On the genus of projective curves not contained in hypersurfaces of given degree, II

Fix integers \(r\ge 4\) $r\ge 4$ and \(i\ge 2\) $i\ge 2$ . Let C be a non-degenerate, reduced and irreducible complex projective curve in \(\mathbb P^r\) ${\mathbb{P}}^r$ , of degree d, not contained in a hypersurface of degree \(\le i\) $\le i$ . Let \(p_a(C)\) $p_a(C)$ be the arithmetic genus of C. Continuing previous research, under the assumption \(d\gg \max \{r,i\}\) $d\gg max\{r,i\}$ , in the present paper we exhibit a Castelnuovo bound \(G_0(r;d,i)\) $G_0(r;d,i)$ for \(p_a(C)\) $p_a(C)$ . In general, we do not know whether this bound is sharp. However, we are able to prove it is sharp when \(i=2\) $i=2$ , \(r=6\) $r=6$ and \(d\equiv 0,3,6\) $d\equiv 0,3,6$ (mod 9). Moreover, when \(i=2\) $i=2$ , \(r\ge 9\) $r\ge 9$ , r is divisible by 3, and \(d\equiv 0\) $d\equiv 0$ (mod \(r(r+3)/6\) $r(r+3)/6$ ), we prove that if \(G_0(r;d,i)\) $G_0(r;d,i)$ is not sharp, then for the maximal value of \(p_a(C)\) $p_a(C)$ there are only three possibilities. The case in which \(i=2\) $i=2$ and r is not divisible by 3 has already been examined in the literature. We give some information on the extremal curves.