Hipersuperfícies com Hessiano nulo

Abstract

Hesse said in one of his articles that a hypersurface in the projective space Pn that has null hessian polynomial is a cone. Later, Gordam and Noether prove that the statement of Hesse is valid only for n 3, presenting counter-examples for n 4. Initially we tried to solve the problem in a direct and elementary form, been well succeeding only in the case of P1, so we set out to study the dual of variety and polar map associated to the hypersurface X = Z(F) Pn. Having mind that X IF , where IF is the polar map image, and that X is a cone if and only if, X is degenerate. Which brings us to display a series of technical results in order to conclude that IF is a linear variety, speci cally a line if n = 2 and a plane or line if n = 3. Thus we prove for a given hypersurface X = Z(F) Pn. If n 3, then X is a Cone () det [Hess (F)] = 0.