On cubic hypersurfaces with vanishing hessian

We prove that for N≤6 an irreducible cubic hypersurface with vanishing hessian in PN is either a cone or a scroll in linear spaces tangent to the dual of the image of the polar map of the hypersurface. We also provide canonical forms and a projective characterization of Special Perazzo Cubic Hypersurfaces, which, a posteriori, exhaust the class of cubic hypersurfaces with vanishing hessian, not cones, for N≤6. Finally we show by pertinent examples the technical difficulties arising for N≥7.

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