Trivariate Splines on Fans of Hyperplane Arrangements and Koszul Homology

Abstract

We study the space of splines $\mathcal{S}^{\mathbf{r}}(Σ^\mathscr{A})$ where ${\mathbf{r}}$ denotes a smoothness distribution and $Σ^\mathscr{A}$ is the fan of a central hyperplane arrangement $\mathscr{A}$ in $\mathbb{R}^3$. This is the first step in the analysis of splines on three-dimensional cross-cut partitions, which naturally generalize planar cross-cut partitions. We show that the Hilbert function of $\mathcal{S}^{\mathbf{r}}(Σ^\mathscr{A})$ is bounded by an expression that involves the dimensions of specific Koszul homology modules constructed from the defining equations of the hyperplane arrangement $\mathscr{A}$ and the smoothness distribution function. By exploiting this connection with Koszul homology, we are able to: 1) compute the dimension of the spline space in high degrees, 2) compute all values of the dimension of the spline space if $\mathscr{A}$ is generic with five or fewer hyperplanes, and 3) compute the Hilbert function of the spline space if $\mathscr{A}$ is a generic arrangement with sufficiently many hyperplanes and ${\mathbf{r}}$ is a constant distribution. As an application of our methods, we compute $\dim \mathcal{S}^0_d(Σ^\mathscr{A})$ and $\dim \mathcal{S}^1_d(Σ^\mathscr{A})$ for all values of $d$ when $\mathscr{A}$ is a generic arrangement.