Preprint A183/2002
Exceptional families of foliations and the Poincare' problem

Alcides Lins Neto

**Keywords: **
algebraic foliation | Poincare' problem

A 1-parameter family of foliations $(\fa_\a)_{\a\in X}$ on a compact complex surface $M$
is called {\it exceptional and elliptic} if it satisfies the following properties :
{\bf (a).} The family has singularities of fixed analytic type;
{\bf (b).} The set $E=\{\a\in X|\, \fa_\a$ has a first integral $\}$ is countable
and non-discrete; {\bf (c).} There is $\a\in E$ such that the generic fibre
of the first integral is elliptic. In this paper we show that, if a surface $M$ admits an exceptional and elliptic family
of foliations, then $M$ is algebraic and biholomorphically equivalent to a torus, to a $K3$
surface, or to $\Cp(2)$ (Theorem 3). In the case of $\Cp(2)$ we classify all possible
equireducible and exceptional
families such that the singularities of the generic foliations in the family are
non-degenerate (Theorem 2). This classification is connected to the Poincaré problem of
deciding if an algebraic foliation on $\Cp(2)$ has a first integral (cf. [P-1]).