Bounds on leaves of one-dimensional foliations

Abstract

Let X be a variety over an algebraically closed field, η:Ω^1_X\to L a one-dimensional singular foliation, and C\subseteq X a projective leaf of η. We prove that 2p_a(C)-2=°(L|C)+λ(C)-°(C\cap S) 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 °(C\cap S), and then improve and extend some recent results of Campillo, Carnicer, and de la Fuente, and of du Plessis and Wall.