Symmetry preserving degenerations of the generic symmetric matrix

Abstract

One considers certain degenerations of the generic symmetric matrix over a field k of characteristic zero and the main structures related to the determinant f of the matrix, such as the ideal generated by its partial derivatives, the polar map defined by these derivatives and its image V (f ), the Hessian matrix, the ideal and the map given by the cofactors, and the dual variety of V (f ). A complete description of these structures is obtained.

Hypersurfaces with vanishing hessian via Dual Cayley Trick

“…As far as we know, the fact that this example was the first member of an infinite series of hypersurfaces with vanishing hessian has been noticed by us for the first time several years ago. The very recent paper [CuRaSi2] deals with similar phenomena treated from a purely algebraic point of view. For n ≥ 1 the subspace M n is not a subalgebra of M (n+1)×(n+1) (K) but A # ∈ M for every A ∈ M .…”

“…The ubiquity of the examples suggests that the classification of hypersurfaces with vanishing hessian for N ≥ 5 might be very intricate, perhaps requiring a completely different approach not based (only) on Gordan-Noether Theory, which worked for N ≤ 4, see [GoNo, Fra, GaRe, Rus]. Other interesting series of examples of hypersurfaces with vanishing hessian such that codim(X * , Z X ) is large has been recently constructed in [CuRaSi1,CuRaSi2] (see also Remark 4.3 for a possible geometrical description of this series of examples).…”

“…The known series of examples of hypersurfaces X ⊂ P N with vanishing hessian, which are not cones, have been constructed by Gordan and Noether, Perazzo, Franchetta, Permutti, see [GoNo,Per,Fra,Pm1,Pm2], and later have been revisited and generalised in [CiRuSi,Section 2], see also [CuRaSi1,CuRaSi2].…”

Coordinate sections of generic Hankel matrices

“…For the first two rows of the above table see [7] and [8], respectively. In regard to the question mark at the end of the third row, one can reduce to the case where I m−1 is replaced by the maximal minors of an (m − 1) × (m + 1) degenerate Hankel matrix as stated in Lemma 1.4 (b) and dealt with in the previous subsection.…”

“…The geometric idea behind this terminology is that one is looking at coordinate hyperplane sections of the associated determinantal varieties. This sort of degeneration has been thoroughly dealt with in [7] and [8], and considered before by other authors ( [12], [11]). The main subject is thus a square Hankel matrix H m [r] of order m, with r zeros along the lower stretch of the its last column.…”