Aravinthan Gopalasingham
email: gopalasingham.aravinthan@nokia.com

Dalia Georgiana Herculea
email: dalia-georgiana.herculea@nokia.com

Chung Shue Chen
email: chungshue.chen@nokia.com

Laurent Roullet
email: laurent.roullet@nokia.com

Virtualization of Radio Access Network by Virtual Machine and Docker: Practice and Performance Analysis

Keywords: 5G, radio access network, SDN, Virtual Machine, Docker container, performance analysis

Software defined networking (SDN) and network function virtualization (NFV) are the embraced technologies for the backhauling of future 5G networks. Virtual Machine (VM) and Docker container based deployments have received much attention. This paper presents the virtualization of a prototyped software defined radio access network (RAN) architecture by using VMs and Docker containers. In addition, it provides an analytical model for the generalized software defined RAN architecture with the practice of VM based and Docker container based implementations. Using measurements obtained from the two testbeds and the introduced queuing model, we compare their performances and analyze the two different architectures. Results verify the superiority of the Docker technology. Some observations from the behavior of the testbeds are concluded for a better understanding of the VM and Docker container based technologies for the future development of 5G SDN controller.

I. INTRODUCTION

5G will enable new user experience and ultra-broadband service [START_REF]Looking ahead to 5G[END_REF]. Future mobile networks is expected to offer a capacity up to a thousand times of legacy 3G and 4G. To meet the network capacity beyond 4G, base stations (BS), also known as eNodeB (eNB), will be deployed in very high density and as heterogeneous cellular networks, consisting of macro cells and small cells of various coverage range. They will support various type of devices and communication setups (e.g., mobile users, vehicular communications for autonomous driving car, Internet of Things) while meeting the requirements of ultra low latency and high quality of experience.

Traditional wireless mobile network infrastructure was originally designed to handle voice traffic. However, today's mobile networks are being overwhelmed by the huge demand for data traffic. It is necessary to upgrade to infrastructures that can support very dense mobile networks as well as the interoperability between vendors, including interference management and dynamic escalation of network bandwidth

The research leading to these results has received funding from the European Union Superfluidity H2020 project, https://5g-ppp.eu/superfluidity. A part of the work was carried out at LINCS (www.lincs.fr).

demand. Besides, low power consumption, agile traffic management and high reliability are important for today's network architectures [START_REF]The benefits of cloud RAN architecture in mobile network expansion[END_REF].

In order to meet these requirements and various service demands, the network should be very flexible and easily reconfigurable. The conceptual evolution of SDN that logically centralizes the control plane of the networks leverages standard protocols for managing and re-programming the entire network on the fly. The evolution of SDN is going on for enabling the programmability and flexibility of the radio access networks (RAN). The major benefit of such approach is to allow complete intelligence to the centralized controller so that the functionality of RAN can be better optimized [START_REF] Boviz | Multi-cell coordination in Cloud RAN: Architecture and optimization[END_REF]. On the other hand, NFV is the process of relocating or migrating network functions from dedicated hardware to generic servers. NFV often involves the usage of virtualization technologies such as VMs and containers for deployments, which are often called Virtual Network Functions (VNFs) in such use cases. SDN and NFV are two closely related technologies that are often used together. They are complementary and can benefit from each other.

Centralized-RAN, Cloud-RAN, or C-RAN architecture can address the requirements of network self-optimization, selfconfiguration and self-adaptation in software control through SDN and NFV. Cloud-RAN can support mobile xHaul (Fronthaul and Backhaul) and provide great benefits such as decreasing the operational cost [START_REF] Boviz | Cost-aware fronthaul rate allocation to maximize benefit of multi-user reception in C-RAN[END_REF] and improving the network security and flexibility. In the context of C-RAN, virtualization plays a key role in deploying and managing the life cycle of mobile network functions and services in the central cloud in order to achieve efficient resource utilization, elasticity, and load balancing. The NFV also results in a significant reduction in the CAPEX as well as OPEX by using automation techniques such as orchestration [START_REF] Gopalasingham | Software-defined mobile backhaul for future train to ground communication services[END_REF] together with SDN.

Nowadays, there are two major methods for achieving the dynamic resource control for the NFV: virtual machines (VM) and Docker containers. In today's practice, virtual machines are extensively used as they permit workloads to be isolated from each other and for the resources to be well controlled [START_REF] Felter | An updated performance comparison of virtual machines and Linux containers[END_REF]. During the last few months, much attention has been given to Docker container based deployments of NFV. Docker containers have the great advantage of allowing applications to run separately from the host infrastructure and to treat the infrastructure like a managed application [START_REF] Preeth | Evaluation of Docker containers based on hardware utilization[END_REF]. In particular, Docker can more easily adapt to the rapidly evolving infrastructure of virtualized RAN, with the possibility to ship code faster, test faster, deploy faster, and shorten the cycle between writing code and running code.

Some experiments have shown that Docker based deployment can offer superior performance compared to VM based deployment [START_REF] Felter | An updated performance comparison of virtual machines and Linux containers[END_REF]. However, there is a lack of explicit model and study to facilitate the analysis and estimation of possible performance gain and address for example when it is of interest to use Docker containers or to use VM and how do these architectures scale. There are many interesting questions around.

A. Contributions

This paper presents a SDN controller for radio access networks built on virtual machines and Docker containers, respectively. We investigate their performance in our testbeds with VM implementation and Docker container implementation. Lessons and experience learned from the VM implementation and container implementation are reported.

The contributions of the paper are summarized below:

• We describe the practice of NFV implementation for 5G radio access networks and in particular the new option of using the emerging technology Docker containers for the generalized SDN platform introduced in

B. State of the Art

Docker container is an emerging technology that has been proposed for the deployment of SDN architecture. There are some studies about the practice and performance based on measurements and experiments. The authors of [START_REF] Slominski | Building a multi-tenant cloud service from legacy code with Docker containers[END_REF] propose Docker containers for building multi-tenant cloud infrastructure. In [START_REF] Preeth | Evaluation of Docker containers based on hardware utilization[END_REF], it is shown that in the described testbed, the performance achieved by using Dockers is the same as the performance in a bare metal system. Today, Docker container is a competitor of the virtual machine technology in the deployment of SDN for RAN. In what regards evaluations based on analytical models, some investigations were done in order to provide accurate models for characterizing SDN controllers. The authors of [START_REF] Azodolmolky | An analytical model for software defined networking: A network calculus based approach[END_REF] use an approach based on network calculus to study SDN architecture. The work in [START_REF] Yen | An SDN-based cloud computing architecture and its mathematical model[END_REF] proposes a SDN architecture with analytical model to study user quality-of-service (QoS) under the SDN based cloud computing architecture. In [START_REF] Mahmood | Modelling of OpenFlow-based software-defined networks: the multiple node case[END_REF], the authors introduce a model to address multiple node OpenFlow based SDN by approximating the data plane as an open Jackson network with the controller modeled as an M/M/1 queue. An investigation of SDN is performed in [START_REF] Metter | Investigating the impact of network topology on the processing times of SDN controllers[END_REF] to analyze the impact of the network topology and size on the performance of the controller.

To the best of our knowledge, no comparison between the performances obtained in VM based architecture and Docker container based architecture has been done by modeling the controller as a queuing system.

II. GENERALIZED SDN PLATFORM FOR RAN

The main objective of our research is to bring programmability and flexibility to the RAN using SDN. The major benefit of such approach is to decouple network intelligence to the logically centralized controller(s) so that the functionality of RAN can be better optimized or re-programmed. Fig. 1 shows the general architecture of our SDN controller. It follows the standard three-layered architectural approach with (i) South-Bound (SB) protocol plug-ins to manage the communication between the controller and distributed data-plane entities of the network, (ii) controller modules to implement the network abstraction, and (iii) vendor agnostic Application Programming Interfaces (APIs) for NorthBound (NB) applications and algorithms to program the network.

A. SDN Platform

We implemented our architecture by extending the Open-DayLight (ODL) SDN controller for RAN features along with the integration of MongoDB to maintain the persistence of information and network intelligence. Radio Net Flow (RNF) is the SB protocol plugin introduced to integrate eNBs to this framework. The RNF is a SDN realization of the proprietary X2 interface [START_REF] Sesia | LTE -The UMTS Long Term Evolution: From Theory to Practice[END_REF] used between base stations to facilitate information exchange for the purpose of RAN optimization. Each base station in the network has also an integrated corresponding SDN-Agent called Radio Net Flow Agent (RNFA) [START_REF] Gopalasingham | Generalized software defined network platform for radio access networks[END_REF]. The purpose of the RNFA is to establish communication with the controller. The RAN Configuration Manager, the RAN Statistics Manager, and the RAN topology Manager are the three new modules introduced in the Service Abstraction Layer (SAL) of ODL in our design to abstract and re-program both PHY and MAC layer functions of RAN using the SB RNF protocol.

Each eNB during the initialization establishes a connection to the SDN controller via RNF protocol. The RAN configuration manager in the controller is responsible for validating the initial connectivity to the eNB. Each eNB sends measurements and configuration related information to the controller using RNFA, which will be taken by the RAN Statistics Manager so as to store RAN statistic information in its database (DB). Since RNF protocol is built using Stream Control Transmission Protocol (SCTP), eNBs and the controller communicate in a round robin manner. RAN topology manager creates a network topology based on the neighbor related information that it receives from the eNB. Network applications and optimizations (e.g., enhanced intercell interference coordination, a.k.a. eICIC [START_REF] Sesia | LTE -The UMTS Long Term Evolution: From Theory to Practice[END_REF]) running on the top of the controller read the network configuration, measurements and topology information, using corresponding HTTP REST (Representational State Transfer) APIs as an input for the execution. Once the application finishes its cycle of execution, it stores the re-configuration parameter in the DB to be sent to corresponding eNBs. RAN configuration manager reads the DB frequently and sends new configuration information to eNB. When the RNFA receives the reconfiguration parameters from the controller, it re-configures the corresponding functions in the eNB (e.g., MAC scheduling, frequency or time radio resource allocation decision, power control settings, etc).

