Differential equations on complex projective hypersurfaces of low dimension

Let n = 2, 3, 4, 5 and let X be a smooth complex projective hypersurface of Pn+1. In this paper we find an effective lower bound for the degree of X, such that every holomorphic entire curve in X must satisfy an algebraic differential equation of order k = n = dim X , and also similar bounds for order k > n. Moreover, for every integer n 2, we show that there are no such algebraic differential equations of order k < n for a smooth hypersurface in Pn+1.