monodromy of the foliation R on any fiber is the representation .
Note that, the action of < −z, 1 z > in P 1 gives two kind of orbits: orbits of order 4 and three special orbits of order 2 given by (−1, 1), (−i, i) and (0, ∞). Therefore, the foliation R has a generic leaf which is a cover of order 4 of C and three special leaves, which is a cover of order 2.
Proposition 2.23. The ruled surface E over C is indecomposable such that its invariant κ = 1. It is the ruled surface S 1 .
Proof. If E is a decomposable ruled surface then its invariant κ = 0. Indeed, let F be a leaf of R and σ 0 a minimal section of E, then we have: Using the fact that E is decomposable, we can find a section σ such that σ.σ 0 = 0. Since F.σ 0 and F 2 = 0, we obtain that σ 0 .σ 0 = 0. Eventually, if we assume that E is a decomposable ruled surface, we have F ≡ 4σ 0 or F ≡ 2σ 0 and then F.σ = 0. The section σ does not meet any leaf of R, which does not make a sense because the foliation is regular. The ruled surface E is then indecomposable. Hence, it is either isomorphic to S 0 or S 1 . By the same arguments above, E is not isomorphic to S 0 , if not there would be a section which not intersects any leaf of R. Thus, the ruled surface E is isomorphic to S 1 by Atiyah's theorem. As up to conjugacy the representation is the unique representation onto the 4-order group Γ =< −z, 1 z >, we deduce that the isomorphism between E and S 1 is the identity.
In summary, we have: Theorem 2.24. The ruled surface S 1 has a Riccati foliation Ric with irreducible monodromy group < −z, 1 z >.
Remark 2.25. The foliation Ric has a generic leaf which is cover of degree 4 over C and three special leaves which are covers of degree 2 over C.
3 Geometry of the ruled surface S 1 Let π : S 1 → C be the stable ruled surface over C.
Proposition 3.1. The automorphism group of S 1 is a group of order 4 which is isomorphic to the 2-torsion group in C.
Proof. Let ψ : S 1 → S 1 be a non trivial automorphism of S 1 . Since the self-intersection is invariant by automorphism ψ preserves the set of +1 selfintersection sections on S 1 . More precisely, for any ∈ C, there exists a unique point r ∈ C such that ψ (σ ) = σ r . The automorphism ψ induces an automorphism ψ of C such that for any point ∈ C we have ψ ( ) = r .
If we define C as the complex torus C/Z + τ Z, we can write for any z ∈ C, If we assume this automorphism has a fix point 0 , then by definition we have ψ (σ 0 ) = σ 0 . Hence, using the lemma 2.20, we obtain that for any p ∈ C, For any fiber, the automorphism ψ is Moebius map which fixes at least three points: it is the trivial automorphism, which does not make sense by hypothesis. Therefore, the automorphism ψ has no fixed points, it is a translation like Conversely, for any point p i of order 2 on C, we can define an automorphism Φ i , on S 1 such that for any point p ∈ C, Φ i restricts to the fiber π −1 (p) is the unique Moebius map which associates the points of the sections (σ p∞ , σ p 0 , σ p 1 , σ pt ) to the points of the sections (σ p∞+p i , σ p 0 +p i , σ p 1 +p i , σ pt+p i ) respectively. It is defined by: There exists a one-to-one correspondance between the automorphisms of the fiber bundle S 1 and the points of order 2 in C which preserves the group structure. Hence we have : Proof. Using the fact that the fundamental group of C is abelian, we can extend the monodromy map over every fiber and regard it as automorphism on S 1 which fixes the basis C. Thus, we obtain that the monodromy group of the foliation Ric is a subgroup of order 4 of Aut C (S 1 ) : they are isomorphic. The group Aut C (S 1 ) preserves the Riccati foliation on S 1 . Proof. Let F 1 be a Riccati foliation on S 1 . As its monodromy group is an abelian subgroup of P GL(C, 2), we have three possibilities for its monodromy representation : • If the conjugacy class of the monodromy is the linear class defined by the group az , bz , there exists two disjoint invariant sections of S 1 . Hence S 1 is a ruled surface related to the direct sum of two line bundles over C. It does not make sense because S 1 is indecomposable.
