On Ziegler's conjectures for logarithmic derivations of arrangements

TL;DR

This paper revisits Ziegler's conjectures on hyperplane arrangements, proving one about generic cuts, disproving another on minimal degree generators, and providing new insights using recent arrangement theory developments.

Contribution

It confirms one of Ziegler's conjectures and refutes another, advancing understanding of logarithmic derivations in hyperplane arrangements.

Findings

Proved Ziegler's conjecture on generic cuts of free arrangements.

Disproved Ziegler's conjecture on minimal degree generators.

Provided positive results to related problems using recent arrangement theory.

Abstract

In his paper and thesis in 1989, Ziegler posed several conjectures regarding commutative algebra related to hyperplane arrangements. In this article, we revisit two of them. One is on generic cuts of free arrangements, and the other has to do with minimal degree generators for the logarithmic differential forms. We prove the first one, and disprove the second one. We also give some positive answers to related problems he posed, using recent developments in arrangement theory.