Abstract
In \cite{as}, Alfeld and Schumaker give a formula for the dimension of the space of piecewise polynomial functions (splines) of degree d and smoothness r on a generic triangulation of a planar simplicial complex Δ (for d≥3r+1) and any triangulation (for d≥3r+2). In \cite{ss}, it was conjectured that the Alfeld-Schumaker formula actually holds for all d≥2r+1. In this note, we show that this is the best result possible; in particular, there exists a simplicial complex Δ such that for any r, the dimension of the spline space in degree d=2r is not given by the formula of \cite{as}. The proof relies on the explicit computation of the nonvanishing of the first local cohomology module described in \cite{ss2}.