Effective algebraic degeneracy

Abstract

We show that for every smooth projective hypersurface X⊂ℙn+1 of degree d and of arbitrary dimension n ≥2, if X is generic, then there exists a proper algebraic subvariety Y ⊊ X such that every nonconstant entire holomorphic curve f :ℂ→X has image f(ℂ) which lies in Y, as soon as its degree satisfies the effective lower bound $d\geqslant 2^{n^{5}}$ .