B. Testbed Implementation

In the proposed architecture, we deployed 4G/5G eICIC optimizer as a RAN application of the controller which is to coordinate MAC scheduling over LTE networks. In the testbed and experimentation, we employed a Matlab based LTE-complaint simulator from TU Wien's Institute of Telecommunications to emulate LTE mobile network scenarios. It simulates typical 3GPP heterogeneous cellular network (HetNet) scenarios consisting of macro and small cells, and mobile users. For the purpose of experimenting the end-to-end functionality of our architecture, we emulated light-weighted eNBs that implement RNFA and are fed with eNB related information (measurements and configuration) by Matlab using standard User Datagram Protocol (UDP) socket connection. Then, the emulated eNB acts like real base station to the SDN controller. The Matlab simulator and SDN environment (emulated eNBs, SDN controller and MongoDB) are implemented in two physically separated machines. We deployed the SDN controller, MongoDB and eNBs as virtual entities in one physical machine and used another dedicated physical machine for the Matlab simulator.

In the deployment, we implemented the two virtualization techniques, virtual machines and Docker containers, under the same hardware configuration of 16-core CPUs and 16 GB RAM operating with Ubuntu Linux 14.04 version. The motivation of validating our framework under two different virtualization techniques is to analyze the performance and behavior of the system under the two different virtual environments and to provide a comparative study of their suitability for enabling 5G SDN/NFV. 1) Testbed using Virtual Machines: As shown in Fig. 2, the SDN controller, eICIC optimizer, MongoDB and emulated eNBs are deployed in separate VMs using lightweight version Ubuntu 14.04 images. We used Kernel-based Virtual Machine (KVM), that uses Quick Emulator (QEMU) to emulate hardwares for virtual machine deployment. We launched a number of virtual machines in parallel by executing VM execution command through Python script. The networking between VMs is achieved through network address translation (NAT) through the virtual switch of the hypervisor. Though VMs use limited number of hypercalls or emulated devices via hypervisor to establish communication with the outside world with a certain level of isolation and security, it is known that the hypervisor would add substantial overheads on the performance. These cannot be bypassed or optimized from higher layer. Note that the performance of VM based testbed is often lower when compared to native machines but this can be compensated by the advantage of virtualizing hardware resources of the same physical machine for several entities.

2) Testbed using Docker Containers: As shown in Fig. 3, the SDN controller, eICIC optimizer, MongoDB and emulated eNBs are deployed in isolated Docker containers using Alpine Linux version 3.2 that is 5 MB in size built around musl libc [START_REF]Standard library musl[END_REF] and busybox [START_REF] Busybox | [END_REF]. In order to achieve the containerization of each entity in our architecture, we have created separate Docker files along with the necessary libraries, including Java 1.7, Maven [START_REF][END_REF], SCTP for the deployment of ODL controller, json [START_REF]JavaScript Object Notation[END_REF], and gcc for eICIC optimizer which is developed using C++. We have used the MongoDB Docker file from Docker hub for the deployment of DB container in our platform. We built all the Docker files using Docker daemon version 1.11 to produce corresponding Docker images and stored them in the local repository before launching our experimentation testbed. During the experimentation, we launched all the containers such as ODL controller, MongoDB, eICIC optimizer and emulated eNBs using a single shell script. The instantiation of all the containers of our experimentation testbed appears very fast: it is less than one minute compared to the VM implementation that may take several minutes.

The advantage of using Docker containers is that a container will virtualize the kernel so that each container uses underlying kernel with its own process and network space which is more light weighted and efficient compared to VMs that run full operating system as shown in Fig. 2. Besides, the communication latency between Docker containers is much lower compared to VMs which have hypervisor overheads. Processes running within containers are more elastic in nature compared to those deployed within VMs which are more static in resource utilization. The entire booting time for our testbed with several Docker containers is of the order of seconds whereas the booting time with VMs is of the order of tens of seconds or even more than a minute. This is due to the fact that Docker containers share the single OS and require to load only the necessary software packages and libraries for particular processes whereas VMs require the entire OS to be loaded irrespective of the requirement of the deployed processes.

III. CONTROLLER QUEUING MODEL

According to the previously described architecture, the SDN controller manipulates the fluxes of information (measurements and configuration information) received from the eNBs. In order to characterize the processing discipline of the SDN controller, we adopt the M/M/1 queuing model [START_REF] Kleinrock | Theory[END_REF]. In a M/M/1 queuing model, the length of packet-in message queue at a SDN controller N (t) is a birth and death process on the countable infinite state set E = {0, 1, 2, . . .}. We denote its instantaneous probability by p j (t) and the stationary probability by p j such that

p j (t) = P {N (t) = j}, (1)
and

p j = lim t→∞ p j (t), j ∈ E. (2)
We denote the traffic intensity of the controller by ρ, which is given by λ/µ, where λ is the inter-arrival rate and µ is the service rate. We can use Little's Law to express the length of the waiting queue:

Nq = ρ 1 -ρ . (3)
Notice that p 0 = 1 -ρ is the probability of the controller in idle state, and ρ is just the probability of the controller in busy state, where 0 ≤ ρ ≤ 1 . It is easy to observe that the controller becomes busier with the increase of ρ.

The controller processes the packets on a first-come-firstserved basis. Therefore, the distribution of the waiting time of a packet-in message is thus given by: W q(t) = P {W q ≤ t} = 1 -ρe -(µ-λ)t , t 0.

(4) Using (4), we can express the average waiting time of a packet-in message as:

Wq = E[W q(t) ] = ρ µ(1 -ρ) . (5)
By adding the expected processing time of a packet, which is given by 1/ µ, we have the sojourn time of a packet in the controller expressible as:

W (t) = Wq + 1 µ = 1 µ -λ . (6)
From [START_REF] Felter | An updated performance comparison of virtual machines and Linux containers[END_REF], we get the processing rate:

µ = 1 W (t) + λ. (7)
The above result will allow us to estimate the processing rate in the testbeds by mapping the behavior of the testbeds to the queuing model and the measurements to metrics in terms of inter-arrival rate and average waiting time.

IV. NUMERICAL EVALUATION A. Fitting the Model to the Measurements

The measurements are performed in the two testbeds described in Section II-B1 and II-B2. Fig. 4 illustrates the interarrival time of the packets collected from the eNBs served by the SDN controller. We increase the number of eNBs gradually. Results show a significant difference in the behavior of the two testbeds. We find that it is due to the fact the hardware resource of physical machine that is virtualized using VMs would easily get over-loaded with the increase in the number of virtual nodes. The VM based architecture has the intrinsic nature that requires more resources to deploy the whole operating system stack comparing to Docker containers that run only the necessary packages and libraries on top of the existing kernel. As a result, the communication latency between Docker containers is significantly lower compared Next, we measure the average waiting time spent in the two systems from the testbeds. Note that this waiting time refers to Wq given by (5). Fig. 5 shows the experimental results. As expected, in both testbeds the average waiting time increases with the number of eNBs due to the increase of the load which will result in a request of more physical resources. However, the Docker container architecture shows clearly its superiority over the VM.

B. VM Architecture vs Docker Container Architecture Service Processing Time

We model the behavior of the two described testbeds with the analytical model described in Section III in order to estimate the processing time required for the eNBs information update. Expressing the processing rate by [START_REF] Preeth | Evaluation of Docker containers based on hardware utilization[END_REF] and using the measured inter-arrival rate (see Fig. 4) and waiting time (see Fig. 5), we can obtain the service rate for the VM and Docker based testbeds and plot the result.

From Fig. 6, we can see that both the average service rates of Docker container testbed and VM testbed decrease with the number of virtual nodes running in the same physical system. However, the performance due to Docker containers is always higher than that of VMs due to its lightweight and lower resource occupancy in achieving the same amount of tasks. Note that the Docker container testbed shows a very high service rate when the system is started for a small number of eNBs. This behavior is a consequence of the container's virtually zero performance overhead and fast kernel resource allocation. On the other hand, the VM testbed's performance is nearly flat since for any number of eNBs the testbed will load the whole operating system with large overhead and the resource allocation is comparatively slow.

V. CONCLUSION

This work presents a practice of virtualization of RAN and describes the experimentation in using VMs and Docker containers for implementing a software defined mobile network. The virtualized RAN architecture is evaluated using an analytical model that aims to derive the processing rate under the two NFV deployments. Measurements and comparisons confirm that the Docker container based architecture can provide a superior performance compared to the VM based architecture. This is helpful for the future development of Software Defined Mobile Networks. Note that we also expect the analytical method introduced here could be also useful for studying other similar systems such as Cloud RAN.

  [START_REF] Gopalasingham | Generalized software defined network platform for radio access networks[END_REF].• We provide a queuing model for analyzing the container based SDN controller in comparison to the VM deployment and show their performance. Using the model, we estimate the server processing time under the two architectures for comparison. • We conduct experiments to evaluate the two deployments and give analytical results for the future development of SDN controller with different technologies.

Fig. 1 :

 1 Fig. 1: Generalized SDN platform for RAN

Fig. 2 :

 2 Fig. 2: (a) Virtual machine vs. (b) Docker container architecture

Fig. 3 :

 3 Fig. 3: Testbed using Docker containers

Fig. 4 :

 4 Fig. 4: Inter-arrival time of packets under VM architecture and Docker architecture

Fig. 5 :Fig. 6 :

 56 Fig. 5: Average waiting time for the VM based architecture and for Docker based architecture


Ë L Ga

L U Cousin
email: cousin@math.univ-angers.fr

Gustavo Ís

AND IV ÁN PAN Mendes
email: mendes@mat.ufrgs.br

BIRATIONAL GEOMETRY OF FOLIATIONS ASSOCIATED TO SIMPLE DERIVATIONS AND GENERALIZATIONS

Keywords: 2010 Mathematics Subject Classification. 37F75, 13N15, 14E07

We propose a study of the foliations of the projective plane induced by simple derivations of the polynomial ring in two indeterminates over the complex field. These correspond to foliations which have no invariant algebraic curve nor singularities in the complement of a line. We establish the position of these foliations in the birational classification of foliations and prove the finiteness of their birational symmetries. Most of the results apply to wider classes of foliations.

Introduction and results

A simple derivation (of the ring C[x, y]) is a polynomial vector field of C 2 without zeroes and without algebraic solutions.

The study of simple derivations is an active area of research in Algebra (e.g. [Sha77, Jor81, Now94, MMON01, Cou03, BLL03, Now08, GL12, Kou12]). Most of these papers are dedicated to the proof of simplicity of (families of) examples.

For any derivation D, the isotropy group Aut(D) is composed by the C-automorphisms ρ : C[x, y] → C[x, y] which verify ρD = Dρ.

