Codimension 2 subvarieties of projective space

The motivation for the present paper is theHartshorne Conjecture on complete intersections inP ⁿ , forn6, and in the codimension 2 case: Any smooth codimension 2 subvarietyX ofP ⁿ is conjectured to be a complete intersection forn6. We prove this conjecture for all varieties with degree below a certain bound, which represents an improvement of the numerical information available untill now.This research has been supported by a grant from theStefi andLars Fylkesaker Foundation