Associated primes of spline complexes

The spline complex whose top homology is the algebra of mixed splines over a fan was introduced by Schenck-Stillman as a variant of a complex of Billera. 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 the algebra of mixed splines over a fan, including cases where vanishing is imposed along arbitrary codimension one faces of the boundary of the fan, extending computations by Geramita-Schenck in the simplicial case and McDonald-Schenck in the polytopal case. The second is a description of the fourth coefficient of the Hilbert polynomial of the algebra of mixed splines over simplicial fans. We use this to re-derive a result of Alfeld-Schumaker-Whiteley on the generic dimension of C1 tetrahedral splines in large degree and indicate via an example how this description may be used to give the fourth coefficient in particular non-generic configurations.