Although there exist derivations with infinite group Aut(D), the main result of [START_REF] Luís | On plane polynomial automorphisms commuting with simple derivations[END_REF] is that Aut(D) is trivial for any simple derivation.

Take ρ ∈ Aut(D), R : C 2 → C 2 the polynomial automorphism associated to ρ and let ω D be the dual 1-form to the vector field D = f ∂ x + g ∂ y (i.e. ω D = g dx -f dy). Then ρD = Dρ is equivalent to

R * (ω D ) = Jac(R) • ω D
where Jac(R) ∈ C * is the Jacobian determinant of R. A less restrictive condition is that

R * (ω D ) = c • ω D ,
for some c ∈ C * (depending on R). This means that R preserves the algebraic foliation F D of C 2 associated to D (or to ω D ), see Remark 2.1.

We denote P ol(F D ) the group consisting of polynomial automorphisms of C 2 which preserve the foliation F D . There is a natural homomorphism Aut(D) ֒→ P ol(F D ).

In Section 8 we show that for each n ≥ 2 and B > 0 there are foliations F D of C 2 associated to simple derivations with an element T ∈ P ol(F D ) of order n and degree greater or equal than B.

Let us denote F the foliation of the projective plane P 2 = C 2 ∪ L ∞ which is the extension of F D in C 2 . All along the paper, if D is a simple derivation, both F D in C 2 and its extension F in P 2 are called foliations associated to simple derivations.

But the reader must be warned that, even if F D has no singularity, some singularity of F along the line at infinity L ∞ is unavoidable, see [Bru00, Prop. 1 p. 21]. Also beware that the line at infinity L ∞ may be invariant by F.

Denote Bir(F) the group of birational transformations of P 2 which preserve a foliation F. If F extends a foliation F D of C 2 , then there is a natural homomorphism P ol(F D ) ֒→ Bir(F) whose meaning is that a (non-linear) polynomial automorphism of C 2 extends to a special type of birational map of P 2 . Namely, a birational map with a unique (proper) point of indeterminacy p ∈ L ∞ , whose net effect in P 2 is to replace L ∞ by the strict transform of the last exceptional curve introduced in the elimination of the indeterminacy point.

Our first result is the following generalization of [MP16, Thm. 1].

Theorem A. Let F be a foliation associated to a simple derivation. The group Bir(F) is finite.

It is actually derived from next result which determines, in particular, the positions that foliations associated to simple derivations may occupy in the birational classification of foliations, cf. [START_REF] Brunella | Birational geometry of foliations, Monografías de Matemática[END_REF][START_REF] Mcquillan | Non-commutative mori theory[END_REF][START_REF] Luís | Kodaira dimension of holomorphic singular foliations[END_REF]. This classification is based on the birational invariant of foliations called Kodaira dimension, denoted κ(F), whose range is κ(F) ∈ {-∞, 0, 1, 2}, see Section 2.

All along the paper we use the expression reduced singularity in the sense of Seidenberg's reduction of singularities, cf. Section 2. By a rational curve of C 2 we mean an algebraic curve whose projective closure has geometric genus zero. And by a Riccati foliation on P 2 we mean a foliation which, up to a birational modification of P 2 , is everywhere transverse to the general fiber of a rational fibration; on says the fibration is adapted to such a foliation. Theorem B. Let F be a foliation of the projective plane such that the restriction F| C 2 has no invariant rational curve. i) Then κ(F) ≥ 0; ii) If F| C 2 has at most reduced singularities, then κ(F) ≥ 1; iii) If F| C 2 has no invariant algebraic curve, then κ(F) = 1 if and only if F is a Riccati foliation. iv) The cases κ(F) ∈ {1, 2} are realized by foliations associated to simple derivations.

Remark that case B -iii) includes the foliations associated to Shamsuddin derivations. Note also that Theorem B applies to a class of foliations which is larger than the one of foliations associated to simple derivations. In Section 6 we study the foliations associated to examples of simple derivations found throughout the literature and discuss their birational equivalence. In Section 8, we propose a construction of simple derivations D with arbitrary large (finite) P ol(F D ).

Acknowledgements. We thank Charles Favre and Jorge Vitório Pereira for usefull discussions.

Preliminaries on foliations

The paper relies on concepts and results of the theory of singularities and birational geometry of foliations on algebraic complex surfaces. We present the basic facts in this preliminary section but along the paper, when necessary, we refer the reader to the corresponding sections of [START_REF] Brunella | Birational geometry of foliations, Monografías de Matemática[END_REF] or [START_REF] Brunella | Foliations on complex projective surfaces[END_REF], where the theory is masterfully explained by Marco Brunella.

First definitions. On a smooth complex surface X, a foliation F is given by an open covering (U i ) of X and local vector fields v i ∈ H 0 (U i , T X) with isolated zeroes such that there exist non vanishing holomorphic functions (g ij ) on the intersections U i ∩ U j satisfying

v i = g ij v j . (1)
The cocycle (g ij ) defines a line bundle T * F on X, its dual is denoted T F. Relation (1) means that the family (v i ) defines a section of T * F ⊗ T X and hence a sheaf map T F → T X. Two data ((U i ), (v i )), ((U ′ j ), (v ′ j )) are said to define the same foliation if the images of the associated sheaf maps are the same. The line bundle T F is called the tangent bundle of the foliation and its dual T * F is the cotangent bundle of F. The locus defined by the vanishing of the local vector fields (v i ) is called the singular locus of F, denoted Sing(F). As defined, the line bundle T F is not canonically attached to F, but only its isomorphism class in Pic(X).

One may also consider foliations on normal singular complex surfaces. They are defined by the datum of a foliation on the complement of the singular locus of the surface.

Rational vector fields and 1-forms. If X is smooth projective, T F possesses a non trivial rational section and F can be given by a rational vector field X , hence in Pic(X) we have

T F = O X (div(X )).
On a suitable (Zariski) open covering the local vector fields v i are obtained by chasing the zeroes and poles of X : v i = hX |U i , for a well chosen regular function h on U i . This is how we associate a foliation to a simple derivation: we have a preferred projective compactification of C 2 , namely

P 2 = C 2 ∪ L ∞ , (x, y) → [1 : x : y],
and a polynomial vector field on C 2 extends to a rational vector field on P 2 . One can also define a foliation by local holomorphic 1-forms with isolated zeroes (ω i ) that vanish on the local vector fields (v i ). If X is projective, such a family (ω i ) is obtained by locally chasing the zeroes and poles of a non trivial rational 1-form. Hence, on a smooth projective surface, a foliation may be defined by either a non trivial rational 1-form or a a non trivial rational vector field.

Curves and foliations.

A curve C is termed invariant by F or F-invariant if it is tangent to the local vector fields defining F. When a compact curve C ⊂ X is not F-invariant we have the very useful formula

T * F • C = tang(F, C) -C • C
, where tang(F, C) is the sum of orders of tangency between F and C, cf. [Bru00, Prop 2 p. 23].

If X = P 2 is the projective plane, the degree of F is deg(F) ∈ Z ≥0 defined as the number of tangencies of F with a general projective line. In this case

T * F = O P 2 (deg(F) -1).
Birational maps and foliations. Let X and Y be projective surface with at most normal singularities and φ : X Y is a birational map. If we have a foliation F on X given by the rational vector field X , we can define a foliation φ * F on Y as the one defined by the rational vector field φ * X . Conversely, from a foliation G on Y , one defines φ * G := (φ -1 ) * G. We say that the foliations F and φ * F are birationally equivalent and that φ * F is a (birational) model of F.

Remark 2.1. In the case φ : P 2 P 2 is induced by a polynomial R automorphism of C 2 , if F is given by a polynomial vector field X on C 2 , with isolated zeroes, the condition φ * F = F is tantamount to R * X = hX ; for a suitable rational function h. However, as R is a polynomial automorphism, the vector R * X is a polynomial vector field on C 2 , with isolated zeroes. In particular, the factor h is a constant c ∈ C * .

Singularities. In a neighborhood of a singular point p ∈ X with local centered coordinates x, y the foliation F is defined by a holomorphic vector field v = f (z, w) ∂ ∂z + g(z, w) ∂ ∂w ; with f (0, 0) = g(0, 0) = 0. Denote λ 1 , λ 2 the eigenvalues of the linear part (first jet) of (z, w) → (f (z, w), g(z, w)). We say p is a reduced singularity of F if at least one of them, say λ 2 is not zero and if λ := λ 1 /λ 2 ∈ Q >0 ; otherwise the singularity is non-reduced. A special case of nonreduced singularity occurs when the linear part is the identity, in this case λ = 1, and we say p is a radial point.

If λ = 0 we say that the singularity is non degenerate; otherwise we call it a saddle-node.

We say p is a Morse point if it is non degenerate and, in suitable coordinates, admits a local holomorphic first integral of the form φ(z, w) = z 2 + w 2 + h.o.t.; note that for Morse points λ = -1.

Reduction of singularities, relatively minimal models. A foliation F on a smooth surface is said to be reduced if all its singularities are reduced. After Seidenberg [START_REF] Seidenberg | Reduction of singularities of the differential equation A dy = B dx[END_REF], foliations on smooth projective surfaces always admit a reduction of singularities: a birational morphism Σ : M → X obtained as a composition of blowing-ups such that F := Σ * F is a reduced foliation. Such a reduced model F is not unique. Indeed, by performing a blowing-up at either a non-singular point or a reduced singularity the transformed foliation remains reduced. Doing such an "unnecessary" blowing-up creates a foliated exceptional curve or F-exceptional curve: a rational curve of selfintersection -1 whose contraction to a point q yields a foliated surface with at most a reduced singularity at q. A reduced model F is called a relatively minimal model when it is free of F-exceptional curves. Kodaira dimension. The Kodaira dimension κ(F) of F is defined by

κ(F) := lim sup n→+∞ 1 log n log h 0 ((T * F) ⊗n ).
This is a birational invariant with values in {-∞, 0, 1, 2} which is independent on the particular reduced model. If κ(F) = 2, we say F is of general type. The birational classification of foliations concerns the cases κ(F) ∈ {-∞, 0, 1}.

