On the genus and Hartshorne-Rao module of projective curves

Abstract.

In this paper optimal upper bounds for the genus and the dimension of the graded components of the Hartshorne-Rao module of curves in projective n-space are established. This generalizes earlier work by Hartshorne [H] and Martin-Deschamps and Perrin [MDP]. Special emphasis is put on curves in ${\bf P}^4$. The first main result is a so-called Restriction Theorem. It says that a non-degenerate curve of degree $d \geq 4$ in ${\bf P}^4$ over a field of characteristic zero has a non-degenerate general hyperplane section if 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 Hartshorne-Rao module of curves in ${\bf P}^n$ 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 ${\bf P}^4$. Then curves of maximal genus are investigated. The Betti numbers of their minimal free resolutions are computed and a description of all reduced curves of maximal genus in ${\bf P}^n$ of degree $\geq n+2$ is given. Finally, all pairs (d,g) of integers which really occur as the degree d and genus g of a non-degenerate curve in ${\bf P}^4$ are described.