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Algebraic properties of binomial edge ideals of Levi graphs associated with curve arrangements

Rupam Karmakar, Rajib Sarkar, Aditya Subramaniam

Abstract

In this article, we study algebraic properties of binomial edge ideals of Levi graphs associated with certain plane curve arrangements. Using combinatorial properties of Levi graphs, we discuss the Cohen-Macaulayness of binomial edge ideals of Levi graphs associated to some curve arrangements in the complex projective plane, like the $d$-arrangement of curves and the conic-line arrangements. We also discuss the existence of certain induced cycles in the Levi graphs of these arrangements and obtain lower bounds for the regularity of powers of the corresponding binomial edge ideals.

Algebraic properties of binomial edge ideals of Levi graphs associated with curve arrangements

Abstract

In this article, we study algebraic properties of binomial edge ideals of Levi graphs associated with certain plane curve arrangements. Using combinatorial properties of Levi graphs, we discuss the Cohen-Macaulayness of binomial edge ideals of Levi graphs associated to some curve arrangements in the complex projective plane, like the -arrangement of curves and the conic-line arrangements. We also discuss the existence of certain induced cycles in the Levi graphs of these arrangements and obtain lower bounds for the regularity of powers of the corresponding binomial edge ideals.
Paper Structure (11 sections, 14 theorems, 24 equations, 3 figures)

This paper contains 11 sections, 14 theorems, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgment
  3. Preliminaries
  4. Plane Curve arrangements and Levi graphs
  5. Some basics from graph theory
  6. Regularity and projective dimension
  7. Cohen-Macaulay binomial edge ideals
  8. Binomial edge ideals associated with Hirzebruch quasi-pencils
  9. Binomial edge ideals associated with curve arrangements
  10. Certain properties of Levi graphs associated with curve arrangements
  11. Bound on Castelnuovo–Mumford regularity

Key Result

Theorem 4.1

(HHHKR) Let $G$ be a simple graph with the vertex set $V(G)$. Then In particular, if $G$ is connected, then $\dim(S/J_G)\geq |V(G)|+1$.

Figures (3)

  • Figure 1: Levi Graph $G_k$ associated to the Hirzebruch quasi-pencil $\mathcal{L}$
  • Figure 2: Cycle of length $6$
  • Figure 3: Cycle of length $6$

Theorems & Definitions (35)

  • Conjecture 1.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Theorem 4.1
  • Definition 4.2
  • Example 4.3
  • Theorem 4.4
  • Theorem 4.5
  • ...and 25 more