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From Pascal's Theorem to the geometry of Ziegler's line arrangements

Alexandru Dimca, Gabriel Sticlaru

Abstract

Günter Ziegler has shown in 1989 that some homological invariants associated with the free resolutions of Jacobian ideals of line arrangements are not determined by combinatorics. His classical example involves hexagons inscribed in conics. Independently, Sergey Yuzvinsky has arrived in 1993 at the same type of line arrangements in order to show that formality is not determined by the combinatorics. In this note we look into the geometry of such line arrangements, and find out an unexpected relation to the classical Pascal's Theorem. Our results give information on the minimal degree of a Jacobian syzygy and on the formality of such hexagonal line arrangements in general, without an explicit choice for the six vertices of the hexagon.

From Pascal's Theorem to the geometry of Ziegler's line arrangements

Abstract

Günter Ziegler has shown in 1989 that some homological invariants associated with the free resolutions of Jacobian ideals of line arrangements are not determined by combinatorics. His classical example involves hexagons inscribed in conics. Independently, Sergey Yuzvinsky has arrived in 1993 at the same type of line arrangements in order to show that formality is not determined by the combinatorics. In this note we look into the geometry of such line arrangements, and find out an unexpected relation to the classical Pascal's Theorem. Our results give information on the minimal degree of a Jacobian syzygy and on the formality of such hexagonal line arrangements in general, without an explicit choice for the six vertices of the hexagon.
Paper Structure (7 sections, 6 theorems, 87 equations, 3 figures)

This paper contains 7 sections, 6 theorems, 87 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of the main results
  3. Proof of Theorem \ref{['PascalThm']}
  4. Proof of Corollary \ref{['PascalCor']}
  5. Proof of Theorem \ref{['thm2']}
  6. The octic $h_{{\mathcal{A}}_Z}$ and a Ziegler pair of degree 10
  7. The octic $h_{{\mathcal{A}}_D}$

Key Result

Theorem 1.4

Let $C:f=0$ be a reduced degree $d$ curve in $\mathbb{P}^2$ with $r= {\rm mdr }(f) <d-1$. Then for $k > 2d-5-r$ and for $k \leq 2d-5-r$. In particular for $k>8$ and

Figures (3)

  • Figure 1: Hexagon in a smooth conic
  • Figure 2: Ziegler's arrangement ${\mathcal{A}}_ Z$
  • Figure 3: Line arrangement ${\mathcal{A}}_D$

Theorems & Definitions (12)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.6
  • ...and 2 more