On The Birational Invariants Of Lins Neto's Foliations

Foliation on algebraic surfaces is a generalization of fibrations.If a foliation is algebraic integrable,it is a pencil of algebraic curves,which birationally equivalent to a fibraiton.A foliation on an algebraic surface is the set of all analytic integral curves of an ordinary differential equation.In 1891,Ponicar?e proposed the famous Poincar?e problem : Is it possible to decide if a differential equation has a rational first integral? Painlev?e also studied the problem: Is it possible to determine the genus(or the upper bound of the genus)of the rational first integral of an algebraically integrable differential equation? In recently 20 years,the theory of the birational geometry theory of foliations is established.Some birational invariants are found,and the rough classification of foliations is finished by using algebraic geometry methods.How to classify foliations isomorphically by using these invariants is still an open problem that has just begun to be studied.In 2002,Lins Neto constructed some families of foliations which are counterexamples to Poincar?e's problem and Painlev?e's Problem.On the other hand,some results indicate that under some conditions,these problems have positive answers.For example,for a foliation with negative Kodaira dimension,it is algebraically integrable if and only if its numerical Kodaira dimension is also negative.For the point of view of classification,what is the difference of the geography between algebraic and non-algebraic foliations is a very important problem.Since Lins Neto's foliations are the first examples which have the same topological type but with different algebraic integrability,a natural problem is to find the minimal models of Lins Neto's foliations,calculate their invariants and determine their geographic distribution information.In this paper,we first find out the minimal models of all Lins Neto's foliations.Then we calculate their Kodaira dimension,numerical Kodaira dimension,and Chern numbers.Furthermore,we give the geography of Chern numbers of Lins Neto's foliations in three different situations.We classify also these foliations of Lins Neto's which are elliptic fibrations.As a byproduct,we also determine the number of new singularities of the pull-back of a foliation by a generic finite cover.