On the degree of algebraic cycles on hypersurfaces

Abstract

Abstract Let X ⊂ ℙ 4 {X\subset\mathbb{P}^{4}} be a very general hypersurface of degree d ≥ 6 {d\geq 6} . Griffiths and Harris conjectured in 1985 that the degree of every curve C ⊂ X {C\subset X} is divisible by d. Despite substantial progress by Kollár in 1991, this conjecture is not known for a single value of d. Building on Kollár’s method, we prove this conjecture for infinitely many d, the smallest one being d = 5005 {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 {\mathbb{Q}} that satisfy the conjecture.