Generalized Splines and Graphic Arrangements

TL;DR

This paper introduces a new chain complex for generalized splines on graphs, providing homological tools to analyze the algebraic properties of graphic arrangements, including criteria for freeness and projective dimension bounds.

Contribution

It develops a novel chain complex framework for generalized splines on graphs and applies it to derive homological criteria for properties of graphic arrangements.

Findings

Homology bounds the projective dimension of spline modules.

Freeness of graphic arrangements characterized by splitting as braid arrangements.

Provides criteria for when a graphic arrangement admits a free constant multiplicity.

Abstract

We define a chain complex for generalized splines on graphs, analogous to that introduced by Billera and refined by Schenck-Stillman for splines on polyhedral complexes. The hyperhomology of this chain complex yields bounds on the projective dimension of the ring of generalized splines. We apply this construction to the module of derivations of a graphic multi-arrangement, yielding homological criteria for bounding its projective dimension and determining freeness. As an application, we show that a graphic arrangement admits a free constant multiplicity iff it splits as a product of braid arrangements.