Algebraic Hyperbolicity and Lang Conjecture

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Lang conjecture says that rational and elliptic curves on a variety of general type are algebraically***degenerate. We study this conjecture for generic hypersurfaces in projective spaces. This problem is***closely related to complex hyperbolicity and Kobayashi conjecture. In the algebraic setting, this amounts***to algebraic hyperbolicity in the sense of Demailly. H. Clemens proved that a generic hypersurface is***algebraically hyperbolic when the degree of the hypersurface is sufficiently large. We want to push his***bound on the degree of hypersurfaces to the conjectured optimum. We propose to approach this***conjecture by constructing the global jet spaces of versal families of hypersurfaces and studying the stable***loci of the corresponding jet bundles.