The maximum genus problem for locally Cohen-Macaulay space curves

TL;DR

This paper investigates the maximum genus of certain space curves in projective 3-space, proposing a conjecture supported by computational evidence that could determine these bounds for a wide range of degrees.

Contribution

The authors construct a family of primitive multiple lines and conjecture their generic elements have desirable properties, advancing understanding of the maximum genus problem for locally Cohen-Macaulay curves.

Findings

Constructed a large family of primitive multiple lines.

Used Macaulay2 to verify the conjecture for s ≤ 100.

Conjecture implies exact maximum genus bounds for specific degrees.

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. With the aid of \emph{Macaulay2} we checked the validity of the conjecture for $s \leq 100$. From the conjecture it would follow that $P(d,s)= P_{\text{MAX}}(d,s)$ for $d=s$ and for every $d \geq 2s-1$.