Bounds on leaves of foliations of the plane

Abstract

This paper contributes to the solution of the Poincare problem, which is to bound the degree of a (generalized algebraic) leaf of a (singular algebraic) foliation of the complex projective plane. The first theorem gives a new sort of bound, which involves the Castelnuovo--Mumford regularity of the singular locus of the leaf. The second theorem gives a bound in terms of two singularity numbers of the leaf: the total Tjurina number, and the number of non-quasi-homogeneous singularities. If such singularities are present, then this bound improves one due to du Plessis and Wall, at least when the curve is irreducible.