Zariski decomposition. If F is not birationally equivalent to a rational fibration, Miyaoka and Fujita's results assure that the cotangent line bundle T * F admits a so-called Zariski decomposition [Bru00, p. 100] T * F ′ ≡ N + P, where

• ≡ means numerical equivalence,

• the positive part P is a nef Q-divisor (i.e. P • C ≥ 0 for every curve C)

• the negative part N = j α j N j is a Q + -divisor (α j ∈ Q + ) and each connected compo- nent of ∪ j N j is contractible to a normal singularity. • P • N j = 0, ∀j.
Nef model. If F is a relatively minimal (reduced) model of F and if F is not a rational fibration, McQuillan's theorem [Bru00, Thm.1, Chap. 8] assures that the support of N in T * F ′ ≡ N + P is a union of the so-called maximal F-chains.

A F-chain is a chain of invariant rational (-n)-curves, with n ≥ 2, which starts with a curve containing just one singularity of F and where other components, if it has more than one, contain two singularities, all singularities being reduced non-degenerate. The contraction of a F -chain produces a rational surface singularity, more precisely, a cyclic quotient singularity. The induced foliation on the resulting singular surface is called a nef model of F .

Proof of Theorem B -i)

The birational classification of foliations with κ(F) = -∞ ( [START_REF] Brunella | Foliations on complex projective surfaces[END_REF] or [START_REF] Mcquillan | Non-commutative mori theory[END_REF]) asserts that this class is composed by rational fibrations and by foliations birationally equivalent to the socalled Hilbert modular foliations. Hence, in order to prove the part (i) of Theorem B, we only need to exclude Hilbert modular foliations.

We recall this notion. A Hilbert modular surface is defined (following [Bru03, p. 25]) as a (possibly singular) projective surface Y containing a (possibly empty) curve C ⊂ Y \ Sing(Y ) such that: i) each connected component of C is a cycle of smooth rational curves, contractible to a normal singularity;

ii) Y \ C is uniformised by the bidisc H × H, i.e. we have an isomorphism of analytic spaces

Y \ C ≃ Y Γ := (H × H)/Γ
where Γ is a discrete subgroup of PSL(2, R) × PSL(2, R) ⊂ Aut(H × H); iii) Γ is irreducible (i.e. does not contain a finite index subgroup of the form Γ 1 × Γ 2 with Γ j ⊂ PSL(2, R), j = 1, 2).

The natural singular foliations of the Hilbert modular surface Y which come from the horizontal and vertical foliations by discs of H × H are called Hilbert modular foliations. Both foliations leave invariant the curve C and, in the desingularization of the surface, they leave invariant the exceptional divisors.

Theorem B -i) follows from the following result.

Proposition 3.1. Let F be an algebraic foliation of P 2 such that F| C 2 has no invariant rational curve. Then F is not birationally equivalent to a Hilbert modular foliation.

Proof. Suppose by contradiction that F is birationally equivalent to a Hilbert modular foliation.

Let F be a relatively minimal model of F on a suitable smooth projective surface M . Then M is the minimal desingularization of a Hilbert modular surface Y (cf. [Bru00, Th. 1 p. 75]) and F is the transform of a Hilbert modular foliation on Y .

The hypothesis that F| C 2 has no invariant rational curve implies that there are no cycles of F-invariant rational curves on M . Then the Hilbert modular surface is

Y = Y Γ = H × H/Γ,
for a discrete cocompact (hence lattice) irreducible subgroup Γ of (PSL(2, R)) 2 .

Let

U := P 2 \ (L ∞ ∪ Sing(F)), where Sing(F) is the singular locus of F. Then U is isomorphic to a non-empty Zariski open set of M ′ and also U is isomorphic to a non-empty Zariski open set of ẎΓ , the complement of the quotient singularities of Y Γ . The morphism

π 1 (U ) → π 1 ( ẎΓ )
induced by the injection is surjective. As U is simply connected, so is ẎΓ .

By irreducibility of Γ, the subset F ⊂ H×H given by the points that have non trivial stabilizer under Γ is discrete (cf. [START_REF] Shimizu | On discontinuous groups operating on the product of the upper half planes[END_REF]). The action of Γ on H × H can be restricted to an action on (H × H) \ F . The quotient map

(H × H) \ F → [(H × H) \ F ]/Γ = ẎΓ
is then a covering, in the strict sense of topologists. As (H × H) \ F is simply connected, this proves that the fundamental group of ẎΓ is isomorphic to Γ. But ẎΓ is simply connected, so Γ is trivial, a contradiction. This result is quite sharp, a Hilbert modular foliation with exactly one invariant rational curve in C 2 is given in [START_REF] Mendes | Hilbert modular foliations on the projective plane[END_REF].

Proof of Theorem B -ii)

Now F denotes a foliation of P 2 = C 2 ∪ L ∞ such that its restriction F| C 2 to C 2 does not admit invariant rational curves and has at most reduced singularities.

Remark 4.1. Recall that a degree zero foliation of the projective plane is a pencil of straight lines. Also recall that a degree one foliation of the projective plane has at least two invariant lines [Bru00, 2) p. 27], so at least one invariant affine line.

After Theorem B -i), it suffices to exclude the possibility κ(F) = 0. By contradiction we suppose κ(F) = 0.

If F is a reduced foliation of the projective plane, then κ(F) = 0 implies T * F = O and deg(F) = 1, which contradicts Remark 4.1. Therefore, we have a non-reduced singularity p ∈ L ∞ . Denote p 1 , . . . , p k ∈ L ∞ the non reduced singularities of F.

Let Σ : M → P 2 be a reduction of singularities of F composed by blowing-ups of points affecting only non-reduced singularities (in order to not introduce unnecessary foliated exceptional curves). Denote the reduced foliation in M by F . Denote by E p i the exceptional line of the first blowing-up affecting p i , i = 1, . . . , k, and by E p the corresponding strict transform in M . Note that the reduction of singularities Σ may include additional blowing-ups.

Now let us denote by

q : M → M ′ a (finite, possibly trivial) sequence of blowing downs of foliated exceptional curves (and only them) and by

F ′ := q * F the foliation obtained in M ′ , which is a relatively minimal model of F in M ′ .
Claim 4.2. There exists a non reduced singularity p ∈ L ∞ of F such that the pencil of lines of P 2 passing through p defines a fibration π ′ : M ′ → P 1 .

Proof. If q : M → M ′ is the identity, then any p i verifies the required property. Then we assume q = id. We assert that q starts by contracting the strict transform L ∞ ⊂ M of the line at infinity L ∞ . To justify this, we first remark that E p i cannot be a F-exceptional curve. Indeed, if the effect of Σ on p i is just one blowing-up, say σ, then E p i = E p i and this is not σ * F-exceptional (remember that p i is not reduced). If the effect of Σ on p i includes extra blowing-ups, then the self-intersection of the (-1)-curve E p i decreases to ≤ -2 and so it cannot correspond to a F-exceptional curve.

In particular L ∞ is F-invariant and that line is affected by exactly two blowing-ups. After the first contraction is done, possibly new foliated exceptional curves are created and contracted by q in a domino effect (compare Example 6.5).

We distinguish two cases:

• Case 1: there are exactly two non-reduced points of F in L ∞ , i.e. k = 2; • Case 2: there is exactly one non-reduced point of F in L ∞ , i.e. k = 1.

In both cases, if some E p i (resp. E p 1 ) is not transformed into a foliated exceptional curve, and so contracted by q, then p = p i is the required point.

Finally, we assume that every E p i is contracted and show it eventually gives a contradiction.

On Case 1:

After E p 1 and E p 2 have been introduced no extra blowing-up composing Σ affects the line at infinity. Then E p 1 and E p 2 become (-2)-curves after blowing up points in E p 1 and E p 2 . Next figure illustrates the situation.

L oo (-1) E p 1 (-1) (-1) E p 2 (1) p p 1 σ σ p 2 E p 1 (-2) (-2) E (-1) (-1) L oo (-1) σ σ L oo 2
By contracting first L ∞ and then one of the E p i , we see the other one becomes a (0)-curve: contradiction.

On Case 2:

In this case one must blow up an infinitely near point of L ∞ in E p 1 , r say. Then E p 1 is a (-n)-curve with n ≥ 2 which intersects the strict transform E r in M of the exceptional line E r associates to r.

For q to contract E p 1 , we need n = 2 and E r to be F -invariant. Moreover, since q contracts L ∞ then at least one of the blowing-ups composing Σ is done at a point t ∈ E r . So E r is a (-2)-curve which is part of a chain of curves contracted by q.

Next Figure illustrates the situation.

L oo E (1) σ L oo (0) (-1) E p (-2) σ (-1) E r p E C (-2) (-1) oo L (-1) L oo (-1) E r p 1 p 1 1 1 (-2)
Hence q contracts L ∞ , E r and E p 1 , and transforms

C into a curve C of self-intersection 1 (see next figure). E (-2) L oo (-1) r q E E r (-1) C C E p q Ep 1 (-1) q C (-1) (0) C (1) p 1 1 (-2) (-2) (-1)
The birational map

φ := (q • σ -1 ) : P 2 / / ❴ ❴ ❴ P 2 is a quadratic Cremona transformation with a unique "proper" indeterminacy point in L ∞ . Since κ(F) = κ(F ′ ) = 0 and F ′ is reduced we get deg(F ′ ) = 1.
In particular F ′ has two invariant projective lines, by Remark 4.1. But the restriction

φ| C 2 = φ| P 2 \L∞ is an isomorphism sending F| C 2 to F ′ | P 2 \C , which contradicts that F| C 2 has no invariant rational curve.
Consider the fibration π ′ : M ′ → P 1 given by Claim 4.2.

Claim 4.3. The foliation F ′ is a Riccati foliation, with π ′ an adapted fibration.

Proof. Denote p the point associated to π ′ in Claim 4.2. Let L be a general fiber of π ′ . We must prove tang(L, F ′ ) = 0. By contradiction, we suppose tang(L, F ′ ) > 0.

On one hand, we know F ′ is not a rational fibration. Hence we have a Zariski decomposition (cf. Section 2)

T * F ′ ≡ N + P.
Since F ′ is a relatively minimal model we know that the support of N = α j N j is a union of maximal F-chains. Since we have κ(F) = 0, from McQuillan's theorem [Bru00, Thm.2, Chap. 9] we deduce

P ≡ 0.
On the other hand, every F ′ -invariant rational curve either coincides with the strict transform of L ∞ in M ′ (if this curve was not collapsed by q) or is the strict transform of some exceptional component of Σ. Note that such a component intersects L if and only if it equals q(E p ).