• If the conjugacy class of the monodromy is the euclidian class defined by the group z + 1 , z + s , there exists an invariant section on S 1 with zero self-intersection. In fact by the Camacho Sad's theorem (in [3]), any invariant curve of regular foliation has a zero self-intersection. This monodromy representation does not make sense in S 1 because we have min {σ.σ, σ : The only remaining possibility is that the monodromy has image the group Thus, the foliation F 1 is conjugated to Ric by an element in Aut C (S 1 ). As this automorphism group of the fibration S 1 preserves the foliation Ric, we have Ric = F 1 .
Lemma 3.4. There exists a ramified double cover of the ruled surface S 1 defined by the map : such that its involution is defined by : Proof. According to the lemma 2.20, three minimal sections cannot meet at the same point, then we deduce for any p ∈ C, the morphism is not constant: it is a ramified cover between Riemann surfaces. Futhermore, by the remark 2.21, we know that at most two minimal sections can pass through a given point, then the map ϕ is a ramified double cover.
The immediate consequence of this lemma is the following : Theorem 3.5. There exists a singular holomorphic 2-web W on S 1 defined by the minimal sections whose discriminant ∆ is a leaf of the foliation Ric.
Proof. By lemma 3.4, for any point P ∈ π −1 (p) there exists a minimal section σ r passing through this point. Likewise, by lemma 2.20 the minimal section σ −p−r intersects transversally σ r at the point P. As the sections σ r and σ −p−r are distinct if and only if 2r = −p, we deduce that there exists a singular holomorphic 2-web on S 1 such that its discriminant is defined by : In order to prove that ∆ is a leaf of the Riccati, we need the following : Lemma 3.6. There exists a linear foliation F on C × Jac (C) such that ϕ * F = Ric.
Proof. Assume that C × Jac (C) (C/Z + τ Z) × (C/Z + τ Z), and let be (x, y) its local coordinates . If we consider the linear foliation F := dx + 2dy on C x × C y , then F is invariant by the action of the lattice Z + τ Z. Thus we can lift the foliation F to a foliation, F on C × Jac (C) such that the monodromy is defined by : The foliation F is transverse to the first projection on C×Jac (C) with a monodromy group isomorphic to the group of points of order 2 {p ∞ , p 0 , p 1 , p t }. Moreover, if F (p,ω) is the leaf passing through the point (p, ω), then by definition we have : where i is the involution of the ramified double cover ϕ. Hence, ϕ * F the direct image of the foliation F by ϕ is a Riccati foliation on S 1 having the same monodromy group than Ric. Using the uniqueness of Ric by the corollary 3.3, we obtain the result.
As by definition the curve G = {(2p, −p) /p ∈ C} is a leaf of the foliation F, using the foregoing lemma we can deduce that ϕ (G) = ∆ is a leaf of Ric. Which completes the proof of theorem 3.5.

Study of special leaves of the Riccati foliation Ric
According to lemma 3.6, if P = σ ω (p) ∈ S 1 then the Riccati leaf passing through at this point is given by Thus, if we use this characterisation of the Riccati leaves on S 1 , we have the following lemma : Lemma 3.7. There exists three special leaves Ric 0 , Ric 1 and Ric t of the foliation Ric which are double cover of C. More precisely, they are respectively the set of fixed points of the automorphisms Φ 0 , Φ 1 and Φ t .
Proof. We just give the proof for the leaf Ric 0 because it is the same process for the other special leaves.
• Let Ric 0 be the Riccati leaf passing through the point z 0 = σ p 0 (p 0 ), then by definition, we have : According to the monodromy of Ric, if the leaf Ric 0 passes through the point z = σ ω (p) then it passes through the points σ ω+p i (p), where p i is a point of C of order 2. Since 2ω = p 0 − p, we deduce from lemma 2.20 that: σ ω+p 0 (p) = σ ω (p) and σ ω+p 1 (p) = σ ω+pt (p) thus, Ric 0 meets any fiber of S 1 twice. It is a double cover over C.
