Curves on threefolds and a conjecture of Griffiths-Harris

Abstract: We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface $X\subset \bbP<sup>{4}$ of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type hypersurfaces in $\bbP<sup>4$. We also verify that certain 1-cycles on a general quintic hypersurface are non-trivial elements of the Griffiths group.