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

Fix integers r≥4 and i≥2. Let C be a non-degenerate, reduced and irreducible complex projective curve in Pr, of degree d, not contained in a hypersurface of degree ≤i. Let pa(C) be the arithmetic genus of C. Continuing previous research, under the assumption d≫max{r,i}, in the present paper we exhibit a Castelnuovo bound G0(r;d,i) for pa(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≡0,3,6 (mod 9). Moreover, when i=2, r≥9, r is divisible by 3, and d≡0 (mod r(r+3)/6), we prove that if G0(r;d,i) is not sharp, then for the maximal value of pa(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.