The Maximum Genus Problem for Locally Cohen-Macaulay Space Curves

Let $P_{\textsc{max}}(d,s)$ denote the maximum arithmetic genus of a locally Cohen-Macaulay curve of degree $d$ in $\PP^3$ that is not contained in a surface of degree $<s$. A bound $P(d, s)$ for $P_{\textsc{max}}(d,s)$ 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 $\mathcal{C}$ of primitive multiple lines and we conjecture that the generic element of $\mathcal{C}$ has good cohomological properties. From the conjecture it would follow that $P(d,s)= P_{\textsc{max}}(d,s)$ for $d=s$ and for every $d \geq 2s-1$. With the aid of \emph{Macaulay2} we checked this holds for $s \leq 120$ by verifying our conjecture in the corresponding range.