Geometry of the Stable Ruled Surface Over an Elliptic Curve

We consider the stable ruled surface S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_1$$\end{document} over an elliptic curve. There is a unique foliation on S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_1$$\end{document} 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.


Introduction
Let C be an elliptic curve on C. In 1955, Atiyah proved in Atiyah (1955) that, up to isomorphism, there are only two indecomposable ruled surfaces over C: the semistable ruled surface S 0 → C and the stable ruled surface S 1 → C. In this article, we A. Diaw: supported by ANR-16-CE40-0008 Project "Foliage".
B Arame Diaw aramdiaw2@gmail.com 1 Univ Rennes, CNRS, IRMAR -UMR 6625, 35000 Rennes, France 646 A. Diaw study the geometry of the stable ruled surface. In fact, the surface S 1 can be seen as the suspension over C of the unique representation onto the dihedral group < −z, 1 z > (see Loray and Pérez 2009, p. 23). Thus, we have a Riccati foliation R on S 1 such that the generic leaf is a cover of degree 4 over C and it is the unique foliation transverse to the fibration. On the other hand, the holomorphic section σ : C → S 1 have selfintersection σ.σ ≥ 1 and those having exactly σ.σ = 1 form a singular holomorphic 2-web W. Finally, taking into account the fibration we have a singular holomorphic 4-web on S 1 . The aim of this article is to study the geometry of this 4-web composed by the Riccati foliation, the 2-web W and the P 1 -fibration π : S 1 → C. Our first result is the following: Proposition 1.1 The discriminant of the 2-web W defined by the +1 self-intersection sections on S 1 is a leaf of the foliation R.
Using the isomorphism between the curve C and its jacobian, we have the main result : Theorem 1.2 There exists a double cover ϕ : C × C → S 1 ramified on on which the lifted 4-web W is parallelizable.
Firstly, we show these results using only the properties of an elliptic curve and its jacobian and after we use the theory of birational geometry to illustrate our results with computations on a trivialization S 1 C × P 1 . This paper is part of my thesis work under the direction of Frank Loray and Frédéric Touzet.

Some Properties on An Elliptic Curve
Let C = (x, y) ∈ C 2 , y 2 = x (x − 1) (x − t) ∪ {p ∞ }, where t ∈ C \ {0, 1} be an elliptic curve. The set of points of C forms an abelian group with p ∞ as the 0 element and the points p i = (i, 0) where i = 0, 1, t, are the points of order 2 on C. Note that the map is an automorphism of C which fixes the points of order 2: it is the standard involution of the curve C.
If we denote J ac(C), the jacobian of C, we have: Lemma 2.1 There exists a group isomorphism between C and its jacobian defined by this following map: From now on, we will use this isomorphism between the additive group structure of (C, p ∞ ) and the group structure of its jacobian.

Ruled Surface Over An Elliptic Curve
Let C be a smooth curve on C.
Definition 2.2 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.

Example 2.3
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 whose detailed proof is in Hartshorne (1977): Theorem 2.4 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 : Definition 2.5 A ruled surface P(E) is decomposable if it has two disjoint sections.
The following Lemma whose proof is in (Maruyama 1970, p. 16) shows the relationship between the ruled surface S = P(E) and the vector bundle E.

Lemma 2.6
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 2.7 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 2.9 The ruled surface P(E) is stable if κ > 0.
Definition 2.10 A minimal section of S is a section σ : C → S such that the selfintersection is minimal. That is to say, σ.σ = κ.
Using Lemma 2.6, 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. Let O C ( p ∞ ) be the line bundle related to the divisor [ p ∞ ] . There are unique nontrivial extensions of invertible sheaves: Recall the following theorem of Atiyah as proved in (Atiyah 1955, Th. 6.1): Theorem 2.11 Up to isomorphism, the unique indecomposable ruled surfaces over C are S 0 = P(E 0 ) and S 1 = P(E 1 ).

Remark 2.12
Equivalenty, 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 2.13
The maximal degree of line subbundles of E 1 is zero. Remark 2.14 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.

