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

Abstract

We prove that a general Brill--Noether curve $C$ of genus $g\ge 2$ 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.