Local cohomology of bivariate splines

Graphs, Syzygies, and Multivariate Splines

“…The following results are generalizations of Theorem 5.3(a) in [13] and Theorem 5.2 in [12], respectively.…”

“…This is because for a topological disk, G always contains a basis of cycles around vertices, and such cycles will always have rank 2. In fact, Schenck and Stillman prove, for any simplicial , that C r (ˆ ) free implies is topologically trivial [13]. The following example shows that this result is false for polyhedral complexes, i.e., it is possible for Syz r (G) to be free without being acyclic.…”

“…. , s+1 s+1 } of degree s. By Lemma 5.1 of [13] these must be linearly independent. Thus r +1 e , s+1 1 , .…”

“…When B r ( ) is graded this decomposition also gives a straightforward way of computing its Hilbert Series. We describe conditions on G, L, and r for such a decomposition to occur, providing alternate proofs for some results of Schenck and Stillman from [12] and [13], and extending those results to polyhedral complexes. Schenk and Stillman claim that these are the best possible results using only local data.…”

“…This idea led to the development of an algebraic theory of spline modules, which has had subsequent applications to hyperplane arrangements and face rings of simplicial complexes. The use of algebro-geometric methods to study C r ( ) can be found in [11]- [13] in the simplicial case, and in the work of Yuzvinsky [17], who provides a criterion for determining the projective dimension of spline modules on polyhedral complexes, using a sheaf-theoretic approach.…”