Generalization of the addition and restriction theorems from free arrangements to the class of projective dimension one

TL;DR

This paper extends Terao's addition theorem from free arrangements to arrangements with projective dimension one, providing criteria for algebraic structures and exploring a new class of arrangements with combinatorial properties.

Contribution

It generalizes the addition theorem to a broader class of arrangements and introduces divisionally SPOG arrangements linked to intersection lattice properties.

Findings

Provides a criterion for the algebraic structure of derivation modules.

Introduces divisionally SPOG arrangements with lattice-dependent properties.

Extends the scope of addition and restriction theorems in arrangement theory.

Abstract

We study a generalized version of Terao's famous addition theorem for free arrangements to the category of those with projective dimension one. Namely, we give a criterion to determine the algebraic structure of logarithmic derivation modules of the addition when the deletion and restrictions are free with a mild condition. Also, we introduce a class of divisionally SPOG arrangements whose SPOGness depends only on the intersection lattice like Terao's famous conjecture on combinatoriality of freeness.