Later Work
The Riemann singularity theorem was extended by George Kempf in 1973, building on work of David Mumford and Andreotti - Mayer, to a description of the singularities of points p = class(D) on Wk for 1 ≤ k ≤ g − 1. In particular he computed their multiplicities also in terms of the number of independent meromorphic functions associated to D (Riemann-Kempf singularity theorem).
More precisely, Kempf mapped J locally near p to a family of matrices coming from an exact sequence which computes h0(O(D)), in such a way that Wk corresponds to the locus of matrices of less than maximal rank. The multiplicity then agrees with that of the point on the corresponding rank locus. Explicitly, if
- h0(O(D)) = r + 1,
the multiplicity of Wk at class(D) is the binomial coefficient
When d = g − 1, this is r + 1, Riemann's formula.
Read more about this topic: Theta Divisor
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