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
Famous quotes containing the word work:
“One should either be a work of art, or wear a work of art.”
—Oscar Wilde (18541900)
“Working women today are trying to achieve in the work world what men have achieved all alongbut men have always had the help of a woman at home who took care of all the other details of living! Today the working woman is also that woman at home, and without support services in the workplace and a respect for the work women do within and outside the home, the attempt to do both is taking its tollon women, on men, and on our children.”
—Jeanne Elium (20th century)