Maximal rank and minimal generation (Q1104372)

Let C be a curve in \({\mathbb{P}}^ 3\); we say that C is of maximal rank if the restrictions map \(\rho _ k: H^ 0({\mathcal O}_{{\mathbb{P}}^ 3}(k))\to H^ 0({\mathcal O}_ C(k))\) are of maximal rank for each k. We say that C is minimally generated if the multiplication map \(\sigma _ k: H^ 0({\mathcal I}_ C(k))\otimes H^ 0({\mathcal O}_{{\mathbb{P}}^ 3}(1))\to H^ 0({\mathcal I}_ C(k+1))\) are of maximal rank for each k. In this paper we give examples of curves (which are in fact unions of skew lines) which satisfy one of these two properties and not the other. The examples for ``maximal rank does not imply minimal generation'' are given using multisecant curves of degree \(>1\).