Proof
If we consider max E 1 = {L → E 1 , deg L = 0} the set of line subbundles of E 1 having maximal degree, we have:

Lemma 2.15
There is a bijection between the jacobian of C and the set max E 1 defined by: To prove this Lemma, we have to use a key lemma of Maruyama in (Maruyama 1970, p. 8): Lemma 2.16 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 2 . Now, we can prove Lemma 2.15: 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 Lemma 2.16, the last case cannot occur because E 1 is not decomposable. Thus, the map M is injective. • Let L ∈ J ac (C) be a line bundle distinct from the trivial bundle. If we apply the functor Hom (L, −−) to the exact sequence and we use Riemann Roch's theorem, we obtain dim Hom (L , E 1 ) = 1. There exists a non zero morphism τ : L → E 1 . Thus, if we denote D the effective divisor of zeros of τ then L ⊗ O C (D) is a line subbundle of E 1 . Since deg (L) = 0, D is a effective divisor of zero degree, that is to say O C (D) = O C . Hence, L is a line subbundle of E 1 .

Remark 2.17
The minimal sections of S 1 have self-intersection equal to 1 and they are parametrised by the jacobian which is isomorphic to C. For any point ∈ C, we denote σ the minimal section corresponding via Lemma 2.15 to the subbundle In the sequel, we use the equivalence between line bundles and divisors on C (Fig. 1).

Lemma 2.18
Let σ and σ be two minimal sections of the ruled surface S 1 . If we set P := σ .σ , the intersection point of σ and σ , we have: where π : S 1 → C is the projection map and the sign − is the inverse of the law group of C.
be the subbundles of E 1 associated with sections σ and σ respectively. Using the fact that S 1 = P(E 1 ) we have for every x ∈ C, σ (x) and σ (x) are respectively the projectivisation of the

Remark 2.19
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.
We also have the following theorem proved by André Weil in Weil (1995):

Theorem 2.20 A holomorphic vector bundle on a compact Riemann surface is flat if and only if it is the direct sum of indecomposable vector bundles of degree 0.
By this theorem, the 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 Loray and Pérez (2009).

Theorem 2.21
The ruled surface S 1 has a Riccati foliation R with irreducible mon-

Remark 2.22
The generic leaf of the foliation R is a degree 4 cover of C and three special leaves are covers of degree 2.

Geometry of the Ruled Surface S 1
Let π : S 1 → C be the stable ruled surface over C and let Aut C (S 1 ) be the group of automorphisms of the fibration over the identity of C. In the sequel, unless otherwise specified by automorphism group of S 1 we mean the group Aut C (S 1 ) .

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 self-intersection 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 2.18, we obtain that for any p ∈ C, ψ σ − p− 0 = σ − p− 0 . 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 ψ ẑ = z + b. As by definition we have: Conversely, for any point p i of order 2 on C we can define an automorphism i on S 1 such that for any point p ∈ C, i restricts to the fiber π −1 ( p) as the unique Moebius map which associates the points of the sections (σ 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 : Remark 3.2 If we consider the bijection : → σ between the curve C and the minimal sections, we can see the set of minimal sections as a riemann surface in such a way that the application is holomorphic. Hence we can deduce that the application ψ = −1 • ψ • is holomorphic.

Proposition 3.3 The automorphism group of S 1 preserves the foliation R.
Proof Using the fact that the fundamental group of C is abelian, we can see that the restriction of the projection π : S 1 → C to a generic leaf F of the Riccati foliation R is a galois covering whose deck transformation group is the restriction of Aut C (S 1 ) to F. Hence the group Aut C (S 1 ) preserves the Riccati foliation on S 1 .

