On the duals of smooth projective complex hypersurfaces

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