Rose-Terao-Yuzvinsky theorem for reduced forms

TL;DR

This paper offers a new proof of the maximum homological dimension of the gradient ideal for generic hyperplane arrangements and extends the result to products of general forms and specific non-generic cases.

Contribution

It provides a novel proof of the Rose-Terao-Yuzvinsky theorem and generalizes the result to products of forms of arbitrary degrees and certain non-generic cases.

Findings

New proof of the maximum homological dimension for generic arrangements

Extension of the theorem to products of forms of arbitrary degrees

Analysis of specific non-generic form cases

Abstract

Yuzvinsky and Rose-Terao have shown that the homological dimension of the gradient ideal of the defining polynomial of a generic hyperplane arrangement is maximum possible. In this work one provides yet another proof of this result, which in addition is totally different from the one given by Burity-Simis-Tohaneanu. Another main drive of the paper concerns a version of the above result in the case of a product of general forms of arbitrary degrees (in particular, transverse ones). Finally, some relevant cases of non general forms are also contemplated.