Hypersurfaces with vanishing hessian via Dual Cayley Trick

TL;DR

This paper introduces a new method to construct hypersurfaces with zero hessian using the Dual Cayley Trick, revealing diverse geometric properties and dual varieties with arbitrary codimension.

Contribution

It provides a general construction for hypersurfaces with vanishing hessian from varieties with hypersurface duals, expanding the known classes of such hypersurfaces.

Findings

Hypersurfaces with vanishing hessian can be constructed from varieties with hypersurface duals.

The dual varieties of these hypersurfaces can have arbitrary codimension.

The geometric properties of these hypersurfaces differ from previously known series.

Abstract

We present a general construction of hypersurfaces with vanishing hessian, starting from any irreducible non-degenerate variety whose dual variety is a hypersurface and based on the so called Dual Cayley Trick. The geometrical properties of these hypersurfaces are different from the known series constructed until now. In particular, their dual varieties can have arbitrary codimension in the image of the associated polar map.