  In this article, we prove that there does not exist a family of entire curves
in the universal family of hypersurfaces of degree $d\geq 2n$ in the complex
projective space ${\mathbb P}^n$. This can be seen as a weak version of the
Kobayashi conjecture asserting that a general projective hypersurface of high
degree is hyperbolic in the sense of Kobayashi.
