Curves on threefolds and a conjecture of Griffiths-Harris

TL;DR

This paper characterizes complete intersection curves on general hypersurfaces in projective 4-space and investigates the Griffiths group elements on a general quintic hypersurface, advancing understanding of algebraic cycles.

Contribution

It proves that arithmetically Gorenstein curves on general degree ≥6 hypersurfaces in P^4 are complete intersections and examines Griffiths group elements on quintic hypersurfaces.

Findings

Arithmetically Gorenstein curves on general hypersurfaces of degree ≥6 are complete intersections.

Certain 1-cycles on a general quintic hypersurface are non-trivial in the Griffiths group.

Provides a characterization of complete intersection curves on general type hypersurfaces in P^4.

Abstract

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