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.

中文翻译：

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局部Cohen-Macaulay空间曲线的最大类问题

让\（{P _ {\ RM MAX}（d，S）} \）表示程度的局部科恩-麦考曲线最大算术属d在\（{\ mathbb {P} ^ 3} \）是不包含在度< s的表面中。一个绑定P（d，S）为\（{P _ {\ RM MAX}（d，S）} \）已在特性零被证明由第一作者，然后在任何特性由第三作者一概而论。在本文中，我们构造了一个由原始多行组成的大族\（{\ mathcal {C}} \}，并且我们推测\（{\ mathcal {C}} \}的通用元素具有良好的同调性质。从猜想中可以得出\（{P（d，s）= P _ {\ rm MAX}（d，s}} \）对于d = s和每个\（{d \ geq 2s-1} \）。在Macaulay2的帮助下，我们通过验证在相应范围内的猜想，检查了该保全\（{s \ leq 120} \）。