A complex algebraic variety is said to be Brody hyperbolic if it contains no entire curves, which are non-constant holomorphic images of the complex line. The Green-Griffiths-Lang conjecture predicts that varieties of (log) general type are hyperbolic outside of a proper subvariety called an exceptional locus. We prove an algebraic version of this Conjecture, with respect to Demailly’s algebraic version of hyperbolicity, for the complement of a very general degree 2n hypersurface in Pn. Moreover, for the complement of a very general quartic plane curve, we completely characterize the exceptional locus as the union of the flex and bitangent lines. Based on joint work with Xi Chen and Eric Riedl.