Symmetry preserving degenerations of the generic symmetric matrix

TL;DR

This paper studies specific degenerations of the generic symmetric matrix over a field of characteristic zero, focusing on the determinant and related algebraic structures such as derivatives, polar map, Hessian, cofactors, and dual varieties.

Contribution

It introduces new classes of degenerations of symmetric matrices and analyzes their geometric and algebraic properties related to the determinant and associated structures.

Findings

Characterization of degenerations preserving symmetry

Analysis of the polar map and its image for these degenerations

Insights into the dual varieties associated with the determinant hypersurface

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 $V(f)$.