On the degree of algebraic cycles on hypersurfaces

Let X subset of P-4 be a very general hypersurface of degree d >= 6 . Griffiths and Harris conjectured in 1985 that the degree of every curve C subset of X is divisible by d. Despite substantial progress by Kollar in 1991, this conjecture is not known for a single value of d. Building on Kollar's method, we prove this conjecture for infinitely many d, the smallest one being d=5005 . The set of these degrees d has positive density. We also prove a higher-dimensional analogue of this result and construct smooth hypersurfaces defined over Q that satisfy the conjecture.