P∈π−1(p) /P=σ (p), 2 = −p In order to prove thatis a leaf of the Riccati, we need the following :
Lemma 3.8 There exists a linear foliationFon C×J ac(C)such thatϕ∗F=R.
Proof Assume thatC×J ac(C)(C/Z+τZ)×(C/Z+τZ).If we consider the linear foliationF:=dx+2dyonCx ×Cy, thenFis invariant by the action of the latticeZ+τZ. Thus we can project the foliationFto a foliation,FonC×J ac(C) such that the monodromy is defined by :
⎧⎨
⎩
ξ:−→ Aut(C) λ−→
p→ p−1 2λ whereis the latticeZ+τZ.
The foliationFis transverse to the first projection onC×J ac(C)with a mon-odromy group isomorphic to the group of points of order 2{p∞,p0,p1,pt}. Moreover, ifF(p, )is the leaf passing through the point(p, ), then by definition we have :
i F(p, )
=F(p,−p− )
whereiis the involution of the ramified double coverϕ. Hence,ϕ∗Fthe direct image of the foliationFbyϕis a Riccati foliation onS1having the same monodromy group thanR. Using the uniqueness ofRby Corollary3.4, we obtain the result.
As by definition the curveG= {(2p,−p) /p∈C}is a leaf of the foliationF, using the foregoing lemma we can deduce thatϕ (G)=is a leaf ofR. Which completes
the proof of Theorem3.6.
3.1 Study of Special Leaves of the Riccati FoliationR According to Lemma3.8, ifP=σ
p
∈S1then the Riccati leaf passing through at this point is given by
RP =
P ∈S1| P=σ (p) ,p=π(P),2 =2 +p−p
Thus, if we use this characterisation of the Riccati leaves onS1, we have the following lemma :
Lemma 3.9 There exists three special leavesR0,R1andRt of the foliationRwhich are double cover of C. More precisely, they are respectively the set of fixed points of the automorphisms0,1andt.
Proof We just give the proof for the leafR0because it is the same process for the other special leaves.
– LetR0 be the Riccati leaf passing through the point P0 = σp0(p0), then by definition, we have :
R0= {P∈ S1| P =σ (p) ,p=π(P),2 = p0−p}
According to the monodromy ofR, if the leafR0passes through the point P = σ (p)then it passes through the pointsσ +pi(p), where pi is a 2-torsion point ofC. Since 2 = p0−p, we deduce from Lemma2.18that:σ +p0(p)=σ (p) andσ +p1(p)=σ +pt(p)thus,Ri c0meets any fiber ofS1twice. It is a double cover overC.
– Let0be the automorphism ofS1related to the pointp0defined by:
0:S1 −→ S1
P=σ (p)−→ P=σ +p0(p) and consider the set of its fix points Fix0= {P/0(P)=P}.
IfP=σ (p)is the fix point of0, then by Lemma2.18, we have 2 = p0−p and thereforeP∈R0. Conversely, ifP ∈R0, we have by definition 2 = p0−p, and according to Lemma2.18, we have σ +p0(p) = σ (p). Thus, we deduce that:
R0= {P /0(P)=P}
According to all the foregoing, we have:
Remark 3.10 The 2-web given by the+1 self-intersection sections, the Riccati foli-ation and theP1-bundleπ: S1 → C form a singular holomorphic 4-webWonS1
whose the discriminant is. 3.2 The Geometry of the 4-Web W
Let(x,y)be the local coordinates ofC2. As the linear foliationsGandHrespectively defined by dy=0 and by dy+dx=0 are invariant by the action of the latticeZ+τZ, we can project them to a decomposable 2-webWonC×J ac(C).
Proposition 3.11 The direct imageϕ∗ W
of the2-webWby the ramified coverϕ is the2-webWon S1defined 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 S1. Let p,
∈C×J ac(C)and consider
• A =
(p, )∈C×J ac(C) / =
• B =
(p, )∈C×J ac(C) / = −p+
p+ ,
the leaves of the 2-webWpassing through this point. Since, using Lemma2.18, we have:ϕ(A )=σ andϕ(B)=σ−p− , then the leaves ofϕ∗
W
are the minimal sections ofS1which are the same along the discriminant. The local study of the 4-webWonS1is the same as the 4-web onC×J ac(C)given by the 2-webW, the foliationFand theJ ac(C)-bundle defined by the first projection onC×J ac(C).
