When does the Hessian determinant vanish identically?

Abstract

In 1851, Hesse claimed that the Hessian determinant of a homogeneous polynomial f vanishes identically if and only if the projective hypersurface V (f) is a cone. We follow the lines of the 1876 paper of Gordan and Noether to give a proof of Hesse’s claim for curves and surfaces. For higher dimensional hypersurfaces, the claim is wrong in general. We review the construction of polynomials with vanishing Hessian determinant but V (f) not being a cone. For three dimensional hypersurfaces the latter gives, again, the complete answer to the question asked in the title.