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.