Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The variety of orthogonal frames

Laura Casabella, Alessio Sammartano

TL;DR

This work studies the variety $V(d,n)$ of orthogonal $n$-frames in a $d$-dimensional quadratic space, together with its defining ideal $I(d,n)$ and the scheme $X(d,n)=\mathrm{Spec}(S/I(d,n))$. By stratifying $V(d,n)$ according to anisotropic and isotropic ranks and analyzing degeneration, smoothness, and dimensions, the authors establish precise thresholds $D_{CI}(n)$, $D_{prime}(n)$, and $D_{UFD}(n)$ that determine when $I(d,n)$ is a complete intersection, prime, or when $R(d,n)$ is a UFD, with $V(d,n)$ normal in the prime regime. They prove generic reducedness of $X(d,n)$ and connect these results to Lovász-Saks-Schrijver ideals, showing that LSS algebras inherit CI, normal, and UFD properties under the same bounds via deformations. The paper also develops a stratification framework, analyzes irreducible components, and outlines open problems on singularities, syzygies, and embedded primes, laying groundwork for further representation-theoretic and geometric investigations. Overall, this work advances the understanding of quadratic frame varieties and their connections to determinantal-like ideals and graph-theoretic constructions.

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 Lovász-Saks-Schrijver ideals.

The variety of orthogonal frames

TL;DR

This work studies the variety of orthogonal -frames in a -dimensional quadratic space, together with its defining ideal and the scheme . By stratifying according to anisotropic and isotropic ranks and analyzing degeneration, smoothness, and dimensions, the authors establish precise thresholds , , and that determine when is a complete intersection, prime, or when is a UFD, with normal in the prime regime. They prove generic reducedness of and connect these results to Lovász-Saks-Schrijver ideals, showing that LSS algebras inherit CI, normal, and UFD properties under the same bounds via deformations. The paper also develops a stratification framework, analyzes irreducible components, and outlines open problems on singularities, syzygies, and embedded primes, laying groundwork for further representation-theoretic and geometric investigations. Overall, this work advances the understanding of quadratic frame varieties and their connections to determinantal-like ideals and graph-theoretic constructions.

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 Lovász-Saks-Schrijver ideals.
Paper Structure (22 sections, 38 theorems, 70 equations, 1 figure)

This paper contains 22 sections, 38 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ground field
  4. Quadratic vector spaces
  5. The ideal, variety, and scheme of orthogonal frames
  6. Stratification of the variety of orthogonal frames
  7. Dimension of strata
  8. Degeneration problem and irreducible components
  9. Smoothness
  10. Proof of the main results
  11. Generically reduced
  12. Complete intersection
  13. Domain
  14. Unique factorization domain
  15. Applications to Lovász-Saks-Schrijver ideals
  16. ...and 7 more sections

Key Result

Theorem 1

The ideal $\mathsf{I}(d,n)$ is a complete intersection if and only if $d \geq D_{\mathrm{CI}}(n)$. In this case, the scheme $\mathsf{X}(d,n)$ is reduced.

Figures (1)

  • Figure 1: The domain $\Delta(d,n)$ for various values of $(d,n)$. Each integer point $(p,q)$ on the upper boundary $\Omega$ is labeled with the dimension $\sigma(p,q)$.

Theorems & Definitions (77)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Corollary 5
  • Conjecture 6: CW
  • Theorem 7
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3: Witt
  • ...and 67 more