Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Computations and Equations for Segre-Grassmann hypersurfaces

Noah S. Daleo, Jonathan D. Hauenstein, Luke Oeding

TL;DR

The paper addresses the problem of finding defining equations for secant varieties of Segre-Grassmann varieties, focusing on hypersurface cases and an infinite family. It combines Numerical Algebraic Geometry (via Bertini) to compute degrees of candidate hypersurfaces with Representation Theory (Schur–Weyl duality and Young symmetrizers) to produce explicit invariants and determinantal equations. A key result is that the hypersurface $\sigma_{5}(\operatorname{Seg}(\mathbb{P}^{2} \times \mathbb{G}(2,5)))$ is defined by a single irreducible degree-6 polynomial arising from a specific Young symmetrizer, and that the entire family $\sigma_{3\ell+2}(\operatorname{Seg}(\mathbb{P}^{2} \times \mathbb{G}(1,4\ell+2)))$ is cut out by $\det\varphi_{T}$ via exterior flattenings. The paper also analyzes irreducibility of determinants of tensor-product flattenings, proving irreducibility for large $s$, and discusses connections to hyperdeterminants and invariant theory, thereby bridging numerical evidence with non-numerical proofs and advancing applications to border rank problems and computational complexity.

Abstract

In 2013, Abo and Wan studied the analogue of Waring's problem for systems of skew-symmetric forms and identified several defective systems. Of particular interest is when a certain secant variety of a Segre-Grassmann variety is expected to fill the natural ambient space, but is actually a hypersurface. Algorithms implemented in Bertini are used to determine the degrees of several of these hypersurfaces, and representation-theoretic descriptions of their equations are given. We answer Problem 6.5 [Abo-Wan2013], and confirm their speculation that each member of an infinite family of hypersurfaces is minimally defined by a (known) determinantal equation. While led by numerical evidence, we provide non-numerical proofs for all of our results.

Computations and Equations for Segre-Grassmann hypersurfaces

TL;DR

The paper addresses the problem of finding defining equations for secant varieties of Segre-Grassmann varieties, focusing on hypersurface cases and an infinite family. It combines Numerical Algebraic Geometry (via Bertini) to compute degrees of candidate hypersurfaces with Representation Theory (Schur–Weyl duality and Young symmetrizers) to produce explicit invariants and determinantal equations. A key result is that the hypersurface is defined by a single irreducible degree-6 polynomial arising from a specific Young symmetrizer, and that the entire family is cut out by via exterior flattenings. The paper also analyzes irreducibility of determinants of tensor-product flattenings, proving irreducibility for large , and discusses connections to hyperdeterminants and invariant theory, thereby bridging numerical evidence with non-numerical proofs and advancing applications to border rank problems and computational complexity.

Abstract

In 2013, Abo and Wan studied the analogue of Waring's problem for systems of skew-symmetric forms and identified several defective systems. Of particular interest is when a certain secant variety of a Segre-Grassmann variety is expected to fill the natural ambient space, but is actually a hypersurface. Algorithms implemented in Bertini are used to determine the degrees of several of these hypersurfaces, and representation-theoretic descriptions of their equations are given. We answer Problem 6.5 [Abo-Wan2013], and confirm their speculation that each member of an infinite family of hypersurfaces is minimally defined by a (known) determinantal equation. While led by numerical evidence, we provide non-numerical proofs for all of our results.

Paper Structure

This paper contains 7 sections, 7 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Symmetry
  4. Computing the degree of a hypersurface with Bertini
  5. Young symmetrizers and explicit polynomial invariants
  6. The Abo-Wan hypersurfaces $\sigma_{3\ell+2}(\operatorname{Seg}(\mathbb{P}^{2} \times \mathbb{G}(1,4\ell+2)))$
  7. Determinants of tensor products of generic matrices

Key Result

Lemma 2.1

Let $X\subset \mathbb{P}^{N}$ be an algebraic variety, and let $[x_{1}],\dots,[x_{k}]$ be general points of $X$. Set $p = \sum_{i=1}^{k}x_{i}$ and suppose that $[p]$ is a general point of $\sigma_{k}(X)$. Then the tangent space of the secant variety is the sum of tangent spaces to the original varie

Figures (2)

  • Figure 1: Summary of procedure for computing $\deg{\mathcal{H}}$
  • Figure 2: An algorithm for evaluating Young symmetrizers

Theorems & Definitions (17)

  • Lemma 2.1: Terracini
  • Proposition 2.2: AboWan
  • proof : Summary of computation.
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Theorem 5.1: AboWan
  • Remark 5.2
  • Theorem 5.3
  • proof
  • ...and 7 more