Singularities of dual hypersurfaces of spacelike hypersurfaces in lightcone and Legendrian duality

The theory of the Legendrian singularity is applied for lightcones that are canonically embedded in the higher-dimensional lightcone and de Sitter space in the Minkowski space-time. The singularities of two classes of hypersurfaces that are dual to space-like hypersurface in the lightcone under Legendrian dualities are analyzed in detail.