Projective curves of degree=codimension+2

Abstract

In this article we study nondegenerate projective curves \({X \subset \mathbb{P}^{d-1}}\) of degree d which are not arithmetically Cohen-Macaulay. Note that \({X = \pi_{P} (\widetilde{X})}\) for a rational normal curve \({\widetilde{X} \subset \mathbb{P}^d}\) and a point \({P \in \mathbb{P}^d \setminus \widetilde{X}^2}\) . Our main result is about the relation between the geometric properties of X and the position of P with respect to \({\widetilde{X}}\) . We show that the graded Betti numbers of X are uniquely determined by the rank \({\hbox{rk}_{\widetilde{X}} P}\) of P with respect to \({\widetilde{X}}\) . In particular, X satisfies property N 2,p if and only if \({p \leq \quad{rk}_{\widetilde{X}} P -3}\) . Therefore property N 2,p of X is controlled by \({\quad{rk}_{\widetilde{X}} P}\) and conversely \({\quad{rk}_{\widetilde{X}} P}\) can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves \({X \subset \mathbb{P}^{d-1}}\) of degree d which are not arithmetically Cohen–Macaulay, there are exactly \({\lfloor \frac{d-2}{2} \rfloor}\) distinct Betti tables.