Curves on Threefolds and a Conjecture of Griffiths-Harris

We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface X ⊂ P^4 of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type hypersurfaces in P^4. We also verify that certain 1-cycles on a general quintic hypersurface are non-trivial elements of the Griffiths group.