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The Complete Intersection property for binomial ideals of collections of cells

Rodica Dinu, Francesco Navarra

Abstract

In this paper, we provide a combinatorial characterization of those collections of cells whose inner $2$-minor ideals are complete intersections. More precisely, given a collection of cells $\mathcal C$ and its associated inner $2$-minor ideal $I_{\mathcal C}$, we prove that $I_{\mathcal C}$ is a complete intersection if and only if $\mathcal C$ is a chessboard.

The Complete Intersection property for binomial ideals of collections of cells

Abstract

In this paper, we provide a combinatorial characterization of those collections of cells whose inner -minor ideals are complete intersections. More precisely, given a collection of cells and its associated inner -minor ideal , we prove that is a complete intersection if and only if is a chessboard.
Paper Structure (3 sections, 4 theorems, 12 equations, 3 figures)

This paper contains 3 sections, 4 theorems, 12 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Collections of cells and Commutative Algebra
  3. A combinatorial characterization of the Complete Intersection property

Key Result

Theorem 3.1

Let $\mathcal{C}$ be a collection of cells. Then $I_\mathcal{C}$ is a complete intersection ideal if and only if $\mathcal{C}$ is a chessboard.

Figures (3)

  • Figure 1: A weakly connected collection of cells and a chessboard.
  • Figure 2: Order of vertices in a chessboard.
  • Figure 5: Two cells sharing an edge

Theorems & Definitions (7)

  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof