On the order of projective curves (Q1263630)

The Hartshorne-Rao module of a locally Cohen-Macaulay curve C in \({\mathbb{P}}_ n\) is by definition the graded module \(H_ C:=\oplus_{t\in {\mathbb{Z}}}H^ 1(I_ C(t)) \) over the graded ring A of \({\mathbb{P}}_ n\). The diameter diam(C) of C is the number \(p-q+1\) if \(H^ 1(I_ C(t))=0\) for \(t>p\) and \(t<q\) and \(H^ 1(I_ C(q))\neq 0\neq H^ 1(I_ C(p))\). The order ord(C) of C is by definition the minimal integer t such that \(m^ t\cdot H_ C=0\), where m denotes the maximal homogeneous ideal of A of all forms of positive degree. In the paper the existence of several components of the Hilbert scheme of curves in \({\mathbb{P}}_ n\) is proved whose general curve has Hartshorne-Rao module with order equal to its diameters. The proof consists in a reduction to several papers of Ballico and Ellia on the maximal rank conjecture.