ON THE DUALS OF SMOOTH PROJECTIVE COMPLEX HYPERSURFACES

Abstract

We first show that a generic hypersurface $V$ of degree $d\ge 3$ in the projective complex space ${\mathrm{\mathbb{P}}}^n$ of dimension $n\ge 3$ 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 $n,d\ge 3$ has at least a very singular point $q$, in particular a point $q$ of multiplicity $\ge n$.