On the genus and Hartshorne-Rao module of projective curves

In this paper optimal upper bounds for the genus and the dimension of the g raded components of the Hartshorne-Rao module of curves in projective n-spa ce are established. This generalizes earlier work by Hartshorne [H] and Mar tin-Deschamps and Perrin [MDP]. Special emphasis is put on curves in P-4. T he first main result is a so-called Restriction Theorem. It says that a non -degenerate curve of degree: d greater than or equal to 4 in P-4 over a fie ld of characteristic zero has a non-degenerate general hyperplane section i f and only if it does not contain a planar curve of degree d - 1 (see Th. 1 .3). Then, using methods of Brodmann and Nagel, bounds for the genus and Ha rtshorne-Rao module of curves in P-n with non-degenerate general hyperplane section are derived. It is shown that these bounds are best possible in a very strict sense. Coupling these bounds with the Restriction Theorem gives the second main result for curves in P-4. Then curves of maximal genus are investigated. The Betti numbers of their minimal free resolutions are comp uted and a description of all reduced curves of maximal genus in pn of degr ee greater than or equal to n + 2 is given. Finally, all pairs (d, g) of in tegers which really occur as the degree d and genus g of a non-degenerate c urve in P-4 are described.