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The Segre Determinant

Elizabeth Pratt

TL;DR

The paper introduces the Segre determinant $\mathrm{Seg}_{k,\ell}$, a bi-homogeneous invariant that encodes when $k\ell$ points in $\mathbb{P}(V)\times\mathbb{P}(W)$ lie on a hyperplane section of the Segre embedding. It develops its invariant-theoretic computation via bracket algebras, establishes its role as the Chow-Lam form of torus orbits in Grassmannians, and connects to Chow quotients and algebraic-vision applications. A central result shows that the Chow-Lam form of a torus orbit in $Gr(k,n)$ equals $\mathrm{Seg}_{k,\ell}(A,B)$ in primal coordinates, tying torus actions to Segre invariants. The paper further defines the Segre coefficient variety, proves a maximal-span property for the coefficients, and demonstrates both a concrete GIT correspondence for $k=2$ and non-uniqueness phenomena for larger $k$, highlighting the Segre determinant’s constructive and geometric utility.

Abstract

The Segre determinant is a polynomial which encodes the condition for points to lie on a bilinear hypersurface in the product of projective spaces. We study Segre determinants and compute them in various coordinate systems. We show that the Segre determinant represents the Chow-Lam form of a generic torus orbit in the Grassmannian. These Chow-Lam forms were introduced as a generalization of Chow forms for projective varieties, and enjoy many similar properties. We also present applications to algebraic vision and to Chow quotients of Grassmannians.

The Segre Determinant

TL;DR

The paper introduces the Segre determinant , a bi-homogeneous invariant that encodes when points in lie on a hyperplane section of the Segre embedding. It develops its invariant-theoretic computation via bracket algebras, establishes its role as the Chow-Lam form of torus orbits in Grassmannians, and connects to Chow quotients and algebraic-vision applications. A central result shows that the Chow-Lam form of a torus orbit in equals in primal coordinates, tying torus actions to Segre invariants. The paper further defines the Segre coefficient variety, proves a maximal-span property for the coefficients, and demonstrates both a concrete GIT correspondence for and non-uniqueness phenomena for larger , highlighting the Segre determinant’s constructive and geometric utility.

Abstract

The Segre determinant is a polynomial which encodes the condition for points to lie on a bilinear hypersurface in the product of projective spaces. We study Segre determinants and compute them in various coordinate systems. We show that the Segre determinant represents the Chow-Lam form of a generic torus orbit in the Grassmannian. These Chow-Lam forms were introduced as a generalization of Chow forms for projective varieties, and enjoy many similar properties. We also present applications to algebraic vision and to Chow quotients of Grassmannians.
Paper Structure (7 sections, 11 theorems, 30 equations, 2 figures)

This paper contains 7 sections, 11 theorems, 30 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Properties and Computation
  4. An Application to Algebraic Vision
  5. The Chow-Lam Form
  6. Torus Orbits in the Grassmannian
  7. The Segre Coefficient Variety

Key Result

Theorem 1.1

Suppose $k, l \geq 2$ and let $n = k\ell.$ Fix a point $A$ in ${\rm Gr}(k,n)$ with non-zero Plücker coordinates. Then the Chow-Lam form of the torus orbit of $A$ in primal Plücker coordinates $B$ on ${\rm Gr}(n-l, n)$ equals the Segre determinant ${\rm Seg}_{k,\ell}(A,B).$

Figures (2)

  • Figure 1: Definition of the Chow-Lam locus
  • Figure 2: Geometry in $\mathbb P^3$ (left) and $\text{Gr}(2,4)$ (right)

Theorems & Definitions (21)

  • Theorem 1.1: Segre Determinant
  • Theorem 2.1: Theorem 3.2.1 of AIT
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Proposition \ref{['prop:brackets']}
  • Remark 2.4
  • Remark 3.1
  • Theorem 3.2
  • proof
  • ...and 11 more