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Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties

Oliver Clarke, Giacomo Masiero, Fatemeh Mohammadi

Abstract

We study point-line configurations through the lens of projective geometry and matroid theory. Our focus is on their realisation spaces, where we introduce the concepts of liftable and quasi-liftable configurations, exploring cases in which an $n$-tuple of collinear points can be lifted to a non-degenerate realisation of a point-line configuration. We show that forest configurations are liftable and characterise the realisation space of liftable configurations as the solution set of certain linear systems of equations. Moreover, we study the Zariski closure of the realisation spaces of liftable and quasi-liftable configurations, known as matroid varieties, and establish their irreducibility. Additionally, we compute an irreducible decomposition for their corresponding circuit varieties. Applying these liftability properties, we present a procedure generate some of the defining equations of the associated matroid varieties. As corollaries, we provide a geometric representation for the defining equations of two specific examples: the quadrilateral set and the $3\times4$ grid. While the polynomials for the latter were previously computed using specialised algorithms tailored for this configuration, the geometric interpretation of these generators was missing. We compute a minimal generating set for the corresponding ideals.

Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties

Abstract

We study point-line configurations through the lens of projective geometry and matroid theory. Our focus is on their realisation spaces, where we introduce the concepts of liftable and quasi-liftable configurations, exploring cases in which an -tuple of collinear points can be lifted to a non-degenerate realisation of a point-line configuration. We show that forest configurations are liftable and characterise the realisation space of liftable configurations as the solution set of certain linear systems of equations. Moreover, we study the Zariski closure of the realisation spaces of liftable and quasi-liftable configurations, known as matroid varieties, and establish their irreducibility. Additionally, we compute an irreducible decomposition for their corresponding circuit varieties. Applying these liftability properties, we present a procedure generate some of the defining equations of the associated matroid varieties. As corollaries, we provide a geometric representation for the defining equations of two specific examples: the quadrilateral set and the grid. While the polynomials for the latter were previously computed using specialised algorithms tailored for this configuration, the geometric interpretation of these generators was missing. We compute a minimal generating set for the corresponding ideals.
Paper Structure (10 sections, 25 theorems, 57 equations, 6 figures, 2 tables)

This paper contains 10 sections, 25 theorems, 57 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matroids and matroid varieties
  4. Point-line configurations
  5. (Quasi-)liftable configurations
  6. Realisation space of liftable configurations
  7. Irreducible decomposition of varieties of (quasi-)liftable configurations
  8. The ideal of quasi-liftable configurations
  9. The quadrilateral set
  10. The $3 \times 4$ grid matroid

Key Result

Theorem 1.5

Let $M$ be a rank-$3$ matroid whose associated point-line configuration $\mathop{\mathrm{\mathcal{C}}}\nolimits_M$ has no triplets of concurrent lines. Then,

Figures (6)

  • Figure 1: (a) 3 concurrent lines; (b) Quadrilateral set $\mathop{\mathrm{\mathcal{L}}}\nolimits_{QS}$; (c) $3\times4$ grid $\mathop{\mathrm{\mathcal{L}}}\nolimits_{G^3_4}$
  • Figure 2: From left to right the figure shows a forest planar configuration with ten points and how it is realised starting from ten collinear points, following the proof of Lemma \ref{['forest lifting']}.
  • Figure 3: In the notation of Example \ref{['example:non-maximal']}, from left to right there are the linear point-line configurations of the matroids $M$, $M_1$, $M_2$, and $M_3$.
  • Figure 4: Examples of liftable and quasi-liftable plane arrangements.
  • Figure 10: From left to right, this figure justify the existence of at least a choice of 6 completing $1, \dots, 5$ to a projective image of a quadrilateral set.
  • ...and 1 more figures

Theorems & Definitions (80)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Theorem 1.5
  • Theorem : Theorems \ref{['Generators']} and \ref{['generators grid']}
  • Definition 2.2: Matroid
  • Remark 2.3
  • Definition 2.4: Realisation space of a matroid
  • Remark 2.5
  • Definition 2.6: Matroid variety
  • ...and 70 more