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Brackets and Projective Geometry in Macaulay2

Dalton Bidleman, Timothy Duff, Jack Kendrick, Michael Zeng

TL;DR

The Brackets package for Macaulay2 provides a practical platform for manipulating bracket rings and Grassmann-Cayley algebras to perform invariant-theoretic and incidence-based computations in projective geometry. By implementing $\mathcal{B}_{n,d}$ and $\mathcal{G}_d(e_1,\dots,e_n)$, along with the straightening algorithm and shuffle product, the approach translates geometric conditions into bracket polynomials and vice versa, enabling automatic verification of classical theorems. The paper demonstrates the workflow with concrete examples such as Desargues' theorem and the problem of four-line transversals in $\mathbb{P}^3$, illustrating how parameterized coefficient rings accommodate extra variables and how discriminants determine solution counts. Overall, the work lays a foundation for symbolic invariant-theoretic computations and geometric theorem proving within Macaulay2, with future directions toward efficiency improvements and broader interfaces.

Abstract

We introduce the Brackets package for the computer algebra system Macaulay2, which provides convenient syntax for computations involving the classical invariants of the special linear group. We describe our implementation of bracket rings and Grassmann-Cayley algebras, and illustrate basic functionality such as the straightening algorithm on examples from projective and enumerative geometry.

Brackets and Projective Geometry in Macaulay2

TL;DR

The Brackets package for Macaulay2 provides a practical platform for manipulating bracket rings and Grassmann-Cayley algebras to perform invariant-theoretic and incidence-based computations in projective geometry. By implementing and , along with the straightening algorithm and shuffle product, the approach translates geometric conditions into bracket polynomials and vice versa, enabling automatic verification of classical theorems. The paper demonstrates the workflow with concrete examples such as Desargues' theorem and the problem of four-line transversals in , illustrating how parameterized coefficient rings accommodate extra variables and how discriminants determine solution counts. Overall, the work lays a foundation for symbolic invariant-theoretic computations and geometric theorem proving within Macaulay2, with future directions toward efficiency improvements and broader interfaces.

Abstract

We introduce the Brackets package for the computer algebra system Macaulay2, which provides convenient syntax for computations involving the classical invariants of the special linear group. We describe our implementation of bracket rings and Grassmann-Cayley algebras, and illustrate basic functionality such as the straightening algorithm on examples from projective and enumerative geometry.

Paper Structure

This paper contains 8 sections, 2 theorems, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The Bracket Ring
  4. Grassmann-Cayley Algebra
  5. Data Types and Basic Usage
  6. Further Examples
  7. Desargues' Theorem
  8. Transversals of Four Lines in Space

Key Result

Theorem 4.1

Two triangles $\triangle abc$ and $\triangle def$ of points in $\mathbb{P}^2$ are perspective from a point if and only if they are perspective from a line.

Figures (2)

  • Figure 1: Two perspective triangles $\triangle abc$ and $\triangle def$. The lines connecting corresponding vertices of each triangle all intersect at the point $O$, whereas pairs of corresponding sides all intersect at points on the line $\ell.$
  • Figure 2: Construction of the transversal $\ell$ using the Grassmann-Cayley algebra.

Theorems & Definitions (9)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Example 3.1
  • Example 3.2
  • Theorem 4.1: Desargues
  • proof
  • Theorem 4.2