Theorem 3.12 Outside the discriminant locus, the4-webWis locally parallelizable.
Proof According to the foregoing, the pull-back of 4-webWby the ramified cover ϕis locally the 4-web defined byW(x,y,y+x,y+2x)onC2. It is a holomorphic
parallelizable web.
Remark 3.13 An immediate consequence of Theorem3.12is that the curvature of the 4-webWis 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×P1.
4 Geometry of 4-Web W After Elementary Transformations
Letπ: S1→Cbe theP1-bundle and{p0,p1,pt,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 bira-tional map given by the composition of the blow-up of the pointP, followed by the contraction of the proper transform of the fiberπ−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 surfaceS1? Lemma 4.3 The ruled surface S1is obtained after three elementary transformations at the points P0=(p0,0) , P1=(p1,1)and P∞=(p∞,∞)on the trivial bundle C×P1.
In fact, if we perform the elementary transformations at the three special pointsP0, P1andP∞ofC×P1: (see Fig.2), we have a ruled surfaceSwith three special points P0,P1andP∞(see Fig.3).
Recall that after 3 elementary transformations, ifσis a section onSsuch thatσis its strict transform on the trivial bundle, we have:σ.σ =σ.σ+rwherer= 0+ 1+ ∞
such that
i =
−1 if Pi ∈σ
+1 if Pi ∈/σ (1)
Fig. 2 Special points of the trivialP1-bundle overC
Fig. 3 Special points and special sections of+1 self-intersection ofS1
in particular,r ∈ {−1,1,−3,3}. Then, we can deduce that the ruled surfaceShas a invariantκ ≤1.
1. If κ = 0, there is a section σ of zero self-intersection on S and then its strict transformσhas odd self-intersection. This cannot hold because all the sections of the trivial bundle have even self-intersection;
2. Ifκ <0, letσ be a section of S such thatσ.σ =κ. Using the fact that its strict transformσhas a strictly positive self-intersection, we can see that eitherκ = −1 orκ = −2. As the ruled surface S admits a sectionσ0 of+1 self-intersection, the case whereκ = −2 cannot occur. If κ = −1,thenσ is a section of +2 self-intersection onC×P1passing through by three pointsP0,P1andP∞. Thus, this section defines a non-constant morphismσ:C →P1of degree 1. It is absurd because the curveCis not rational.
According to these two cases, after three elementary transformations on the trivial bundleC×P1, we obtain a ruled surface such that its invariantκ =1. Therefore, it is the stable ruled surface over an elliptic curve.
4.1 The Riccati Foliation onS1After Elementary Transformations
Proposition 4.4 After elementary transformations of the three special points P0, P1
and P∞ on S1, the Riccati foliationRinduces a Riccati foliationRon the trivial bundle C×P1such that the pointsP0=(p0,0),P1=(p1,1)and P∞=(p∞,∞)
whereM(C)is the field of meromorphic functions onC.By Puiseux parametrisation of the curveCwe can see that meromorphic functions1
yandx
ybelong toH0(OC(D)), and then we can deduce that the family
by Riemann-Roch theoremH0(OC(D))is a vector space of dimension 3,we have:
H0(OC([p0] + [p1] + [p∞]))=C<1,1 y,x
y > .
