| Special Divisor |
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The condition to be a special divisor ''D'' can be formulated in Sheaf Cohomology terms, as the non-vanishing of the ''H''1 cohomology of the sheaf of the sections of the Invertible Sheaf or Line Bundle associated to ''D''. This means that, by the Riemann-Roch Theorem , the ''H''0 cohomology or space of holomorphic sections is larger than expected. Alternatively, by Serre Duality , the condition is that there exist Holomorphic Differential s with divisor ≥ −''D'' on the curve. BRILL-NOETHER THEORY Brill-Noether theory in Algebraic Geometry is the theory of special divisors on ''generic'' algebraic curves. It is of interest mainly in the case of Genus g In conceptual terms, for ''g'' given, the ) of a given degree ''d'', as a function of ''g'', that ''must'' be present on a curve of that genus. The theory is named for the German geometers Ludwig Brill and Max Noether . The results were given in Nineteenth Century style; the whole theory was updated and modern proofs given by Phillip Griffiths and others. These formulations can be carried over into higher dimensions, and there is now a corresponding Brill-Noether theory for some classes of Algebraic Surface s. |
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