For a simplicial subdivison Δ of a region in kn (k algebraically closed) and r ∈ N, there is a reflexive sheaf script K sign on Pn such that H0 (script K sign (d)) is essentially the space of piecewise polynomial functions on Δ, of degree at most d, which meet with order of smoothness r along common faces. In , Elencwajg and Forster give bounds for the vanishing of the higher cohomology of a bundle ε on Pn in terms of the top two Chern classes and the generic splitting type of ε. We use a spectral sequence argument similar to that of  to characterize those Δ for which script K sign is actually a bundle (which is always the case for n = 2). In this situation we can obtain a formula for H0(script K sign (d)) which involves only local data; the results of  cited earlier allow us to give a bound on the d where the formula applies. We also show that a major open problem in approximation theory may be formulated in terms of a cohomology vanishing on P2 and we discuss a possible connection between semi-stability and the conjectured answer to this open problem.
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