Hypertetrahedral arrangements

TL;DR

This paper introduces complete hypertetrahedral arrangements in projective space, analyzes their local freeness using graph theory, and provides bounds on the initial degree of syzygies of their Jacobian ideals.

Contribution

It defines complete hypertetrahedral arrangements, characterizes their local freeness, and establishes bounds on syzygy degrees, extending previous geometric and algebraic understanding.

Findings

General arrangements are not locally free.

Local freeness relates to smaller arrangements and graph properties.

Bounds on the initial degree of syzygies are established.

Abstract

In this paper, we introduce the notion of a complete hypertetrahedral arrangement $\mathcal{A}$ in $\mathbb{P}^{n}$. We address two basic problems. First, we describe the local freeness of $\mathcal{A}$ in terms of smaller complete hypertetrahedral arrangements and graph theory properties, specializing the Musta\c{t}\u{a}-Schenck criterion. As an application, we obtain that general complete hypertetrahedral arrangements are not locally free. In the second part of this paper, we bound the initial degree of the first syzygy module of the Jacobian ideal of $\mathcal{A}$.