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

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