Stability of normal bundles of canonical curves

Normal bundles of curves embedded in projective space have been studied extensively only in a few cases: among them are rational space curves, elliptic curves of high degree and low genus canonical curves. In this talk we focus on the latter case and the question of stability. As is well known, for canonical curves of genus up to 6, the normal bundle decomposes and hence is not a stable vector bundle. On the other hand, the general canonical curve of genus 7 was proven to have stable normal bundle and conjecturally this also holds for general canonical curves of higher genus. Using Mukai's Grassmannian construction, we prove the conjecture in genus 8 and make some way towards the genus 9 case. We also discuss possible characterizations of the locus (in M_g) of curves having poly-, semi- or unstable normal bundle.