Free Multiplicities on the moduli of $X_3$

TL;DR

This paper investigates the conditions under which the module of derivations is free for all multiplicities on the $X_3$ arrangement, employing advanced homological techniques from recent multi-arrangement research.

Contribution

It extends the understanding of freeness in multi-arrangements by applying homological methods to the $X_3$ arrangement with arbitrary multiplicities.

Findings

Identifies criteria for freeness of derivation modules on $X_3$ arrangements.

Utilizes homological techniques to analyze multi-arrangements.

Provides a framework for future studies on arrangement freeness.

Abstract

In this note we study the freeness of the module of derivations on all moduli of the $X_3$ arrangement with multiplicities. We use homological techniques stemming from work of Yuzvinsky, Brandt, and Terao which have recently been developed for multi-arrangements by the first author, Francisco, Mermin, and Schweig.