Thus, it means thata =a0+a1x+a2y
y , whereaiare constant. If we write the same relation for the functionsb andc, we obtain that the foliation Ris defined by the following 1-form:
yd z−
(a0+a1x+a2y)z2+(b0+b1x+b2y)z+(c0+c1x+c2y)d x 2y As the foliation is invariant by the involution I: (x,y)→(x,−y)onC, the coeffi-cientsa2,b2,andc2are zero. Futhermore, if we use the relation on an elliptic curve, y2 =x(x−1) (x−t), and the fact that the points(0,0,0) , (1,0,1) , (p∞,∞)are the radial singularities, we haveRis defined by the 1-form :
w:=d z+
Proposition 4.5 The monodromy group of the foliation R along a generic fiber π−1(x0,y0)is an abelian group given by these automorphisms:0:z → z−x0
z−1, 1:z→ x0
z ,t:z→ x0(z−1) z−x0
,∞:z→z.
Proof Letσ∞ := {z= ∞},σ0 := {z=0},σ1 := {z=1}andσd := {z=x}be the four special sections obtained after elementary transformations. By definition of the monodromy group ofR, we can see that for the point p0of order 2, the automor-phism0restricted to any fiber is the unique Moebius transformation which sends respectively the points of the sections(σ0,σ1,σ∞,σd)to the points of the sections (σd,σ∞,σ1,σ0). Using the same process for the other automorphisms, we obtain the result.
We can also describe the special leaves of the foliation R. In fact, if we consider φi:C×P1→C×P1;(x,y,z)→(x,y,i(z)), then according to Lemma3.9, the special leaves are defined by :
1. R0:= {(x,y,z) , φ0(x,y,z)=(x,y,z)} =
(x,z) ,−z2−x+2z=0 2. R1:= {(x,y,z) , φ1(x,y,z)=(x,y,z)} =
(x,z) ,−z2+x=0 3. Rt := {(x,y,z) , φt(x,y,z)=(x,y,z)} =
(x,z) ,z2−2x z+x=0
Now it is natural to ask if we can find the expression of the generic leaf ofR. To do this, we use the special leaves to find a first integral. Let
f0:= −z2+2z−x, f1:= −z2+x, ft :=z2−2x z+x
be the polynomials which define respectively the leavesR0,R1,Rt and consider the functionγ:Cx×P1z →Cx×P1y;(x,z)→(x,F(x,z)), where
F(x,z)=x f0
ft
2
The pull-backγ∗d yof the 1-formd ybyγis a foliation onCx×P1zhaving the function F(x,z)as a first integral and such that the curvesR0,R1andRtare invariant. Hence, the foliationγ∗d ycoincide with the Riccati foliationR. Thus, we can deduce that : Lemma 4.6 The foliationR on C ×P1 has a rational first integral defined by the following function :
F(x,z)= x(z2−2z+x)2 (−z2+2x z−x)2 4.2 The 2-Web After Elementary Transformations
After elementary transformations at the three special points on S1, the generic +1 self-intersection sections (i.e not passing through the three special points) becomes a
Fig. 4 Generic+4 self-intersection section
+4 self-intersection sections ofC×P1passing through the points(0,0,0),(1,0,1) and(p∞,∞): see Fig.4.
Lemma 4.7 A+4self-intersection section passing through the pointsP0,P1and P∞ is either given by a graph z=(1−a0)(b0x−a0y)
b0(x−a0) , or a graph z=x.
Proof Ifσ:C → P1 is a+4 self-intersection section on the trivial bundle, then it defines a rational map of degree 2 generated by two sections σ1 andσ2 of a line bundle of degree 2 over C; more precisely, for any point (x,y) ∈ C,σ (x,y) = (σ1(x,y):σ2(x,y)). Since for any line bundle of degree 2 overC, there exists a pointp=(a0,b0)∈Csuch thatL =[p]+[p∞], we have two cases:
• if p = p∞, due to the Riemann Roch’s theorem, we have the vector space H0(L)=Cy−b0,x−a0and then,σis a graph given by
z= a(y−b0)+b(x−a0)
c(y−b0)+d(x−a0), a,b,c,d∈C
Using the fact that the section passes through the points P0, P1 and P∞ and the Puiseux parametrisation of elliptic curve at the infinity point is given by ε → (1
ε2, 1
ε3), we obtain a system of equations which solutions are
a=d a0(a0−1)
b0 ,b= −d(a0−1),c=0,d =d
whered =0;
• ifp = p∞,then we haveH0(L)=C<1,1
x >, likewise using the fact that the section passes through the points(0,0,0),(1,0,1)and(p∞,∞), we obtain that σis the graphz=x.
