The variety of orthogonal frames

TL;DR

This paper studies the algebraic structure of the variety of orthogonal frames in quadratic spaces, classifying its components and properties, with applications to Lovász-Saks-Schrijver ideals.

Contribution

It classifies irreducible components, determines when the ideal is prime or a complete intersection, and analyzes normality and factoriality of the variety V(d,n).

Findings

Classified irreducible components of V(d,n)

Provided criteria for ideal I(d,n) to be prime or complete intersection

Established conditions for V(d,n) to be normal and factorial

Abstract

An orthogonal n-frame is an ordered set of n pairwise orthogonal vectors. The set of all orthogonal n-frames in a d-dimensional quadratic vector space is an algebraic variety V(d,n). In this paper, we investigate the variety V(d,n) as well as the quadratic ideal I(d,n) generated by the orthogonality relations, which cuts out V(d,n). We classify the irreducible components of V(d,n), give criteria for the ideal I(d,n) to be prime or a complete intersection, and for the variety V(d,n) to be normal. We also give near-equivalent conditions for V(d,n) to be factorial. Applications are given to the theory of Lov\'asz-Saks-Schrijver ideals.