Since

T * F ′ • L = N • L ≥ 1 then N j 0 = q(E p ) for some j 0 , so N • L = α j 0 . But according to [Bru00, pp. 109 -110], one has 0 < α j 0 < 1; yielding 0 < tang(F ′ , L) < 1.
This contradicts the integrality of tang.

Let σ : (X, R) → (P 2 , F) be the blowing-up of the point given by Claim 4.2, denote π : X → P 1 the ruling of X, E the exceptional divisor and L σ ∞ the strict transform of L ∞ . For convenience, we normalize π to have π

-1 ([0 : 1]) = L σ ∞ .
Claim 4.4. The foliation R is a Riccati foliation, the fibration π is adapted to R.

Proof. We have a factorization Σ = σ • Σ 0 . Let F be a fiber of π. If L is sufficiently general, it possesses a neighborhood on which q • Σ -1 0 is a biholomorphic map. By Claim 4.3, we conclude tang(L, R) = 0.

Claim 4.5. The only possible R-invariant rational curves in X are E and

L σ ∞ . Proof. Recall that X \ (L σ ∞ ∪ E) is isomorphic to C 2 via σ, so that no other rational curve is R-invariant.
After Brunella [START_REF] Brunella | Foliations on complex projective surfaces[END_REF], by a rational P 1 -bundle map (over the identity of the basis), biholomorphic outside the fiber L σ ∞ , we transform the Riccati foliation (X, R, π) into a nef Riccati foliation (X 0 , R 0 , π 0 ). The wording "nef Riccati foliation" means the local structure of R 0 around any fiber of π 0 is of one of the types

1 q 2 (c) p 1 p 2 p (b) (a) p 1 p 2 (d) (e) q 1
q 2 p (a) the foliation is transverse to π and S is smooth; (b) the surface S admits two quotient singular points q 1 , q 2 of the same order k ≥ 2 and the fiber is not invariant by the foliation; (c) there are two possibilities: either p 1 , p 2 are non degenerate singularities, or p is an unique saddle-node with Milnor number 2 whose strong separatrix is transverse to the vertical fiber, S being smooth in both cases; (d) there are two saddle-nodes with Milnor number m whose strong separatrix is given by the vertical fiber, S being smooth; (e) the surface S admits two quotient singularities with same order equal to 2 and F nef admits a saddle node with Milnor number m whose strong separatrix is in the vertical fiber.

Claim 4.6. i) The fiber π -1 0 ([0 : 1]) is an R 0 -invariant fiber, of type (d), with multiplicity m = 2.

ii) The bundle T * R 0 is trivial. Proof. By Claim 4.5, π -1 0 ([0 : 1]) is the only possibly invariant fiber. We review the possible types for F := π -1 0 ([0 : 1]). If F is of type (a), the foliation R 0 is everywhere transversal to the rational fibration and trivializes the corresponding P 1 -bundle. The basis of the fibration being P 1 , R 0 should be a rational pencil, which is impossible.

As all the other fiber are of type (a) and P 1 \ {[0 : 1]} is simply connected, the fiber F has trivial local monodromy. The fibers of type (b) or (c) have non-trivial monodromy. Consequently F cannot be of these types.

To consider the last two cases (d) and (e), we recall the following formula for the degree of the Q-divisor π 0 * (T * R 0 ) (see [START_REF] Brunella | Foliations on complex projective surfaces[END_REF]p. 20]):

deg (π 0 * (T * R 0 )) = -2 + m+1 2 in case (e) -2 + m in case (d) (2)
There is a correspondence between deg(π 0 * (T * R 0 )) being negative, zero or positive and κ(F) = -∞, 0, 1, respectively.

Since in our case deg(π * (T * R 0 )) = 0, in the case (e) we obtain m = 3, an odd number, contradicting [Bru00, p. 56].

In case (d) we obtain Milnor number m = 2 for both saddle-nodes. We have just proved i).

For ii), we use the formula for the cotangent bundle in [Bru00, p. 57] and obtain

T * R 0 = π * O P 1 (-2) + O X 0 (2F ) = O X 0 .
The surface X 0 is a smooth rational surface equipped with a regular ruling π 0 , it is therefore a Hirzebruch surface X 0 = F n with n ≥ 0. Claim 4.7. i) The fibration π 0 is not the trivial P 1 -bundle over P 1 .

ii) The negative section is R 0 invariant, it is the transform of E p ⊂ X in X 0 .

Proof. i). Suppose by contradiction X 0 ≃ P 1 × P 1 . Call vertical the fibration π 0 adapted to the Riccati foliation, the second ruling is termed horizontal. Denote by H 1 , H 2 the two horizontal lines passing through the two saddle-nodes in π -1 0 ([0 : 1]). One of them, say H 1 , is not R 0invariant. Passing by a singularity of a foliation produces a tangency, so

1 ≤ tang(R 0 , H 1 ) = tang(R 0 , H 1 ) -H 1 • H 1 = T * R 0 • H 1 ,
contradicting the triviality of T * R 0 .

ii). If the negative section s is not invariant, using the tangency formula as above, we get 0 ≤ tang(R 0 , s) = s • s; contradiction. Claim 4.5 says that the only possible invariant section is the transform of E p in X 0 , whence the conclusion.

Claim 4.8. By rational P 1 -bundle maps (over the identity of the basis), biholomorphic outside the fiber over [0 : 1], we can transform (X 0 , R 0 , π 0 ) in (X 1 , R 1 , π 1 ), a nef Riccati foliation with T * R 1 = O X 1 and X 1 a Hirzebruch surface with a (-1)-section invariant by R 1 .

Proof. This is performed by induction, using the following flip: start by blowing up the saddlenode on F which lies outside the negative section; denote by D the exceptional curve produced by this. The intersection point of D with the strict transform F of F is a Morse point. Then we contract F obtaining a Hirzebruch surface on which D becomes an invariant fiber, again with two saddle-nodes of Milnor number 2, exactly as before, see the picture below. The selfintersection of the invariant section diminishes of 1 in this process. As the multiplicity of the fiber of type (d) does not change, the tangent bundle remains trivial.

(-1) (-n) σ (-n+1) D (0) sn sn C n-1 C n C n nd
Now denote by F 0 the foliation of P 2 obtained from R 1 by contracting the (-1)-curve in X 1 to a point r ∈ P 2 . The sequence of transformations P 2 σ -1 X → X 0 → X 1 → P 2 , extends an automorphism of C 2 , so that F 0 should have no invariant rational curve in C 2 . We reach a contradiction and conclude the proof of Theorem B -ii) by the conjunction of Remark 4.1 and next claim.

Claim 4.9. The foliation F 0 is a degree one foliation of P 2 .

Proof. Let δ be a general line in P 2 , denote δ its strict transform in X 1 . If δ does not pass through r, we have tang(R 1 , δ) = tang(F 0 , δ). The first member is

tang(R 1 , δ) = δ • δ + T * R 1 • δ = 1 + 0 = 1
and the second is deg(F 0 ). Whence the conclusion.

Proof of Theorem B -iii)

In this section we assume κ(F) = 1. According to the birational classification of foliations [Bru00, Th. 1. p. 118], and taking into account that F is not birationally conjugate to a fibration, we know that F is either a Riccati foliation or a turbulent foliation; recall that the definition of turbulent foliation is obtained from the one of Riccati foliation by replacing "rational fibration" with "elliptic fibration" (see page 2).

Therefore the proof of Theorem B -iii) is equivalent to the exclusion of the turbulent case.

This will be done in Proposition 5.3, using the notion of a transversely affine foliation. Consider a foliation F on a surface X given by ω = 0 where ω is a rational 1-form on X. We say F is transversely affine if there exists a closed rational 1-form η such that dω = ω ∧ η.

Remark 5.1. If ω = gω is another 1-form defining F, then η := η -dg/g is closed and satisfies dω = ω ∧ η, so that this definition is independent of the defining 1-form ω.

The following has already been noticed in [Per03, Prop. 22], we give a slightly different proof.

Proposition 5.2. Every turbulent foliation is a transversely affine foliation.

Proof. As transverse affine structures may be transported by birational transformations, stable reduction [Bru00, Prop. 6 p. 69] and [CLNL + 07, Th. 2.21 p. 37] reduce the proof to the case where the foliation F is transverse to the general fiber of an elliptic fiber bundle π : X → C. Let X * be the complement of the set of invariant fibers, and F = π -1 (b) ⊂ X * a fiber. Using the foliation to identify nearby fibers, we obtain a multiform submersion X * → F which defines the foliation; it lifts to a submersion f : X * → C to the universal cover of F ≃ C/Λ. By construction the monodromy group of f fixes the lattice Λ and must lie in Aff(C). Hence the monodromy of df is linear (contained in C * ). In particular, if v is a rational vector field on X which is not tangent to F, the meromorphic function

g = df (v) : X * → C
has the same monodromy as df and ω := df g is a well defined meromorphic 1-form on X * , tangent to F. We have

dω = - df ∧ dg g 2 = ω ∧ η,
with η = -dg g a well defined closed meromorphic one form on X * . It remains to show that the pair (ω, η) extends meromorphically in the neighborhood of any F-invariant fiber of π. Let U ≃ D × F be such a neighborhood, D a disc. Let (z, w) ∈ D × C represent the elements of U , z = 0 corresponding to the invariant fiber. We have a local equation of the form

dw = dz A(z) ,
for F, with A(z) holomorphic in D. Let b be a point in D, if the coordinate w is well chosen, in D * × F , the submersion f expresses as

f (z, w) = w - z b ds A(s)
and df = dw -dz A(z) is meromorphic at z = 0, so as g; we have the required extension property.

Proposition 5.3. Let F be a foliation on P 2 ≃ C 2 ∪L ∞ . If F is a turbulent foliation of Kodaira dimension 1, then it possesses an invariant curve outside L ∞ .

Proof. By contradiction, suppose F possesses no invariant algebraic curve in C 2 = P 2 \ L ∞ . By Proposition 5.2, F is transversely affine. The assumption on non existence of invariant curves allows to use [CP14, Corollary B], to infer that F is given by the pullback ω of a 1-form

ω 0 = dy + (a(x) + b(x) y) dx, a, b ∈ C[x]
under a polynomial map C 2 → C 2 , which extends as a rational map H 0 : P 2 P 2 .

