Interpolation for Brill-Noether space curves (Q1744812)

The purpose of the note under review is to answer the following question: Fix a general collection of \(n\) points in \(\mathbb{P}^3\) over \(\mathbb{C}\). When does there exist a curve \(C\hookrightarrow \mathbb{P}^3\) of degree \(d\) and genus \(g\) and general moduli passing through those points? The author proved that there exist such curves through up to \(2d\) general points with exactly two exceptions: \[ (d,g)\in \{(5,2),(6,4)\}. \] These cases were already noticed by \textit{A. V. Atanasov} [Interpolation and vector bundles on curves. Harvard University (PhD Thesis) (2015)] and \textit{J. Stevens} [Indag. Math. 51, No. 4, 485--494 (1989; Zbl 0724.14017)] respectively. From Larson Theorem (Theorem 1.4, [\textit{E. Larson}, ``The generality of a section of a curve'', Preprint, \url{arXiv:1605.06185}) we may deduce the existence of such curves unless \[ (d,g)\in\{(4,1),(5,2),(6,2),(6,4),(7,5),(8,6)\}. \] Atanasov in [loc. cit.] solved cases \((4,1)\) and \((6,2)\), and in this note the author proves the existence of curves in the two remaining cases \((7,5)\) and \((8,6)\).