Theta divisor

In mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers (and principally polarized) by the zero locus of the associated Riemann theta-function. It is therefore an algebraic subvariety of A of dimension dim A − 1.

Classical theory

Classical results of Bernhard Riemann describe Θ in another way, in the case that A is the Jacobian variety J of an algebraic curve (compact Riemann surface) C. There is, for a choice of base point P on C, a standard mapping of C to J, by means of the interpretation of J as the linear equivalence classes of divisors on C of degree 0. That is, Q on C maps to the class of Q − P. Then since J is an algebraic group, C may be added to itself k times on J, giving rise to subvarieties W_k.

If g is the genus of C, Riemann proved that Θ is a translate on J of W_{g − 1}. He also described which points on W_{g − 1} are non-singular: they correspond to the effective divisors D of degree g − 1 with no associated meromorphic functions other than constants. In more classical language, these D do not move in a linear system of divisors on C, in the sense that they do not dominate the polar divisor of a non constant function.

Riemann further proved the Riemann singularity theorem, identifying the multiplicity of a point p = class(D) on W_{g − 1} as the number of linearly independent meromorphic functions with pole divisor dominated by D, or equivalently as h⁰(O(D)), the number of linearly independent global sections of the holomorphic line bundle associated to D as Cartier divisor on C.

Later work

The Riemann singularity theorem was extended by George Kempf in 1973,^[1] building on work of David Mumford and Andreotti - Mayer, to a description of the singularities of points p = class(D) on W_k 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).^[2]

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 W_k 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 W_k at class(D) is the binomial coefficient

${\displaystyle {g-k+r \choose r}.}$

When k = g − 1, this is r + 1, Riemann's formula.

Notes

1. G. Kempf (1973). "On the geometry of a theorem of Riemann". Ann. of Math. 98 (1): 178–185. doi:10.2307/1970910. JSTOR 1970910.
2. Griffiths and Harris, p.348

References

• P. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. ISBN 0-471-05059-8.