Denote G the foliation of P 2 induced by ω 0 . From its equation, observe that G| C 2 has no singularities. As F| C 2 , G| C 2 possesses no algebraic invariant curves.

There exist sequences of blowing-ups Σ X : X → P 2 , Σ Y : Y → P 2 in the source and the target of H 0 such that • The foliations F red := Σ * X F and R := Σ * Y G have at most reduced singularities. • There exists an elliptic fibration f X : X → P 1 adapted to the turbulent foliation F red .

• There exists a rational fibration f Y : Y → P 1 adapted to the Riccati foliation R.

• The rational map H : X Y such that Σ Y • H = H 0 • Σ X is actually a morphism (i.e. holomorphic).
By the already proved items i) and ii) of Theorem B, we must have κ(R) ≥ 1. As for every Riccati foliation κ(R) ≤ 1, we have κ(R) = 1.

By Lemma 5.4 below, q Y is the Iitaka fibration ([Bru00, p.116]) of the cotangent divisor T * R. Similarly q X is the Iitaka fibration of the cotangent divisor T * F red .

From the remark in [Bru00, p. 29] it follows T * F red = H * (T * R) ⊗ O X (D) for D an effective divisor on X (see also the proof of [Tou03, Lemme 3.2.8]) Then, Lemma 5.5 below yields that H maps the fibers of f X in the fibers of f Y : for general c ∈ P 1 , there exists r(c) ∈ P 1 such that H f -1 X (c) ⊂ f -1 Y (r(c)). Consider, for general c, the following restriction of H,

H c : f -1 X (c) → f -1 Y (r(c)). Denote R ⊂ X and B ⊂ Y the ramification and branching curves of H, namely R := {x ∈ X; det(d x H) = 0}, B := H(R).
The map H c is étale outside R.

Let L ∞ be the strict transform of L ∞ in the sequence of blowing-ups p Y . If B has R-invariant components, they must be contained in L ∞ or in the exceptional divisor of p Y , because G possesses no invariant curve in C 2 . Denote B inv the union of these components. Notice that the general fiber of f Y intersects B inv at most once, because f Y is induced by the coordinate fibration x on C 2 . We Assert that for a general c, the curve f -1 Y (r(c)) does not intersect B \ B inv . After proving this assertion we obtain that, for general c, the map H c ramifies at most over one point of f -1 Y (r(c)) ≃ P 1 , contradicting that f -1 X (c) is elliptic.

We conclude by proving this Assertion. First, remark that any non R-invariant component of B is a curve transverse to the general fiber of f Y and has a finite number of tangencies with R. Therefore we have, for general c:

(1) For any point

p ′ ∈ f -1 Y (r(c)) \ B inv ∩ B, B is transverse to both f -1 Y (r(c)
) and R at p ′ ; (2) for any point p ∈ f -1 X (c) ∩ R, H writes as (s, t) → (S, T ) = (s ℓ , t), with ℓ > 1, in suitable local coordinates centered at p. Take c such that we have (1) and (2). Suppose we have a point p ′ = H(p) in f -1 Y (r(c))∩ B \B inv . In the adapted coordinates (S, T ) of (2), the leaf of R through p ′ has a local equation T = λ 1 S+o(S), λ 1 ∈ C and the fiber f -1 Y (r(c)) passing through p ′ expresses as T = λ 2 S+o(S), λ 2 ∈ C. Thus in the neighborhood of p, their pull-backs have equation t = λ i s ℓ + o(s ℓ ) and are tangent at (s, t) = (0, 0) because ℓ > 1. Meaning p is a tangency point between F red and f -1 X (c). As f X is an adapted fibration for F red , this cannot happen for c general enough.

For the reader's convenience, we prove two facts that belong to the birational theory of foliations and varieties.

Lemma 5.4. Let F be a reduced foliation on a projective manifold X, with κ(F) = 1. Suppose F is a Riccati or a turbulent foliation, with adapted fibration f : X → C. Then f is the Iitaka fibration of T * F.

Proof. Let F be the general fiber for a fibration f adapted to F. Lemma 5.5 shows that the Iitaka fibration associated to F is the fibration f . The proof of [Bru00, Theorem 1 p. 118] shows T * F ⊗m = O(nF + D) for an effective divisor D and suitable integers m, n > 0. Lemma 5.5 (with L = id X ) allows to deduce that both divisors F and T * F have the same Iitaka fibration, yielding the conclusion.

In our context, next Lemma should be applied in the case of (foliated) Kodaira dimension 1.

Lemma 5.5. Let L : X 1 → X 2 be a morphism between projective manifolds. Let D 1 , D 2 be divisors on X 1 and X 2 , respectively. Suppose these divisors have equal positive Iitaka dimension. Take k > 0 big enough so that p i :

X i PΓ(X i , O(D i ) ⊗k ) * is the Iitaka fibration of D i , i = 1, 2. Suppose D 1 = L * D 2 + D with D effective. Let r • q be the Stein factorization of p 2 • L. Then q is the Iitaka fibration of D 1 .
Proof. Choosing a nontrivial global section s ∈ Γ(X 1 , O(D)) we have an injection

φ k : Γ(X 2 , O(D 2 ) ⊗k ) → Γ(X 1 , O(D 1 ) ⊗k ) σ → (L * σ) ⊗ s ⊗k
and the following diagram commutes, with φ * k onto.

X 1 L p 1 / / ❴ ❴ ❴ PΓ(X 1 , O(D 1 ) ⊗k ) * φ * k X 2 p 2 / / ❴ ❴ ❴ PΓ(X 2 , O(D 2 ) ⊗k ) *
Restricting the maps, with S = p 1 (X 1 ), T = p 2 (X 2 ), we get the following.

X 1 L p 1 / / ❴ ❴ ❴ S X 2 p 2 / / ❴ ❴ ❴

T

The map S → T is onto. As dim S = dim T , it must be a generically finite map. Uniqueness in Stein factorization yields q = g • p 1 for some birational map g : S → S ′ . This yields the conclusion, because the Iitaka fibration is defined only up to birational transforms in the target.

To explains the limits of Proposition 5.3, we present a turbulent foliation with κ(F) = 1 having exactly one rational invariant curve in C 2 .

Example 5.6. Consider the pencil of cubics E in the projective plane generated by the cuspidal cubic C : y 2 + x 3 = 0.

and the line at infinity L ∞ taken with multiplicity 3, whose general element is an elliptic curve. Also consider the pencil of rational cuspidal cubics (C λ ) in the projective plane

C λ : y 2 = λx 3 .
For E there is a unique indeterminacy point q ∈ L ∞ , which is an inflexion point for the general elements of the cubic pencil. After nine blowing-ups at q and at suitable infinitely near points we obtain a minimal elliptic fibration π : M → P 1 , having exactly two singular fibers. One fiber is the strict transform of the cuspidal cubic (type II in Kodaira's notation) and the other fiber has nine rational components (type II * ): the strict transforms L ∞ of L ∞ and E 1 , E 2 , . . . , E 8 of the exceptional lines of the first eight blowing-ups. Denote by M ′ the surface obtained from M after three blowing-ups, first at the point coming from (0, 0) ∈ C 2 , which produces the strict transform C of C, and the other ones in order to separate C from its tangent line.

For (C λ ) there are two base points, one (also) at q ∈ L ∞ and the other at (0, 0) ∈ C 2 . Three blowing-ups at infinity are enough to separate the cuspidal cubics of (C λ ) at infinity. And two additional blowing-ups at points infinitely near to (0, 0) are enough to produce a rational fibration. Therefore the surface M ′ is endowed with both: i) a rational fibration obtained from (C λ ) and ii) a non-minimal elliptic fibration.

Consider now the degree 4 foliation F on P 2 associated to the 1-form on C 2 given by ω = d(y 2 + x 3 ) + (y 2 + x 3 )(3ydx -2xdy).

Note that F leaves invariant L ∞ . From [Lor94, Lemme IV.2] it follows that the unique algebraic leaves of F are C and L ∞ . We have ω ∧ d(y 2 + x 3 ) = 6 (y 2 + x 3 ) 2 dx ∧ dy, which means F and E are tangent exactly along C ∪ L ∞ . The transform F ′ of F in M ′ is then a turbulent foliation whose adapted elliptic fibration is obtained from E.

On the other hand, an explicit computation shows that F ′ is tangent to the rational fibration of M ′ along the (-2)-curve E 3 ⊂ M ′ . Hence, if F r is a general fiber of the rational fibration,

T * F ′ • F r = tang(F, F r ) -F r • F r = tang(F ′ , F r ) > 0.
We assert that κ(F) = 1. Otherwise, T * F ′ ≡ N is supported on the union of F ′ -chains. More precisely, there are two F ′ -chains consisting of the following two chains of (-2)-curves:

L ∞ , E 1 , E 2 and E 4 , . . . , E 8 .
We then check T * F ′ • F r = N • F r = 0: contradiction.

Proof of Theorem B -iv) and birational geometry of examples

In this section we give examples of foliations F of P 2 which are associated to simple derivations and whose Kodaira dimension satisfies κ(F) ∈ {1, 2}. In every case we describe the corresponding reduction of singularities and give a nef model. Moreover, we show some birational (non-)equivalences between examples.

We will provide diagrams to illustrate the reduction of singularities and nef models. The following conventions are used in the examples.

• The affine coordinates (x, y) ∈ C 2 correspond to the point (x 0 : x 1 : x 2 ) = (1 : x : y) ∈ P 2 .

• We denote as (σ i ) the sequence of blowing-ups of points composing of a given foliation F and (p j ) the sequence of contractions composing the morphism ρ : M → S to a nef model F nef of F. • In the figures, we use nd, sn, m, r, nil for non-degenerate, saddle-node, Morse, radial and nilpotent singularities, respectively (cf. Section 2). • The line at infinity x 0 = 0 will be denoted by L and any strict transform always by L, except in the figures, see next point. • In the figures, we use the same symbol for each exceptional curve and its strict transforms under other blowing-ups, but the self-intersection numbers indicated in parentheses (n) will help to avoid confusions. • The bracket [m] denotes the multiplicity of the fiber (z = 0) of the Riccati foliation locally defined by z m dw + (a(z)w 2 + b(z)w + c(z)) dz, with a, b, c holomorphic at z = 0.