Corollary 3.4
The Riccati foliation R is the unique regular Riccati foliation on the ruled surface S 1 .
Proof Let F 1 be a smooth Riccati foliation on S 1 . As its monodromy group is an abelian subgroup of PG L(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 theorem of Camacho Sad (see Camacho and Neto 1985), 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 < −z, 1 z >.
Thus, the foliation F 1 is conjugated to R by an element in Aut C (S 1 ). As this automorphism group of the fibration S 1 preserves the foliation R, we have R = F 1 .
Lemma 3.5 With the identification J ac(C) C given by Lemma 2.1 and the notation considered in Remark 2.17 we have: 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 Lemma 2.18, three minimal sections cannot meet at the same point, then we deduce that for any p ∈ C, the morphism is not constant: it is a ramified cover between Riemann surfaces. Futhermore, by Remark 2.19, 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 3.6 There exists a irreducible singular holomorphic 2-web W on S 1 defined by the minimal sections whose discriminant is a leaf of the foliation R.

Remark 3.7
The 2-web W defined by minimal sections cannot be decomposed because there exists a unique smooth Riccati foliation on S 1 .
Proof By Lemma 3.5, for any point P ∈ π −1 ( p) there exists a minimal section σ passing through this point. Likewise, by Lemma 2.18, unless they are equal, the minimal section σ − p− intersects transversally the minimal section σ at the point P.
As the sections σ and σ − p− are distinct if and only if 2 = −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 3.8 There exists a linear foliation F on C × J ac (C) such that ϕ * F = R.
Proof Assume that C × J ac (C) (C/Z + τ Z) × (C/Z + τ Z). 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 project the foliation F to a foliation, F on C × J ac (C) such that the monodromy is defined by : The foliation F is transverse to the first projection on C × J ac (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, ) is the leaf passing through the point ( p, ), 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 R. Using the uniqueness of R by Corollary 3.4, we obtain the result.
As by definition the curve G = {(2 p, − p) / p ∈ C} is a leaf of the foliation F, using the foregoing lemma we can deduce that ϕ (G) = is a leaf of R. Which completes the proof of Theorem 3.6.

Study of Special Leaves of the Riccati Foliation R
According to Lemma 3.8, if P = σ 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 3.9 There exists three special leaves R 0 , R 1 and R t of the foliation R which are double cover of C. More precisely, they are respectively the set of fixed points of the automorphisms 0 , 1 and t .

A. Diaw
Proof We just give the proof for the leaf R 0 because it is the same process for the other special leaves.
-Let R 0 be the Riccati leaf passing through the point P 0 = σ p 0 ( p 0 ), then by definition, we have : According to the monodromy of R, if the leaf R 0 passes through the point P = σ ( p) then it passes through the points σ + p i ( p), where p i is a 2-torsion point of C. Since 2 = p 0 − p, we deduce from Lemma 2.18 that: σ + p 0 ( p) = σ ( p) and σ + p 1 ( p) = σ + p t ( p) thus, Ric 0 meets any fiber of S 1 twice. It is a double cover over C. -Let 0 be the automorphism of S 1 related to the point p 0 defined by: and consider the set of its fix points Fix 0 = {P / 0 (P) = P} . If P = σ ( p) is the fix point of 0 , then by Lemma 2.18, we have 2 = p 0 − p and therefore P ∈ R 0 . Conversely, if P ∈ R 0 , we have by definition 2 = p 0 − p, and according to Lemma 2.18, we have σ + p 0 ( p) = σ ( p). Thus, we deduce that: According to all the foregoing, we have:

Remark 3.10
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 whose the discriminant is .

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 project them to a decomposable 2-web W on C × J ac (C).

Proposition 3.11
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 2-web on S 1 . Let p , ∈ C × J ac (C) and consider the leaves of the 2-web W passing through this point. Since, using Lemma 2.18, 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 × J ac (C) given by the 2-web W , the foliation F and the J ac (C)-bundle defined by the first projection on C × J ac (C).
Theorem 3.12 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 3.13
An immediate consequence of Theorem 3.12 is that the curvature of the 4-web W is zero.
The second part of this paper aims to use the theory of birational 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.
Definition 4.1 An elementary transformation at the point P ∈ π −1 ( p) is the birational map given by the composition of the blow-up of the point P, followed by the contraction of the proper transform of the fiber π −1 ( p).

