Singularities of orthogonal and symplectic determinantal varieties

TL;DR

This paper investigates the singularities and defining equations of orthogonal and symplectic determinantal varieties, which are orbit closures under certain group actions, revealing their normality and singularity properties.

Contribution

It characterizes the singularities and normality of these orbit closures, providing new insights into their algebraic and geometric structure, especially in the symplectic case.

Findings

Symplectic orbit closures are normal with good filtrations.

In characteristic zero, symplectic orbit closures have rational singularities.

Most orthogonal orbit closures share these properties, with specific exceptions.

Abstract

Let either $GL(E)\times SO(F)$ or $GL(E)\times Sp(F)$ act naturally on the space of matrices $E\otimes F$. There are only finitely many orbits, and the orbit closures are orthogonal and symplectic generalizations of determinantal varieties, which can be described similarly using rank conditions. In this paper, we study the singularities of these varieties and describe their defining equations. We prove that in the symplectic case, the orbit closures are normal with good filtrations, and in characteristic $0$ have rational singularities. In the orthogonal case we show that most orbit closures will have the same properties, and determine precisely the exceptions to this.