Bounds on leaves of one-dimensional foliations

Abstract.

Let X be a variety over an algebraically closed field, $\eta :\Omega ^{1}_{X} \to {\cal L}$ a onedimensional singular foliation, and $C \subseteq X$ a projective leaf of η. We prove that

$$ 2p_{a} {\left( C \right)} - 2 = \deg {\left( {{\cal L}\left| C \right.} \right)} + \lambda {\left( C \right)} - \deg {\left( {C \cap S} \right)} $$

where p _a (C) is the arithmetic genus, where λ(C) is the colength in the dualizing sheaf of the subsheaf generated by the Kähler differentials, and where S is the singular locus of η. We bound λ(C) and $\deg {\left( {C \cap S} \right)}$, and then improve and extend some recent results of Campillo, Carnicer, and de la Fuente, and of du Plessis and Wall.