Homaloidal determinants

Abstract

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 object of this work is the search for homaloidal polynomials that are the determinants of sufficiently structured matrices. We focus on generic catatalecticants, with special emphasis on the Hankel matrix. An additional focus is on certain degenerations or specializations thereof. In addition to studying the homaloidal nature of these determinants, one establishes several results on the ideal theoretic invariants of the respective gradient ideals, such as primary components, multiplicity, reductions and free resolutions.

Hypersurfaces with vanishing hessian via Dual Cayley Trick

“…The previous result has been discovered for n = 2 in [GoRu], see also [Rus,Example 7.6.11]. Part (ii) has been proved algebraically also in [MoSi,Proposition 4.9]. By passing to a suitable linear section of X n obtained by putting some more variable equal to zero in the matrix X such that f = det(X), Cunha, Ramos and Simis produced explicit irreducible polynomials with vanishing hessian and such that codim(X * , Z X ) is a function of n. These examples can be also described geometrically as the duals of some explicit projections of Y 1 .…”

Degenerations of the generic square matrix, the polar map and the determinantal structure

“…In the present case, the inner corners have indices satisfying the equation Remark 3.17. It may of some interest to note that degenerating the matrix DG(m − 2) all the way to a Hankel matrix, recovers the so-called sub-Hankel matrix thoroughly studied in [2] from the homaloidal point of view and in [13], from the ideal theoretic side. The situation is ever more intriguing since in the sub-Hankel case the determinant is homaloidal.…”

“…As f m−1,m−1 = ∆ m−1,m−1 + ∆ m,m , adding (11) to (12), (13) to (14) and (15) to (16), respectively, outputs three new linear syzygies of the partial derivatives of f . Thus one has a total of (m − 2)…”

“…, m−2 the block ϕ r+1 r comes from the relation (8) (setting j = r − 1) and ϕ m m−1 comes from the relations (9) and (10). Finally, the lower right corner 3 × 3 block of the matrix of linear syzygies comes from the three last relations obtained by adding (11) to (12), (13) to (14) and (15) to (16). This proves the claim about the large matrix above.…”

Symmetry preserving degenerations of the generic symmetric matrix

“…Next, we show that P is the minimal component in a primary decomposition of the gradient ideal J; more precisely, J is a double structure on the variety defined by P , with a unique embedded component of codimension 2(m − 1) supported on a linear space. Algebraically, this is quite a common situation where one can ask whether J is actually a reduction of the prime ideal P (see, e.g., [22,Conjecture 3.16 and Corollary 3.17 (iii)]). The answer is negative and in order to prove this we first show that the image of the (birational) map defined by the cofactors is a hypersurface of degree m − 1.…”