On the duals of smooth projective complex hypersurfaces

We show first that a generic hypersurface V of degree d ≥ 3 in the complex projective space P^n of dimension n ≥ 3 has at least one hyperplane section V ∩H containing exactly n ordinary double points, alias A_1 singularities, in general position, and no other singularities. Equivalently, the dual hypersurface V^∨ has at least one normal crossing singularity of multiplicity n. Using this result, we show that the dual of any smooth hypersurface with n, d ≥ 3 has at least a very singular point q, in particular a point q of multiplicity ≥ n.