Smooth planar r-splines of degree 2r

Alfeld and Schumaker [Numer. Math. 57 (1990) 651-661] give a for mula 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 Schenck and Stiller [Manuscripta Math. 107 (2002) 43-58], 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 Alfeld and Schumaker [Numer. Math. 57 (1990) 651-661]. The proof relies on the explicit computation of the nonvanishing of the first local cohomology module described in Schenck and Stillman [J. Pure Appl. Algebra 117 & 118 (1997) 535-548].