The Maximum Genus Problem for Locally Cohen-Macaulay Space Curves

Abstract

Let $${P_{\rm MAX}(d, s)}$$ 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 $${P_{\rm MAX}(d, s)}$$ 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 $${P(d, s) = P_{\rm MAX}(d, s)}$$ 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.