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Combinatorial data of a free arrangement and the Terao conjecture

Tran Quoc Cong

Abstract

We present a combinatorial structure of generators of $D(\mathcal{A}).$ This structure permits us to detect the relationship between the combinatorial determined property and the singularity of vector field. Consequently, by using only combinatorial data, we have a basis of the module in free case and that yields a proof for the Terao's conjecture. We also verify the example of Ziegler and give a sufficient condition on combinatorial determined property of generators.

Combinatorial data of a free arrangement and the Terao conjecture

Abstract

We present a combinatorial structure of generators of This structure permits us to detect the relationship between the combinatorial determined property and the singularity of vector field. Consequently, by using only combinatorial data, we have a basis of the module in free case and that yields a proof for the Terao's conjecture. We also verify the example of Ziegler and give a sufficient condition on combinatorial determined property of generators.

Paper Structure

This paper contains 3 sections, 7 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Combinatorial data of module $D(\mathcal{A})$
  3. Combinatorial structure on generators

Key Result

Theorem \oldthetheorem

For any $\theta\in D(\mathcal{A})$, with some polynomials $k_1, \cdots, k_n$, then $\bar{\theta }$ satisfies the following equation: where $\alpha_i= a_{i1}x_1 +\cdots + a_{i\ell }x_\ell, \forall i=\overline{1,n.}$

Theorems & Definitions (9)

  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem: Combinatorial Structure
  • Example \oldthetheorem
  • Theorem \oldthetheorem: Sufficient Condition
  • Remark \oldthetheorem
  • Corollary \oldthetheorem: Terao Theorem