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Set-theoretic defining equations of the tangential variety of the Segre variety

Luke Oeding

TL;DR

This work proves a set-theoretic version of the Landsberg--Weyman conjecture for the tangential variety of a Segre product by translating the problem to the geometry of the variety of principal minors $Z_n$ of symmetric matrices and introducing the exclusive rank $E$-rank. The author constructs and analyzes cubic and quartic polynomials arising from Schur-module representations, showing that, after pulling back to the space of symmetric matrices, the relevant equations cut out the tangential variety set-theoretically via exclusive minors and their principal minors. The key steps include identifying the pullbacks of LW conjecture polynomials, proving the invariance of exclusive minors under the natural group action, and proving that the image of $E$-rank-one symmetric matrices under the principal minor map equals the tangential variety. This advances understanding of tangential varieties of Segre products and connects their equations to the geometry of principal minors and exclusive rank.

Abstract

We prove a set-theoretic version of the Landsberg--Weyman Conjecture on the defining equations of the tangential variety of a Segre product of projective spaces. We introduce and study the concept of exclusive rank. For the proof of this conjecture we use a connection to the author's previous work \cite{oeding_pm_paper, oeding_thesis} and re-express the tangential variety as the variety of principal minors of symmetric matrices that have exclusive rank no more than one.

Set-theoretic defining equations of the tangential variety of the Segre variety

TL;DR

This work proves a set-theoretic version of the Landsberg--Weyman conjecture for the tangential variety of a Segre product by translating the problem to the geometry of the variety of principal minors of symmetric matrices and introducing the exclusive rank -rank. The author constructs and analyzes cubic and quartic polynomials arising from Schur-module representations, showing that, after pulling back to the space of symmetric matrices, the relevant equations cut out the tangential variety set-theoretically via exclusive minors and their principal minors. The key steps include identifying the pullbacks of LW conjecture polynomials, proving the invariance of exclusive minors under the natural group action, and proving that the image of -rank-one symmetric matrices under the principal minor map equals the tangential variety. This advances understanding of tangential varieties of Segre products and connects their equations to the geometry of principal minors and exclusive rank.

Abstract

We prove a set-theoretic version of the Landsberg--Weyman Conjecture on the defining equations of the tangential variety of a Segre product of projective spaces. We introduce and study the concept of exclusive rank. For the proof of this conjecture we use a connection to the author's previous work \cite{oeding_pm_paper, oeding_thesis} and re-express the tangential variety as the variety of principal minors of symmetric matrices that have exclusive rank no more than one.

Paper Structure

This paper contains 6 sections, 3 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. The variety of principal minors of symmetric matrices
  3. The variety of principal minors of symmetric matrices
  4. The pull-back of polynomials in the Landsberg--Weyman Conjecture symmetric matrices via $Z_{n}$
  5. Background and notation
  6. Constructing polynomials for the Landsberg--Weyman Conjecture

Key Result

Theorem 1.2

Let $Z_{n} \subset \mathbb P (V_{1}^{*} \otimes\dots\otimes V_{n}^{*})$ be the variety of principal minors of symmetric matrices and let $HD$ be the module of quartic polynomials with three $S_{2,2}$'s and all other factors $S_{4,0}$. Then, as sets, $\mathcal{V}(HD) = Z_{n}$.

Theorems & Definitions (5)

  • Conjecture 1.1: Conjecture 7.6. LWtan
  • Theorem 1.2: oeding_pm_paperoeding_thesis
  • Theorem 1.3
  • Remark 1.4
  • Proposition 3.1: Landsberg--Manivel LM04 Proposition 4.1