Theta characteristic
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In mathematics, a theta characteristic of a non-singular algebraic curve C is a divisor class Θ such that 2Θ is the canonical class, In terms of holomorphic line bundles L on a connected compact Riemann surface, it is therefore L such that L2 is the canonical bundle, here also equivalently the holomorphic cotangent bundle. In terms of algebraic geometry, the equivalent definition is as an invertible sheaf, which squares to the sheaf of differentials of the first kind.
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[edit] History and genus 1
The importance of this concept was realised first in the analytic theory of theta functions, and geometrically in the theory of bitangents. In the analytic theory, there are four fundamental theta functions in the theory of Jacobian elliptic functions. Their labels are in effect the theta characteristics of an elliptic curve. For that case, the canonical class is trivial (zero in the divisor class group) and so the theta characteristics of an elliptic curve E over the complex numbers are seen to be in 1-1 correspondence with the four points P on E with 2P = 0 (this is clear when E is treated as a complex torus.
[edit] Higher genus
For C of genus 0 there is no interest in the concept, since the divisor class group is trivial. In case of higher genus g, assuming the field over which C is defined does not have characteristic 2, the theta characteristics can be counted as
- 22g
in number.
This comes about because the solutions of the equation on the divisor class level will form a single coset of the solutions of
- 2D = 0.
In other words, with K the canonical class and Θ any given solution of
- 2Θ = K,
any other solution will be of form
- Θ + D.
This reduces counting the theta characteristics to finding the 2-rank of the Jacobian variety J(C) of C. In the complex case, again, the result follows since J(C) is a complex torus of dimension 2g. Over a general field, see the theory explained at Hasse-Witt matrix for the counting of the p-rank of an abelian variety. The answer is the same, provided the characteristic of the field is not 2.
[edit] Classical theory
Classically the theta characteristics were divided into two kinds, syzygetic and asyzygetic, according to the value on them of a certain quadratic form Q with values mod 2. Thus in case of g = 3 and a plane quartic curve, there were 28 of one type, and the remaining 36 of the other; this is basic in the question of counting bitangents. The geometric construction of Q as an intersection form is with modern tools possible algebraically. In fact the Weil pairing applies, in its abelian variety form.
[edit] Spin structures
There is a direct connection, for a connect compact Riemann surface, between theta characteristics and spin structures.
