Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian

We introduce various families of irreducible homaloidal hypersurfaces in projective space Pr, for all r⩾3. Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared to the dimension of the ambient projective space. The existence of such a family solves a question that has naturally arisen from the consideration of the classes of homaloidal hypersurfaces known so far. The result relies on a fine analysis of hypersurfaces that are dual to certain scroll surfaces. We also introduce an infinite family of determinantal homaloidal hypersurfaces based on a certain degeneration of a generic Hankel matrix. The latter family fit non-classical versions of de Jonquières transformations. As a natural counterpoint, we broaden up aspects of the theory of Gordan–Noether hypersurfaces with vanishing Hessian determinant, bringing over some more precision into the present knowledge.