Remark 4.2
After elementary transformation at the point P, we obtain a new ruled surface with a point P which is the contraction of the proper transform of the fiber π −1 ( p).
How many elementary transformations do you need to trivialize the ruled surface S 1 ?
In fact, if we perform the elementary transformations at the three special points P 0 , P 1 and P ∞ of C × P 1 : (see Fig. 2), we have a ruled surface S with three special points P 0 , P 1 and P ∞ (see Fig. 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, there is a section σ of zero self-intersection on S and then its strict transform σ has odd self-intersection. This cannot hold because all the sections of the trivial bundle have even self-intersection; 2. If κ < 0, let σ be a section of S such that σ.σ = κ. Using the fact that its strict transform σ has a strictly positive self-intersection, we can see that either κ = −1 or κ = −2. As the ruled surface S admits a section σ 0 of +1 self-intersection, the case where κ = −2 cannot occur. If κ = −1, then σ is a section of +2 self-intersection on C × P 1 passing through by three points P 0 , P 1 and P ∞ . Thus, this section defines a non-constant morphism σ : C → P 1 of degree 1. It is absurd because the curve C is not rational.
According to these two cases, after three elementary transformations on the trivial bundle C × P 1 , we obtain a ruled surface such that its invariant κ = 1. Therefore, it is the 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 R induces a Riccati foliation R 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.
If ((x, y), z) are cordinates of the trivial bundle C × P 1 , then the foliation R is Thus, 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 R is defined by the following 1-form: As the foliation is invariant by the involution I : (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, , and the fact that the points (0, 0, 0) , (1, 0, 1) , ( p ∞ , ∞) are the radial singularities, we have R is defined by the 1-form : Lemma 4.7 A +4 self-intersection section passing through the points P 0 , P 1 and P ∞ is either given by a graph , or a 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 = (a 0 , b 0 ) ∈ C such that L = [ p] + [ p ∞ ], we have two cases: • if p = p ∞ , due to the Riemann Roch's theorem, we have the vector space H 0 (L) = C y − b 0 , x − a 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 given , we obtain a system of equations which solutions are 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, we denote S the set of +4 self-intersection sections of the trivial bundle which pass through the points P 0 , P 1 and P ∞ . (u, v, z) ∈ C × P 1 such that v = 0, there exists a +4 self-intersection section in S which passes through this point.

Proposition 4.8 For any point
Proof Let (u, v, z) ∈ C × P 1 such that v = 0, we have to find the points Using the fact that b 2 0 = a 0 (a 0 − 1)(a 0 − t) and v 2 = u(u − 1)(u − t), we have the following equation: As by hypothesis v = 0, we obtain two solutions given by the point (a 0 , b 0 ) such that a 0 = t(u − 1) u − t and the point p ∞ ; 2. if (u − z) = 0, then the solutions verify the following second degree equation: The +4 self-intersection sections in S define a singular holomorphic 2-web W on C × P 1 such that the discriminant is the union of , the discriminant of the equation ( ) and the singular fibers at the points p 0 , p 1 and p ∞ . We have: := (t − u)z 4 − 4(t − 1))uz 3 + 2u(2tu + t − u − 2)z 2 − 4u 2 (t − 1)z + u 2 (t − u) = 0 Lemma 4.9 The discriminant is a leaf of order 4 of the Riccati foliation R.

Proof
In fact, by the definition of the first integral of the foliation R, we have: Therefore, the first integral is constant along of the discriminant .
According to the foregoing, on the birational trivialisation of S 1 , we have a 4-web W 4 defined by the 2-web W, the Riccati foliation R and the trivial fiber bundle.

Geometry of the 4-web W 4
We want to find the slopes of the leaves of W 4 in order to represent it by a differential equation. Let (x 0 , y 0 , z 0 ) ∈ C × P 1 be a generic point. As the leaves of the 2-web W passing through this point are respectively the graph z = (1 − a 0 )(b 0 x − a 0 y) where μ = z 2 − z 0 z 2 − z 1 , then the pull-back of the first integral of R by ψ is the following meromorphic function : Finally, the foliation ψ * R is locally defined by the 1-form d Z = 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.
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