Parole chiave
Abstract. Let $P_{\text{max}}(d,s)$ denote the maximum arithmetic genus of a locally Cohen-Macaulay curve of degree $d$ in $\mathbb{P}^3$ that is not contained in a surface of degree $< s$. A bound $P(d, s)$ for $P_{\text{max}}(d,s)$ 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 $P(d,s)= P_{\text{max}}(d,s)$ for $d=s$ and for every $d \geq 2s-1$. With the aid of _Macaulay2_ we checked this holds for $s \leq 120$ by verifying our conjecture in the corresponding range.