Multiple lines of maximum genus in \(\mathbb{P}^3\) (Q6537107)

The study of invariants of space curves has a long history in algebraic geometry. \textit{L. Gruson} and \textit{C. Peskine} gave sharp upper bounds on the genus of a smooth curve \(C \subset \mathbb P^3\) of degree \(d\) not lying on a surface of degree \(< s\) [Lect. Notes Math. 687, 31--59 (1978; Zbl 0412.14011)], completing work initiated by \textit{G. Halphen} more than 100 years earlier [C. R. 70, 380 (1870; JFM 02.0559.01)]. \textit{V. Beorchia} et al. investigated sharp upper bounds for the \textit{arithmetic} genus of locally Cohen-Macaulay curves of degree \(d\) not lying on a surface of degree \(< s\) [Milan J. Math. 86, 137--155 (2018; Zbl 1405.14009)]. They conjecture that the upper bounds \(B(d,s)\) established in [\textit{V. Beorchia}, Math. Nachr. 184, 59--71 (1997; Zbl 0884.14012)] and \textit{E. Schlesinger} [Math. Nachr. 194, 197--203 (1998; Zbl 0929.14019)], namely\N\[\NB(d,s) = \left\{\begin{array}{cc} (s-1)d+1-\binom{s+2}{3} & s \leq d \leq 2s \\\N\binom{d-s}{2}-\binom{s-1}{3} & 2s+1 \leq d \end{array}\right.\N\]\Nare sharp. They give evidence of sharpness and provide a framework to prove it.\N\NThe author proposes a program to classify sharp examples of the genus bound when \(d=s\). This case is of special importance to the conjecture, because if the bound \(B(s-1,s-1)\) is sharp, then so is \(B(d,s)\) for \(d \geq 2s-1\). Sharp examples for the bound \(B(d,d)\) necessarily consist of curves either supported on one line or two disjoint lines. Looking at conditions on the curves supported on the two lines, the author defines a \(C_{d,l}\) to be a degree \(d\) multiplicity structure \(C\) supported on a line \(L\) with \(g(C) = -(d-1)-\binom{d}{3} - l\binom{d}{2}\) and \(H^0 (\mathcal I_C (l+d-1)) = H^0 (\mathcal I_L^d (l+d-1))\). The point of this definition is that if \(C\) is a \(C_{k,d-k}\) and \(D\) is a \(C_{d-k,k}\) for \(0 < k < d\), then \(C \cup D\) is a curve exhibiting sharpness of the genus bound \(B(d,d)\). The first main result uses the classifications of \textit{S. Nollet} [Ann. Sci. Éc. Norm. Supér 30, 367--384 (1997; Zbl 0892.14004] and \textit{S. Nollet} and \textit{E. Schlesinger} [Compos. Math. 139, 169--196 (2003; Zbl 1053.14035)] to determine the curves \(C_{d,l}\) for \(d \leq 4\), showing that the corresponding families are irreducible. The author determines the sharp examples for the genus bounds \(B(4,4)\) and \(B(5,5)\) and concludes that the corresponding families are not irreducible.