On the connectedness of the Hartshorne-Rao module of curves in \({\mathbb{P}}^ 3\) (Q1325101)

The Hartshorne-Rao module of a locally Cohen-Macaulay curve \(C\) in \(\mathbb{P}^ 3_ k\) is defined by \(M(C)= \bigoplus_{t \in \mathbb{Z}} H^ 1({\mathcal I}_ C(t))\), where \({\mathcal I}_ C\) is the ideal sheaf of \(C\). \(M(C)\) is a graded \(k[x_ 0,x_ 1,x_ 2,x_ 3]\)-module of finite length. This invariant characterizes the liaison classes of curves in \(\mathbb{P}^ 3\) [see \textit{A. P. Rao}, Invent. Math. 50, 205-217 (1979; Zbl 0406.14033)] and thus plays a central role in the classification problem [see for example \textit{M. Martin-Deschamps} and \textit{D. Perrin}, ``Sur la classification des courbes gauches'', Astérisque 184-185 (1990; Zbl 0717.14017)]. In the paper under review the notion of connectedness of \(M(C)\) is defined as follows: \(M(C)\) is connected if there are no gaps in the graduation -- that is, if there is no \(t \in \mathbb{Z}\) such that \(h^ 1({\mathcal I}_ C (t-1)) \neq 0\), \(h^ 1({\mathcal I}_ C (t))=0\) and \(h^ 1({\mathcal I}_ C (t+k)) \neq 0\) for some \(k>0\). It is clear by Rao's theorem [theorem 2.6 in the above mentioned paper] that there exist integral curves with a not connected module. In this paper a sufficient condition is proved for the Hartshorne-Rao module of an integral curve to be connected, namely, if \(C\) has index of speciality \(e_ C\) with \(e_ C \leq 3\), then \(M(C)\) is connected. -- As a corollary, one gets that if \(d\) and \(g\) are respectively the degree and genus of \(C\) and \(d>(g-1)/2\), then \(M(C)\) is connected. The construction of two counterexamples shows that the condition is sharp.