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We show that for every smooth generic projective hypersurface X in P^{n+1} there exists a proper subvariety Y in X of codimension at least 2 such that for every non-constant holomorphic map f: C--->X one has f(C) is contained in Y, provided that the degree of X is greater than 2^{n^5}. In particular we obtain an affirmative confirmation of the Kobayashi conjecture for threefolds in P^4

A remark on the codimension of the Green–Griffiths locus of generic projective hypersurfaces of high degree