Projective curves of degree=codimension+2. II. (Q2796979)

Let \(\mathcal{C} \subset \mathbb{P}^r\) denote a nondegenerate projective integral curve. Then \(\deg \mathcal{C} \geq r\). In case of equality \(\mathcal{C}\) is a rational normal curve and it is arithmetical Cohen-Macaulay with known Betti-numbers. In the case of almost minimal degree (i.e., \(\deg \mathcal{C} = r+1\)) and \(\mathcal{C}\) not linearly normal it follows that it is the isomorphic projection of a rational normal curve \(\tilde{\mathcal{C}} \subset \mathbb{P}^{r+1}\) from a point \(P \in \mathbb{P}^{r+1} \setminus \tilde{\mathcal{C}}\). In this situation, there is a large variation of the Betti table. The results of the present paper are a continuation of those of the second author (see [Math. Z. 256, No. 3, 685--697 (2007; Zbl 1139.14027)]), where he introduced \(\mathrm {rk}_{\tilde{\mathcal{C}}} (P) = \min\{k | P \in \mathcal{C}^k\}\), where \(\mathcal{C}^k\) denotes the \(k\)-the join of \(\mathcal{C}\) with itself. Let \(\ell = \mathrm {rk}_{\tilde{\mathcal{C}}} (P)\), then there is a complete description of the Betti numbers \(\beta_{i,2}\) for \(i \leq r-\ell\) and \(i \geq r-3\) and an estimation from below and above for \(r-\ell +1 \leq i \leq r-4\). A conjecture about the precise value of \(\beta_{i,2}\) in the range of \(r-\ell +1 \leq i \leq r-4\) is discussed and proved for \(r \geq 7\) and \(\ell \in \{3,4\}\) and \(7 \leq r \leq 13\). The last result follows by the second author's results [loc. cit.] and computational examples.