Convergent normal form for real hypersurfaces at a generic Levi-degeneracy

We construct a complete convergent normal form for a real hypersurface in , , at a generic Levi-degeneracy. This seems to be the first convergent normal form for a Levi-degenerate hypersurface. As an application of the convergence result, we obtain an explicit description of the moduli space of germs of real-analytic hypersurfaces with a generic Levi-degeneracy. As another application, we obtain, in the spirit of the work of Chern and Moser [ ], distinguished curves inside the Levi-degeneracy set that we call