Let PMAX(d,s) denote the maximum arithmetic genus of a locally Cohen-Macaulay curve of degree d in P3 that is not contained in a surface of degree < s. A bound P(d, s) for PMAX(d,s) has been proven by the first author in characteristic zero and then generalized in any characteristic by the third author. In this paper, we construct a large family of primitive multiple lines and we conjecture that the generic element has good cohomological properties. From the conjecture it would follow that P(d,s)=PMAX(d,s) for d = s and for every d≥2s−1. With the aid of Macaulay2 we checked this holds for s≤120 by verifying our conjecture in the corresponding range.
