  We provide sharp lower bounds for the multiplicity of a local holomorphic
foliation defined in a complex surface in terms of data associated to a germ of
invariant curve. Then we apply our methods to invariant curves whose branches
are isolated, i.e. they are never contained in non-trivial analytic families of
equisingular invariant curves. In this case we show that the multiplicity of an
invariant curve is at most twice the multiplicity of the foliation. Finally, we
apply the local methods to foliations in the complex projective plane.
