Trivariate Splines on Fans of Hyperplane Arrangements and Koszul Homology

We study the space of splines ${\mathcal{S}}^{\mathbf{r}}({\mathrm{\Sigma }}^{\mathcal{A}})$ where $\mathbf{r}$ denotes a smoothness distribution and ${\mathrm{\Sigma }}^{\mathcal{A}}$ is the fan of a central hyperplane arrangement $\mathcal{A}$ 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 ${\mathcal{S}}^{\mathbf{r}}({\mathrm{\Sigma }}^{\mathcal{A}})$ is bounded by an expression that involves the dimensions of specific Koszul homology modules constructed from the defining equations of the hyperplane arrangement $\mathcal{A}$ 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 $\mathcal{A}$ is generic with five or fewer hyperplanes, and 3) compute the Hilbert function of the spline space if $\mathcal{A}$ is a generic arrangement with sufficiently many hyperplanes and $\mathbf{r}$ is a constant distribution. As an application of our methods, we compute $\dim {\mathcal{S}}_d^0({\mathrm{\Sigma }}^{\mathcal{A}})$ and $\dim {\mathcal{S}}_d^1({\mathrm{\Sigma }}^{\mathcal{A}})$ for all values of $d$ when $\mathcal{A}$ is a generic arrangement.