Bounds on leaves of one-dimensional foliations
Abstract
Let X be a variety over an algebraically closed field, η:1X L a one-dimensional singular foliation, and C⊂eq X a projective leaf of η. We prove that 2pa(C)-2=(L|C)+λ(C)-(C S) where pa(C) is the arithmetic genus, where λ(C) is the colength in the dualizing sheaf of the subsheaf generated by the K\"ahler differentials, and where S is the singular locus of η. We bound λ(C) and (C 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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