Associated Primes of Spline Complexes

The spline complex $\mathcal{R}/\mathcal{J}[\Sigma]$ whose top homology is the algebra $C^\alpha(\Sigma)$ of mixed splines over the fan $\Sigma\subset\mathbb{R}^{n+1}$ was introduced by Schenck-Stillman in [Schenck-Stillman 97] as a variant of a complex $\mathcal{R}/\mathcal{I}[\Sigma]$ of Billera [Billera 88]. In this paper we analyze the associated primes of homology modules of this complex. In particular, we show that all such primes are linear. We give two applications to computations of dimensions. The first is a computation of the third coefficient of the Hilbert polynomial of $C^\alpha(\Sigma)$, including cases where vanishing is imposed along arbitrary codimension one faces of the boundary of $\Sigma$, generalizing the computations in [Geramita-Schenck 98,McDonald-Schenck 09]. The second is a description of the fourth coefficient of the Hilbert polynomial of $HP(C^\alpha(\Sigma))$ for simplicial fans $\Sigma$. We use this to derive the result of Alfeld, Schumaker, and Whiteley on the generic dimension of $C^1$ tetrahedral splines for $d\gg 0$ [Alfeld-Schumaker-Whiteley 93] and indicate via an example how this description may be used to give the fourth coefficient in particular nongeneric configurations.