We start with examples having κ(F) = 1. Part of these examples are associated to Shamsuddin derivations, see [START_REF] Shamsuddin | Automorphisms and skew polynomial rings[END_REF]. These are derivations of the form

D = ∂ x + (a(x)y + b(x))∂ y , a, b ∈ C[x].
The associated foliation is given by ω = dy -(a(x)y + b(x)) dx. It is a special Riccati foliation.

Example 6.1. Consider the foliation F of the projective plane associated to ω = (xy + 1) dx -dy, called Bergman's example in [START_REF] Coutinho | On the differential simplicity of polynomial rings[END_REF]. The extended foliation F of P 2 has degree 2. The point at infinity (1 : 0 : 0) is a saddle-node with Milnor number m = 3 whose strong separatrix is the line at infinity L : x 0 = 0; in particular L is F-invariant. At (0 : 1 : 0) there is a a quadratic singularity: the blowing-up at this point produces a Riccati foliation on F 1 , leaving invariant the exceptional curve E 1 . There is just one singular point along E 1 , a saddle-node with Milnor number m = 3 with strong separatrix L and weak separatrix E 1 . This is already a nef model in F 1 . The multiplicity of L as an invariant fiber and the formula for the cotangent bundle of [Bru00, p. 57] enable us to compute

T * F nef = -2L + 3L = O(L) and κ(F) = 1. L σ 1 L(0) E 1 ( -1) [3]
(1) sn sn Example 6.2. From [START_REF] Giné | A note on Liouvillian integrability[END_REF] we consider the foliation of degree 2 in the projective plane associated to ω = (1 + x (2x + y))dy + 2x(2x + y)dx = 0 The singularities along L are a saddle-node at (0 : 1 : 0) and a non-reduced (quadratic) singularity at (-1 : 2 : 0). The affine lines 2x + y = c are completely transverse to the foliation. One blowing-up at (-1 : 2 : 0) is enough to reduce the singularity and produces a Riccati foliation. The exceptional line is invariant and has a saddle-node. L is the unique invariant fiber. This is already a nef model. The multiplicity of L as a invariant fiber is 3 and we compute

T * F nef = O(-2L) + 3L = O(L), κ(F) = 1.
Example 6.3. From [START_REF] Nowicki | Polynomial derivations and their rings of constants[END_REF] we consider the foliation of degree 3 in the projective plane of Shamsuddin type given by ω = (yx 2 + xy + x 2 )dx -dy = 0. At (1 : 0 : 0) there is a saddle-node with Milnor number m = 4 whose strong separatrix is L. At the vertical infinity there is a cubic singularity. The blowing-up at this point produces a Riccati foliation relatively to the vertical lines. The exceptional line E 1 is invariant and the unique singularity along E 1 is a saddle-node with Milnor number m = 4 with weak separatrix E 1 and strong separatrix L. This curve is the unique invariant fiber and its multiplicity as invariant fiber is 4. The foliation on F 1 is already a nef model. The cotangent line bundle can be computed as above,

T * F nef = O(-2L) + 4L = O(2L) and κ(F) = 1.
Example 6.4. Consider the Shamsuddin type foliation of degree 4 in the plane associated to ω = ((x 3 + 1)y + 5x 4 -x 3 -2x 2 + 4x)dx -dy = 0.

At (1 : -5 : 0) there is a saddle node with Milnor number m = 5. At (0 : 1 : 0) there is a quartic singular point (algebraic multiplicity = 4). The foliation obtained after blowing up this point is Riccati, having just one singular point along E 1 which is saddle-node with Milnor number m = 5, with weak separatrix E 1 and strong separatrix L.

The foliation on F 1 is already a nef model. The multiplicity of L as invariant fiber is 5 and again the cotangent line bundle can be computed as

T * F nef = O(-2L) + 5L = O(3L) and κ(F) = 1.
Example 6.5. From [Now94, Ex 13.3.7 p. 154] we have a foliation of degree 8 of Shamsuddin type ω = ((x 3 + 1)y + x 8 + 3x 5 + 1)dx -dy = 0.

At (0 : 1 : 0) there is a highly degenerate singularity (with algebraic multiplicity = 8). The blow up of the foliation at this point produces a Riccati foliation, but the reduction of singularities is not completed yet. It needs four additional blowing-ups. From the second blown up point to the fifth the algebraic multiplicity is = 2. Along the fifth exceptional line E 5 there are three singular points: two saddle-nodes with Milnor number m = 5 and one Morse point. The foliation obtained is reduced but not a relatively minimal model.

To obtain a relatively minimal model we contract L, E 2 , E 3 and E 4 , in this order.

L(0) L(1) σ 1 E 1 (-1) σ 2 L(-1) E 1 (-2) [9] [9] [8] E 2 (-1) (-2) (-2) (-2) [5] E 1 sn sn m m m m 1 2 1 E E (0) [5] sn sn m E 5 L(-1) (-5) (-5) 4 σ σ 3 σ 5 [6] [7] [8] E 4 E 3 E 2 4 3 L(-1)
[9] 5 q q q q As we contract foliated exceptional curves with Morse points, the strict transform of E 5 contains just two saddle-nodes. In this example the relatively minimal model is already a nef model (on the Hirzebruch surface F 5 ).

The multiplicity of E 5 as invariant fiber and the formula for the cotangent bundle gives

T * F nef = O(-2) + 5E 5 = O(3L) and κ(F) = 1.
Example 6.6. This example is not of Shamsuddin type but is of Riccati type. From [START_REF] Maciejewski | Simple quadratic derivations in two variables[END_REF] (see also [START_REF]On the classification of simple quadratic derivations over the affine plane[END_REF]) we consider the foliation of degree 2 in the projective plane associated to

ω = dx -(x 2 + y)dy = 0.
At the horizontal infinity point there is a radial point (Milnor number m = 1), whose blowingup produces a Riccati foliation completely transverse to the exceptional line E 1 . There is a nilpotent singularity at the vertical infinity point whose Milnor number is m = 6 (thanks to Darboux's formula in the plane). The blowing-up at the nilpotent point produces an invariant exceptional curve E 2 having just one quadratic singularity, at the intersection with L at infinity. The blowing-up at this quadratic singularity produces three singularities along E 3 : two of them being non-degenerate and reduced singularities, placed at the intersections of E 3 with L and E 2 , and a third one being a saddle-node, with strong separatrix E 3 . We assert that the Milnor number of this saddle-node is m = 4: indeed, it follows from the diagram on the top of [Bru00, p. 56] and the fact that a nilpotent point in P 2 has m = 6.

The foliation obtained is not a nef model. The morphism ρ = ρ 2 •ρ 1 contracts two (-2)-curves and produces a singular surface with two quotient singularities q 1 , q 2 along the strict transform of E 3 (where there is also a saddle-node with Milnor number m = 4).

nil r σ 1 E 1 E 2 [3] L (-1) (-1) (-1) 2 σ σ 3 sn E (-2) E 3 (-1) L (-2) nd nd [3] 2 1 2 E 1 (-1) q 1 sn L(1) 2 q E 1 (-1) ρ ρ
In this nef model of the foliation, the computation of the degree of π * (T * F) ∈ Pic(P 1 ) ⊗ Q gives: As there is no curve with negative self intersection in P 2 , F is its own nef model. The group P ol(F| C 2 ) contains the linear automorphism L j (x, y) → (j • x, j 2 • y) where j is a primitive cubic root of the unity and L * j (ω) = j 2 • ω. The affine line x = 0 is transverse to the foliation and, for c = 0, the affine lines x = c have one movable tangency.

deg(π * (T * F nef )) = -2 + 4 + 1 2 = 1 2 > 0
Example 6.8. From [START_REF]An example of a simple derivation in two variables[END_REF], we have a family of foliations F k of degree k ≥ 2 in the plane associated to simple derivations. It is defined by the family of 1-forms

ω k = (y k + x)dx -dy, k ∈ N * .
For k = 2 this coincides with Example 6.6, up to permutation of (x, y).

We assert that κ(F k ) = 2 for all k ≥ 3. At the vertical infinity point, each F k has a radial point p. The exceptional line of the blow up at p belongs to the contact divisor between the transformed foliation and the rational fibration of Σ 1 (p). For simplicity, let's focus on the case k = 3. The reduction of singularities of F 3 is made up of 4 blowing-ups at quadratic singularities of the foliation. The fourth blowing-up introduces E 4 having a saddle-node and 2 extra nondegenerated reduced points (at the intersections of E 4 with the strict transforms of E 2 and E 3 ). The Zariski decomposition of the cotangent line bundle is

T * F 3 ≡ P + 1 4 E 3 + 1 2 L + 1 3 E 2 .
The nef model is obtained after contraction of the support of N and introduces two quotient singularities of the surface q 1 , q 2 . L(0)

L(1)

σ 1 r E 1 (-1) L(-1) E 2 (-1) E 1 (-1) σ 3 1 E (-1) E 2 (-2) 2 σ E 3 (-1) E 1 (-1) L(-2) E 3 (-2) 4 E 2 (-3) (-1) 4 σ L(-2) nd nd nd E nd 1 2 E 4 E 1 q 1 q 2 sn sn ρ ρ
Since the 4 blown up singularities were quadratic (l(F) = 2), we can compute

T * F 3 • T * F 3 = (deg(F 3 ) -1) 2 - 4 i=1 (l p i (F 3 ) -1) 2 = 4 -4 = 0
Combining this with

N • N = ( 1 4 E 3 + 1 2 L + 1 3 E 2 ) 2 = - 3 8 - 1 3 < 0
and the fact that P • N = 0 in the Zariski decomposition, the conclusion is that P • P > 0. Therefore the so called numerical Kodaira dimension is 2 and also κ(F 3 ) = 2.

Example 6.9. From [START_REF] Jordan | Differentially simple rings with no invertible derivatives[END_REF] we have a foliation F of degree 3 in the projective plane associated to ω = y 3 dy -(1 -xy) dx. At (0 : 1 : 0), there is a quadratic singularity whose reduction is composed by 3 blow ups. The foliation has κ(F) = 2 and its nef model is not much different than the one of Example 6.8. At last, we remark that the affine lines y = c, c = 0, exhibit one movable tangency point with the foliation (at the intersection of y = c and y = 1

x ). The affine line y = 0 is completely transverse to the foliation, a property that will be useful in Section 8. Example 6.10. From [START_REF] Kour | Some simple derivations of k[x,[END_REF] we have examples of foliations F r,s,g in the projective plane with degrees deg(F r,s,g ) = s + 1, defined for r, s ∈ N such that r + 2 ≤ s and g ∈ C * by ω D := (xy s + g) dx -y r dy.

