On the duals of smooth projective complex hypersurfaces

TL;DR

This paper investigates the singularities of duals of smooth projective hypersurfaces, establishing the existence of specific singularities and their implications for the dual hypersurface structure.

Contribution

It proves that generic hypersurfaces have hyperplane sections with ordinary double points and that their duals possess highly singular points, advancing understanding of hypersurface duality.

Findings

Existence of hyperplane sections with exactly n ordinary double points.

Dual hypersurfaces have at least one very singular point of multiplicity ≥ n.

Duals of smooth hypersurfaces with n,d ≥ 3 have significant singularities.

Abstract

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 $\geq n$.