• Let Φ 0 be the automorphism of S 1 related to the point p 0 defined by: and consider the set of its fix points is the fix point of Φ 0 , then by lemma 2.20, we have 2ω = p 0 − p, and therefore z ∈ Ric 0 . Conversely, if z ∈ Ric 0 , we have 2ω = p 0 − p, and according to lemma 2.20, we have, σ ω+p 0 (p) = σ ω (p). Thus, we deduce that: According to all the foregoing, we have: Remark 3.8. The 2-web given by the +1 self-intersection sections, the Riccati foliation and the P 1 -bundle π : S 1 → C form a singular holomorphic 4-web W on S 1 .

The geometry of the 4-web W
Let (x, y) be the local coordinates of C 2 . As the linear foliations G and H respectively defined by dy = 0 and by dy + dx = 0 are invariant by the action of the lattice Z + τ Z, we can lift them to a decomposable 2-web W on C × Jac (C).
Proposition 3.9. The direct image ϕ * (W ) of the 2-web W by the ramified cover ϕ is the 2-web W on S 1 defined by the minimal sections.
Proof. As by definition the 2-web W is invariant by the involution of the ramified double cover ϕ, its direct image is also a singular holomorphic 2web on S 1 . Let (p , ) ∈ C × Jac (C) and consider the leaves of the 2-web W passing through this point. Since, using lemma 2.20, we have: ϕ(A ) = σ and ϕ(B ) = σ −p − , then the leaves of ϕ * (W ) are the minimal sections of S 1 which are the same along the discriminant ∆.
The local study of the 4-web W on S 1 is the same as the 4-web on C × Jac (C) given by the 2-web W , the foliation F and the Jac (C)-bundle defined by the first projection on C × Jac (C).
Theorem 3.10. Outside the discriminant locus ∆, the 4-web W is locally parallelizable .
Proof. According to the foregoing, the pull-back of 4-web W by the ramified cover ϕ is locally the 4-web defined by W (x, y, y + x, y + 2x) on C 2 . It is a holomorphic parallelizable web.
Remark 3.11. An immediate consequence of theorem 3.10 is that the curvature of the 4-web W is zero.
The second part of this paper aims to use the theory of birationnal geometry in order to find the theoretic results of the first part by computations on the birational trivialisation C × P 1 .

Geometry of 4-web W after elementary transformations
Let π : S 1 → C be the P 1 -bundle and {p 0 , p 1 , p t , p ∞ }, the set of points of order 2 in C.  How many elementary transformations do you need to trivialize the ruled surface S 1 ? Lemma 4.3. The ruled surface S 1 is obtained after three elementary transformations at the points P 0 = (p 0 , 0) , P 1 = (p 1 , 1) and P ∞ = (p ∞ , ∞) on the trivial bundle C × P 1 . In fact, if we perform the elementary transformations of the three special points P 0 , P 1 and P ∞ of C × P 1 : (see figure 2), we have a ruled surface S with three special points P 0 , P 1 and P ∞ (see figure 3). Recall that after 3 elementary transformations, if σ is a section on S such that σ is its strict transform on the trivial bundle, we have: σ.σ = σ .σ + r where r = 0 + 1 + ∞ such that in particular, r ∈ {−1, 1, −3, 3}. Then, we can deduce that the ruled surface S has a invariant κ 1.
1. If κ = 0, then there is σ.σ = 0 on S, then its strict transform σ .σ = odd on C×P 1 . This cannot hold because all the sections of the trivial bundle have even self-intersection; 2. If κ < 0, then there exists a +2 self-intersection section of C × P 1 passing through by three points. It is absurd because there exists a effective divisor equivalent to its normal bundle which contains at least three points.
According to these two cases, after our elementary transformations on the trivial bundle C × P 1 , we obtain a ruled surface such that its invariant κ = 1. Therefore, it is a stable ruled surface over an elliptic curve.

The Riccati foliation on S 1 after elementary transformations
Proposition 4.4. After elementary transformations of the three special points P 0 , P 1 and P ∞ on S 1 , the Riccati foliation Ric induces a Riccati foliation Ric on the trivial bundle C × P 1 such that the points P 0 = (p 0 , 0), P 1 = (p 1 , 1) and P ∞ = (p ∞ , ∞) are radial singularities.
