Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian

Abstract

We prove the existence of various families of irreducible homaloidal hypersurfaces in projective space ${\mathbb{P}}^r$, for all $r\ge 3$. 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 dual hypersurfaces to certain scroll surfaces. We also introduce an infinite family of determinantal homaloidal hypersurfaces based on a certain degeneration of a generic Hankel matrix. These examples 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 to the present knowledge.

56 pages. v2: Some material added in section 1; minor changes. v3: typos corrected in Propositions 1.1 and 1.7

References in corpus (1)

• On the degree of Polar Transformations -- An approach through Logarithmic Foliations

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