We have a singular point at (0 : 1 : 0) with algebraic multiplicity ℓ(F r,s,g , p) = s. At (0 : 0 : 1), there is a singular point with algebraic multiplicity 3. Except for y = 0, all horizontal affine lines exhibit one movable contact point with the restricted foliation F r,s,g | C 2 . However, the affine line y = 0 is completely transversal to the foliation. Example 6.11. According to [START_REF] Odani | The limit cycle of the van der Pol equation is not algebraic[END_REF], the foliations in the plane associated to

ω := (f (x) • y + g(x)) dx + y dy = 0, f, g ∈ C[x]
do not have algebraic solutions in C 2 if three conditions are satisfied: i) f, g = 0, ii) deg(f ) ≥ deg(g) and iii) f g is not constant. These foliations do not have singularities in C 2 exactly when g(x) = c ∈ C * . Therefore, to produce foliations associated to simple derivations it suffices to take g(x) = c ∈ C * and f ∈ C[x]. In this case, the affine line y = 0 is everywhere transverse to the foliation while general horizontal lines y = c exhibit deg(f ) tangencies with the foliation. And the line at infinity is invariant by the extended foliation. Now we establish some birational (non)-equivalences among the Examples. Proposition 6.12. i) Example 6.2 is linearly equivalent to a foliation of Shamsuddin type.

ii) Examples 6.4 and 6.5 are equivalent by a polynomial automorphism of degree five.

iii) The foliations of Example 6.1 and Example 6.3 are not birationally equivalent. ii).We start with the nef model of Example 6.5 in the Hirzebruch surface F 5 . After an elementary transformation we pass to F 4 keeping the saddle-nodes and the multiplicity [5] of the unique invariant fiber. This is shown in next figure. F (-1) E q (-1)

(-4) E E q (0) q sn q σ σ [5] sn sn sn nd
After this we perform three more elementary transformations to obtain a foliation of F 1 and then, after contraction of the (-1)-section, we arrive in the projective plane. The pair of saddlenodes on the invariant vertical fiber of F 5 is transformed in a pair of singularities along an invariant projective line of a degree 4 foliation.

The net effect in the plane can be described concretely by means of a polynomial isomorphism of C 2 . Indeed, η := ((x 3 + 1)y + 5x 4 -x 3 -2x 2 + 4x)dx -dy defines the foliation of Example 6.4 and, if R : C 2 → C 2 is (x, y) → (x , y + x 5 + 2x 2 + 1) then R * (η) is the 1-form of Example 6.5.

iii). A way to see this is to use the birational invariant g(F) of [Men00, p. 139]. In Example 6.1 and Example 6.3 the invariants are g(F) = 2 and g(F) = 3, respectively. Proposition 6.12 -ii) illustrates the general issue of finding the simplest (of least degree) plane birational model for foliations of the plane or derivations. Algorithmic procedures toward this objective would be of great utility.

Proof of Theorem A

In the language of foliations Theorem A takes the following form.

Theorem 7.1. Let F be a foliation of P 2 that the restriction F| C 2 has neither algebraic invariant curves nor non-reduced singularities. The group Bir(F) is finite.

Proof. We proceed by contradiction: suppose Bir(F) is infinite. From our assumption on algebraic invariant curves, F cannot have a rational first integral. We have two cases: either (1) there exists a birational model (X, F ) of (P 2 , F) such that Aut( F ) = Bir( F) or (2) for every birational model (X, F ), the inclusion Aut( F ) ⊂ Bir( F) is strict.

In case (2), we may apply [CF03, Th. 1.1] to (X, F ), as fibrations are excluded we have one of the following situations.

(i) There exists a non trivial holomorphic vector field on X defining a one parameter subgroup of Aut( F ) or (ii) the surface X is generalized Kummer surface (see [CF03, Ex. 1.1]), X is a quotient of an abelian surface A and F lifts to A as a linear foliation G.

In case (1), we may apply [CF03, Th. 1.2]. Since fibrations are excluded, we are in the situation of [CF03, Ex. 1.3], in particular up to passing to a birational model (X, F ), (iii) the surface X is a finite quotient of S = P 1 × P 1 and F lifts to a foliation G of S given by a differential form αwdz + βzdw, for some α, β ∈ C.

In cases (ii) and (iii) the foliation G has Kodaira dimension 0. By the remark in [Bru00, p. 29] or the proof of [Tou03, Lemme 3.2.8], this forces κ( F ) ≤ 0 and contradicts Theorem B.

So we only need to derive a contradiction from situation (i) to complete the proof. Moreover, if the vector field is tangent to F, we may use [Bru00, Prop. 6.6 iii)] to see that F is a Riccati foliation with two distinct adapted fibrations and consequently κ(F) = 0, contradicting Theorem B.

The conclusion is given by the following argument, proposed by Jorge Pereira. If the vector field is not tangent to F consider its image X in P 2 , consider also a rational 1-form ω defining the foliation. The form η = ω ω( X) is closed, by the computation [PS02, Proof of Cor. 2] inherited from [START_REF] Cerveau | Formes intégrables holomorphes singulières[END_REF]. The poles of η give F-invariant algebraic curves, so that η has no poles in C 2 . Subsequently, the first integral x ⋆ η has no monodromy and gives a rational first integral for F, contradiction.

Polynomial symmetries of foliations associated to simple derivations

Consider a foliation F of the projective plane P 2 = C 2 ∪ L ∞ with two properties: a) F is associated to a simple derivation, b) there exists some affine straight line completely transversal to the foliation F| C 2 . Proposition 8.1. Given n ≥ 2 and B > 0, there exists a foliation G associated to a simple derivation and an element in P ol(G| C 2 ) of order n and degree greater than B.

Actually the construction below starts from any foliation F satisfying a) and b). These properties are verified in all examples of Section 6, so that we have plenty of examples.

Proof. Let F be a foliation satisfying a) and b). Up to an affine transformation, we can suppose that the line in b) is x = 0. Consider the n to 1 rational map φ n : C 2 → C 2 given by φ n (x : y) = (x n , y). The foliation F is defined by the (rational extension to P 2 of the) polynomial 1-form ω = a(x, y)dx + b(x, y)dy. The map φ n extends to a birational map φ n : P 2 P 2 .

Let G 0 = φ * n F. It is defined by (the extension to P 2 of) φ * n (ω) = a(x n , y)d(x n ) + b(x n , y)dy. Since the affine line x = 0 is supposed to be completely transverse to F, we know G has no singularity along x = 0. Note also that in C 2 \ {x = 0} the map φ n (x, y) = (x n , y) is a local isomorphism, so the pull-back does not introduce any singularity. Any algebraic G 0 | C 2 -invariant curve would descend to an F| C 2 -invariant curve, so that no such curve exists. Example 8.2. We examplify the construction given in the proof of Proposition 8.1. Starting from Example 6.1 ω = (xy + 1)dx -dy and taking n = 2, so φ 2 (x, y) = (x 2 , y). We obtain a foliation G 0 defined by the 1-form φ * 2 (ω) = (2x 3 y + 2x) dx -dy and admiting the linear automorphism T (x, y) = (-x, y). Now, we consider the automorphism P τ (x, y) = (x, y + x 3 ) and define Ω := P * τ (φ * 2 (ω)) = (2x 3 y -2x 6 + 3x 2 + 2x)dx -dy. Then the involution Γ ξ,τ = Γ -1,x 3 : (x, y) → (-x, y -2x 3 ) preserves Ω.

At last, note that the foliations we have constructed in the proof of Proposition 8.1 do not have minimal degree in their birational classes, due to the fact that the automorphism P τ has positive degree. In other terms, such foliations are not primitive in the sense of [START_REF] Cerveau | Action of the Cremona group on foliations on P 2 C : some curious facts[END_REF]. On the other side, the foliation of Example 6.7 is primitive but the exhibited automorphism is linear. This raises the following question.

  (a), (b), (c), (d), (e) below (Figure adapted from [Bru03, p. 20]).

  q

F

  and κ(F) = 1. All the remaining examples have κ(F) = 2. Example 6.7. Consider the foliation F of degree deg(F) = 2 in the projective plane associated to the following equation in C 2 : ω = x (1 + xy) dx -(1 + xy + x 3 ) dy = 0 which was taken from [CDGBM10, Prop. 1.3]. The line at infinity is not invariant and there is just one saddle-node with Milnor number m = 7 at infinity. So the singularity of F is reduced. The cotangent line bundle is T * F = O(1) and κ(F) = 2.

  Proof. i). With the linear change of variables y = u -2v, x = v, from the equation of Example 6.2 we obtain η = 2dv -(1 + vu)du which is of Shamsuddin type.

Hence G 0

 0 is associated to a simple derivation.Let ξ be a primitive n-th root of unity. The linear automorphism T ξ (x, y) = (ξ • x, y) clearly preserves G 0 and has order n. Note however thatT ξ ∈ Aut(D n ) for the derivation D n dual to φ * n (ω), because T * ξ (φ * n (ω)) = φ * n (ω) and Jac(T ξ ) = ξ = 1.Now, for any polynomial τ ∈ C[x], consider the automorphism of C 2 defined byP τ (x, y) = (x , y + τ (x))and setG := P * τ G 0 . If the polynomial τ (ξ • x) -τ (x) has degree d ≥ 1, (e.g. if deg x (τ ) = d and (d, n) = 1) the map Γ ξ,τ (x, y) := P -1 τ • T ξ • P τ (x, y) = (ξ • x , y + τ (ξ • x) -τ (x)) defines a polynomial automorphisms of degree d and order n in Aut(G), which completes the proof.

  Are there non-linear polynomial automorphisms of primitive foliations associated to simple derivations ? By [Bru99, Cor. p 293], such automorphisms would necessarily be conjugated to automorphisms of the form (x, y) → (ax + P (y), by + c) with a, b ∈ C * , c ∈ C and P ∈ C[y].

G. Cousin was partially funded by FIRB RBFR12W1AQ, Labex IRMIA and ANR-13-JS01-0002-01. I. Pan was partially supported by ANII and PEDECIBA of Uruguay.