On the genus and Hartshorne-Rao module of projective curves
In this paper the authors give upper estimates for the Hilbert function of the Hartshorne-Rao module of curves in projective \(n\)-space. They generalize earlier results for curves in 3-space obtained by \textit{R. Hartshorne} [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 40 (1994), 207-223 (1996; Zbl 0865.14013)] and by \textit{M. Martin-Deschamps} and \textit{D. Perrin} [C. R. Acad. Sci., Paris, Sér. I 317, No. 12, 1159-1162 (1993; Zbl 0796.14029)]. The authors first consider a restriction problem, namely they show that over a field of characteristic 0 on a non-degenerate curve \(C\) of degree \(d\) in 4-space, a general hyperplane section is non-degenerate if and only if \(C\) does not contain a planar curve of degree \(d-1\). This result is obtained by using syzygy arguments as well as an idea from Strano's paper [\textit{R. Strano}, in: Complex Projective Geometry, Trieste and Bergen 1989, Lond. Math. Soc. Lect. Note Ser. 179, 284-293 (1992; Zbl 0789.14032)]. In addition they construct examples of non-degenerate curves in 4-space with degenerate general hyperplane sections. In the second main theorem of this paper they give a sharp bound for the Hilbert function of the Hartshorne-Rao module of a curve in \(n\)-space, \(n\geq 3\), whose general hyperplane section is non-degenerate in arbitrary characteristic. Furthermore they apply their results to give a description of all reduced curves of maximum genus in \(n\)-space of degree \(\geq n+2\), as well as to determine all pairs \((d,g)\) for which there exists a non-degenerate curve in 4-space , over characteristic 0, of degree \(d\) and genus \(g\).
scientific article; zbMATH DE number 1336827