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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References
Atiyah, M.F.: Complex fibre bundles and ruled surfaces. Proc. Lond. Math. Soc.3(5), 407–434 (1955) Atiyah, M.F.: Vector bundles over an elliptic curve. Proc. Lond. Math. Soc.3(7), 414–452 (1957) Camacho, C., Neto, A.L.: Geometric theory of foliations. Birkhäuser, Boston (1985)
Friedman, R.: Algebraic surfaces and holomorphic vector bundles. Universitext. Springer, New York (1998) Hartshorne, R: Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New
York-Heidelberg (1977)
Loray, F., Pérez, D.M.: Projective structures and projective bundles over compact Riemann surfaces.
Astérisque323, 223–252 (2009)
Maruyama, M.: On classification of ruled surfaces. Kinokuniya, Tokyo (1970)
Mukai, S.: An introduction to invariants and moduli, vol. 81. Cambridge University Press (2003) Pirio, L.: Sur les tissus plans de rang maximal et le problème de Chern. C. R. Math. Acad. Sci. Paris339(2),
131–136 (2004)
Ripoll, O.: Détermination du rang des tissus du plan et autres invariants géométriques. C. R. Math. Acad.
Sci. Paris341(4), 247–252 (2005)
Weil, A.: Variété de Picard et variétés jacobiennes. In: Séminaire Bourbaki, Vol. 2, pages Exp. No. 72, 219–226. Soc. Math. France, Paris, (1995)
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