Deletion-Restriction for Logarithmic Forms on Multiarrangements

TL;DR

This paper studies how logarithmic differential forms on hyperplane arrangements behave under deletion and restriction, revealing new algebraic properties and implications for freeness and conjectures.

Contribution

It extends the understanding of logarithmic forms under deletion-restriction, introduces the dual strongly plus-one generated property, and proves related conjectures.

Findings

Restriction of logarithmic forms may not be surjective, measured via algebraic invariants.

Arrangements obtained by adding a hyperplane to a free arrangement are dual strongly plus-one generated.

Provides a new proof for when adding a hyperplane preserves freeness and addresses Ziegler's conjectures.

Abstract

We consider the behaviour of logarithmic differential forms on arrangements and multiarrangements of hyperplanes under the operations of deletion and restriction, extending early work of G\"unter Ziegler. The restriction of logarithmic forms to a hyperplane may or may not be surjective, and we measure the failure of surjectivity in terms of commutative algebra of logarithmic forms and derivations. We find that the dual notion of restriction of logarithmic vector fields behaves similarly but inequivalently. A main result is that, if an arrangement is free, then any arrangement obtained by adding a hyperplane has the "dual strongly plus-one generated" property. One application is another proof of a main result of a paper by the first author characterizing when adding a hyperplane to a free arrangement remains free. A further application is to resolve two conjectures due to Ziegler, which we defer to a sequel.