Singularities of the dual varieties associated to exterior representations: 1. Dual Grassmannian

TL;DR

This paper classifies the singularities of the dual Grassmannian, identifying cusp and node types related to degeneracies of a Hessian matrix and tangent planes with multiple critical points, reaffirming its normality.

Contribution

It provides a detailed classification of the irreducible components of the singular locus of the dual Grassmannian, expanding understanding of its geometric structure.

Findings

Dual Grassmannian has two main singular components: cusp and node types.

The cusp component relates to Hessian degeneracies.

The node component involves tangent planes with multiple critical points.

Abstract

For a given irreducible projective variety $X$, the closure of the set of all hyperplanes containing tangents to $X$ is the projectively dual variety $X^{\vee}$. We study the singular locus of projectively dual varieties of certain Segre-Pl\"{u}cker embeddings in series of papers. In this work we give a classification of the irreducible components of the singular locus of the dual Grassmannian. Basically, it admits two components: cusp type and node type which are degeneracies of a certain Hessian matrix, and the closure of the set of tangent planes having more than one critical point, respectively. In particular we reproduce the result about the normality of the dual Grassmannian varieties.