### Abstract

For a simplicial subdivison Δ of a region in k^{n} (k algebraically closed) and r ∈ N, there is a reflexive sheaf script K sign on P^{n} such that H^{0} (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 [9], Elencwajg and Forster give bounds for the vanishing of the higher cohomology of a bundle ε on P^{n} 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 [16] 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 H^{0}(script K sign (d)) which involves only local data; the results of [9] 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 P^{2} and we discuss a possible connection between semi-stability and the conjectured answer to this open problem.

Original language | English (US) |
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Pages (from-to) | 43-58 |

Number of pages | 16 |

Journal | Manuscripta Mathematica |

Volume | 107 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2002 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Manuscripta Mathematica*,

*107*(1), 43-58. https://doi.org/10.1007/s002290100222