Homaloidal determinants

A form in a polynomial ring over a field is said to be homaloidal if its polar map is a Cremona map, i.e., if the rational map defined by the partial derivatives of the form has an inverse rational map. The overall object of this work is the search for homaloidal polynomials that are the determinants of sufficiently structured square matrices. We focus particularly on the generic square Hankel matrix and other specializations or degenerations of the fully generic square matrix. In addition to studying the homaloidal nature of these determinants, one establishes several results on the ideal theoretic invariants of the respective Jacobian ideals, such as primary components, multiplicity, reductions and free resolutions.