Proof. As the problem is local, we can prove it on the surface C 2 . In our context, after elementary transformation at the origin, a regular Riccati foliation becomes the pull-back of this holomorphic foliation dz dx = az 2 + bz + c where a, b, c ∈ C by the birational map C 2 C 2 ; (x, z) → (x, xz). Thus, we obtain that a Riccati foliation such that the linear part looks like xdz −zdx = 0. Therefore the origin is a radial singularity. Finally, we can say the foliation Ric on S 1 is after elementary transformations a Riccati foliation on C × P 1 having three radial singularities at the points P 0 , P 1 and P ∞ .
If ((x, y), z) are cordinates of the trivial bundle C × P 1 , then the foliation where a, b, c are the meromorphic functions with pole of order 1 at the points p 0 , p 1 and It means that a = a 0 + a 1 x + a 2 y y , where a i are constant. If we write the same relation for the functions b and c, we obtain that the foliation Ric is locally defined by the following 1-form: As the foliation is invariant by the involution (x, y) → (x, −y) on C, the coefficients a 2 , b 2 , and c 2 are zero. Futhermore, if we use the relation on an elliptic curve, y 2 = x (x − 1) (x − t), and the fact that the points (0, 0, 0) , (1, 0, 1) , (p ∞ , ∞) are the radial singularities, we have Ric is defined by the 1-form : Lemma 4.6. The foliation Ric on C × P 1 has a rational first integral defined by the following function :

The generic 2-web after elementary transformations
After elementary transformations at the three special points on S 1 , the generic +1 self-intersection sections (i.e not passing through the three special points) become the +4 self-intersection sections of C × P 1 passing through the points (0, 0, 0), (1, 0, 1) and (p ∞ , ∞): see figure 4.
Lemma 4.7. A +4 self-intersection section passing through the points P 0 , P 1 and P ∞ is either given by the graph z = , or the graph z = x. Proof. If σ : C → P 1 is a +4 self-intersection section on the trivial bundle, then it defines a rational map of degree 2 generated by two sections σ 1 and σ 2 of a line bundle of degree 2 over C; more precisely, for any point (x, y) ∈ C, σ (x, y) = (σ 1 (x, y) : σ 2 (x, y)). Since for any line bundle of degree 2 over C, there exists a point p = (x 0 , y 0 ) ∈ C such that L = [p] + [p ∞ ], we have two cases: • if p = p ∞ , according to the Riemann Roch's theorem, H 0 (L) = C y − y 0 , x − x 0 and then, σ is a graph given by Using the fact that the section passes through the points P 0 , P 1 and P ∞ and the puiseux parametrisation of elliptic curve at the infinity point is , we obtain a system of equations which solutions x >, likewise using the fact that the section passes through the points (0, 0, 0), (1, 0, 1) and (p ∞ , ∞), we obtain that σ is the graph z = x.
From now on, unless otherwise mentionned, we will assume that a +4 self-intersection section is a section which passes through the points P 0 , P 1 and P ∞ . Now using the birational trivialisation of S 1 , we can give another proof to show that the minimal sections of S 1 define a singular 2-web and its discriminant is a leaf of the Riccati foliation on S 1 . Proof. Let (u, v, z) ∈ C × P 1 such that v = 0, we have to find the points (x 0 , y 0 ) = (u, v) of C such that z = (1 − x 0 )(y 0 u − x 0 v) y 0 (u − x 0 ) .
Proof. In fact, by the definition of the first integral of the foliation Ric, we have: Finally, the foliation ψ * Ric is locally defined by the 1-form dZ = 0. Likewise, the pull-back of the slopes Z 1 and Z 2 by ψ defines a 2-web such that the leaves verify the following differential equation : In summary, the 4-web ψ * W (∞, Z 0 , Z 1 , Z 2 ) is locally equivalent to the web W (∞, 0, β, −β), where β is a solution of ( ). As the 4-web W (∞, 0, β, −β) has a constant cross-ratios equal to −1 and all the 3 subweb are hexagonal, we can deduce that it is locally parallelizable.