Stability of normal bundles of Brill–Noether curves

We prove that the normal bundle of a general Brill–Noether curve of genus \(g \ge 1\) $g\ge 1$ and degree d in \(\mathbb {P}^r\) ${\mathbb{P}}^r$ is semistable if \(g=1\) $g=1$ or or d is larger than an explicit function of g and r. We further prove that the normal bundle is in fact stable if \(g\ge 2\) $g\ge 2$ and either g or d satisfy slightly stronger bounds. In particular, for each r, there are at most finitely many (d, g) with \(g \ge 1\) $g\ge 1$ (respectively, \(g \ge 2\) $g\ge 2$ ) for which the normal bundle of the general Brill–Noether curve in \(\mathbb {P}^r\) ${\mathbb{P}}^r$ is not semistable (respectively, stable).