From now on, we denoteSthe set of+4 self-intersection sections of the trivial bundle which pass through the points P0,P1andP∞.
Proposition 4.8 For any point(u, v,z)∈C×P1such thatv=0, there exists a+4 self-intersection section inSwhich passes through this point.
Proof Let(u, v,z)∈C×P1such thatv=0, we have to find the points(a0,b0)= (u, v)ofCsuch thatz= (1−a0)(b0u−a0v)
b0(u−a0) .
Using the fact thatb02=a0(a0−1)(a0−t)andv2 =u(u−1)(u−t), we have the following equation:
(♣):(u−z)2a03− [(2(uz−u))(−z+u)−(−z+u)2t−v2]a02+ [(uz−u)2− (2(uz−u))(−z+u)t+v2]a0−(uz−u)2t =0
1. if(u−z)=0, then(♣)becomes(a0−u)
a0−t(u−1) u−t
=0. As by hypothesis v=0, we obtain two solutions given by the point(a0,b0)such thata0= t(u−1)
u−t and the pointp∞;
2. if(u−z)=0, then the solutions verify the following second degree equation:
():(u−z)2a20+ [(−t−u)z2+2u(t+1)z−u(t+u)]a0+t u(z−1)2=0 The+4 self-intersection sections inSdefine a singular holomorphic 2-webWon C×P1such that the discriminant is the union of,the discriminant of the equation ()and the singular fibers at the points p0,p1andp∞.We have:
:=(t−u)z4−4(t−1))uz3+2u(2t u+t−u−2)z2−4u2(t−1)z+u2(t−u)=0
Lemma 4.9 The discriminantis a leaf of order4of the Riccati foliationR.
Proof In fact, by the definition of the first integral of the foliationR, we have:
F(x,z)−t= −(t−u)z4−4(t−1))uz3+2u(2t u+t−u−2)z2−4u2(t−1)z+u2(t−u) (2x z−z2−x)2
Therefore, the first integral is constant along of the discriminant. According to the foregoing, on the birational trivialisation ofS1, we have a 4-webW4
defined by the 2-webW, the Riccati foliationRand the trivial fiber bundle.
4.3 Geometry of the 4-webW4
We want to find the slopes of the leaves ofW4in order to represent it by a differential equation. Let(x0,y0,z0) ∈ C×P1be a generic point. As the leaves of the 2-web W passing through this point are respectively the graphz = (1−a0)(b0x−a0y)
b0(x−a0)
andz= (1−a0)(b0x−a0y) Thus, the irreducible 2-webWis defined by the following differential equation:
dz
Proof The pull-back of the foliationRby the multiplication of order 2 onCis another Riccati foliationR2onC×P1with trivial monodromy.
LetM2:C→C,be the multiplication of order 2 onCthen, for any point(x,y)∈ Cthe first projection ofM2(x,y)is given by the following formula :
pr1◦M2(x,y)= (3x2−2(t+1)x+t)2
4x(x−1)(x−t) +(1+t)−2x Using the pull-back of the special leaves, we can choose three curves by:
1. C0:=
whereμ= z2−z0
z2−z1
, then the pull-back of the first integral ofRbyψis the following meromorphic function :
(ψ∗F) (X,Z)=(Z2−2Z+2)2 Z2(Z−2)2
Finally, the foliationψ∗Ris locally defined by the 1-formd Z = 0. Likewise, the pull-back of the slopesZ1andZ2byψdefines a 2-web such that the leaves verify the following differential equation :
(): d Z
d X 2
+(t−1)Z4+(−4t+4)Z3+(4t−8)Z2+8Z−4
4X(X−1)(X−t) =0
In summary, the 4-webψ∗W(∞,Z0,Z1,Z2)is locally equivalent to the web W(∞,0, β,−β), whereβ is a solution of(). As the 4-webW(∞,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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