Theta characteristic
Encyclopedia
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.
s. 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 counting of the solutions is clear from the group structure, a product of two circle groups, when E is treated as a complex torus
.
in number if the base field is algebraically closed.
This comes about because the solutions of the equation on the divisor class level will form a single coset
of the solutions of
In other words, with K the canonical class and Θ any given solution of
any other solution will be of form
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.
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, as it corresponds to the 28 bitangents of a quartic. 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.
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a theta characteristic of a non-singular algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
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
Compact Riemann surface
In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. Riemann surfaces are generally classified first into the compact and the open .A compact Riemann surface C that is a...
, it is therefore L such that L2 is the canonical bundle
Canonical bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n is the line bundle\,\!\Omega^n = \omegawhich is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V.This is the dualising...
, here also equivalently the holomorphic cotangent bundle. In terms of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, the equivalent definition is as an invertible sheaf
Invertible sheaf
In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...
, which squares to the sheaf of differentials of the first kind.
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 bitangentBitangent
In mathematics, a bitangent to a curve C is a line L that touches C in two distinct points P and Q and that has the same direction to C at these points...
s. 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
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
. 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 counting of the solutions is clear from the group structure, a product of two circle groups, when E is treated as a complex torus
Complex torus
In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense...
.
Higher genus
For C of genus 0 there one such divisor class, namely the class of -P, where P is any point on the curve. 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 if the base field is algebraically closed.
This comes about because the solutions of the equation on the divisor class level will form a single coset
Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
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
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...
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
Hasse-Witt matrix
In mathematics, the Hasse–Witt matrix H of a non-singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping with respect to a basis for the differentials of the first kind...
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.
Classical theory
Classically the theta characteristics were divided into two kinds, syzygetic and asyzygetic, according to the value on them of a certain quadratic formQuadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
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, as it corresponds to the 28 bitangents of a quartic. The geometric construction of Q as an intersection form
Intersection form
Intersection form may refer to:*Intersection theory *intersection form...
is with modern tools possible algebraically. In fact the Weil pairing
Weil pairing
In mathematics, the Weil pairing is a construction of roots of unity by means of functions on an elliptic curve E, in such a way as to constitute a pairing on the torsion subgroup of E...
applies, in its abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
form.
Spin structures
There is a direct connection, for a connected compact Riemann surface, between theta characteristics and spin structureSpin structure
In differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
s.
External links
- Dolgachev, Lectures on Classical Topics, Ch. 5 (PDF)