Theta Divisor - Later Work

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 ≤ kg − 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.

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