  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 $\Delta$ (for $d
\ge 3r+1$) and any triangulation (for $d\geq 3r+2$). In \cite{ss}, it was
conjectured that the Alfeld-Schumaker formula actually holds for all $d \ge
2r+1$. In this note, we show that this is the best result possible; in
particular, there exists a simplicial complex $\Delta$ 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}.
