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- Title
On the genus and Hartshorne-Rao module of projective curves.
- Authors
Chiarli, Nadia; Greco, Silvio; Nagel, Uwe
- 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.
- Publication
Mathematische Zeitschrift, 1998, Vol 229, Issue 4, p695
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/PL00004678