Stability of Normal Bundles of Space Curves

In this paper, we prove that the normal bundle of a general Brill-Noether space curve of degree $d$ and genus $g geq 2$ is stable if and only if $(d,g) otin { (5,2), (6,4) }$. When $gleq1$ and the characteristic of the ground field is zero, it is classical that the normal bundle is strictly semistable. We show that this fails in characteristic $2$ for all rational curves of even degree.