Integral arithmetically Buchsbaum curves in \({\mathbb{P}}^ 3\)
Let X be an arithmetically Buchsbaum curve in complex projective three- space. Two sequences of integers are naturally associated to X. The hypothesis ensures that the Hartshorne-Rao module, M(X), of X is isomorphic to \(\oplus^{b}_{j=1}{\mathbb{C}}(-n_ j) \) for some integers \(n_ 1\leq...\leq n_ b\). Let n(X) represent \(n_ 1\leq...\leq n_ b\). If \(f_ 1,...,f_ c\) is a minimal generating set for the homogeneous ideal \(I_ X\) for X, with \(\deg (f_ 1)\leq...\leq \deg (f_ c)\), then let d(X) represent \(\deg (f_ 1)\leq...\leq \deg (f_ c)\). In a previous work [Publ. Res. Inst. Math. Sci. 20, 793-837 (1984; Zbl 0574.14030)], the author found necessary and sufficient conditions on sequences n: \(n_ 1\leq...\leq n_ b\) and d: \(d_ 1\leq...\leq d_ c\) in order to that there exists an arithmetically Buchsbaum curve X with \(n(X)=n\) and \(d(X)=d\). In particular, it is necessary that \(d_ 1\geq 2b.\) The present paper finds necessary and sufficient conditions on sequences n and d in order to that there exists an integral arithmetic Buchsbaum curve X with \(n(X)=n\) and \(d(X)=d\). In particular, \(d_ 1-2b\geq n_ b- n_ 1\) is a necessary condition. This last inequality has also been discovered by \textit{G. Bolondi} and \textit{J. Migliore} [J. Algebra 123, No.2, 426-456 (1989; Zbl 0695.14022)] and \textit{A. V. Geramita} and \textit{J. Migliore} [J. Pure Appl. Algebra 58, No.2, 147-167 (1989; Zbl 0699.14062)].
minimal generating set for the homogeneous ideal
scientific article; zbMATH DE number 4150399