Preprint A161/2002
Bounds on leaves of one-dimensional foliations

Steven Kleiman | Esteves, Eduardo

**Keywords: **
foliations | curves | singularities

Let $X$ be a
variety over an algebraically closed field,
$\eta\:\Og^1_X\to\c L$ a one-dimensional singular foliation, and
$C\subseteq X$ a projective leaf of $\eta$. We prove that
$2p_a(C)-2=\deg(\c L|C)+\lambda(C)-\deg(C\cap S)$ where $p_a(C)$ is the
arithmetic genus, where $\lambda(C)$ is the colength in the dualizing
sheaf of the subsheaf generated by the Káhler differentials, and where
$S$ is the singular locus of $\eta$. We bound $\lambda(C)$ and
$\deg(C\cap S)$, and then improve and extend some recent results of
Campillo, Carnicer, and de la Fuente, and of du Plessis and Wall.