Normal bundles of hyperelliptic curves and uniform propagation of semistability

Abstract: Let $C$ be a smooth hyperelliptic curve of genus $g\ge2$ over an algebraically closed field of characteristic zero. We prove that the normal bundle of the embedding of $C$ in $\mathbb{P}<sup>n$, $n\ge3$, by the complete linear series of a general line bundle of degree $n+g$ is semistable if and only if $n\ge3g+1$. By openness, general complete embeddings of general curves of genus $g$ also have semistable normal bundle when $n\ge3g+1$. We then show that a general nonspecial curve of genus two and degree $d$ in $\mathbb{P}<sup>n$ has semistable normal bundle except when $(n,d)$ is $(3,5)$, $(4,6)$, $(5,7)$ or $(6,8)$. The degenerations used in genus two extend to every genus $g\ge3$: semistability in all nonspecial degrees and dimensions follows from the cases $0\le d-n-g\le4g-4$. Together with a separate estimate for large $n$, this reduces semistability in each fixed genus to finitely many cases. The proofs combine degenerations to reducible nodal curves, explicit bounds on the degrees of subsheaves, and, in genus two, finitely many exact computations, which are provided as ancillary files.