Singularities of duals of Grassmannians

TL;DR

This paper investigates the singularities of dual varieties of Grassmannians, providing conditions for maximal dimension components and criteria for normality, extending previous work on hyperdeterminants.

Contribution

It offers new criteria to identify maximal components of singular loci in dual Grassmannians and determines their normality, advancing understanding of their geometric properties.

Findings

Identifies conditions for maximal dimension of singular components.

Provides criteria for normality of dual Grassmannians.

Extends methods from hyperdeterminants to Grassmannian duals.

Abstract

Let $X$ be a smooth irreducible nondegenerate projective variety and let $X^*$ denote its dual variety. It is well known that $\sigma_2(X)^*$, the dual of the 2-secant variety of $X$, is a component of the singular locus of $X^*$. The locus of bitangent hyperplanes, i.e. hyperplanes tangent to at least two points of $X$, is a component of the sigular locus of $X^*$. In this paper we provide a sufficient condition for this component to be of maximal dimension and show how it can be used to determine which dual varieties of Grassmannians are normal. That last part may be compared to what has been done for hyperdeterminants by J. Weyman and A. Zelevinski (1996).