Stability of normal bundles of Brill--Noether curves in $\mathbb{P}^4$

Abstract

We prove that a general Brill--Noether curve $C$ of genus $g \geq 2$ and degree $d$ in $\mathbb{P}^4$ has stable normal bundle $N_C$ if and only if $$(g, d) \notin \{(2,6), (3,7), (5,8), (6,9), (7,10)\}.$$ Moreover, $N_C$ is strictly semistable if $(g, d) \in \{(3, 7), (5, 8)\}$, and is unstable if $(g, d) \in \{(2, 6), (6, 9), (7, 10)\}$. Our results are valid in any characteristic. Along the way, we also generalize previous results of Larson--Vogt on interpolation for $N_C(-1)$, from characteristic zero to arbitrary characteristic.