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.

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