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In algebraic geometry, the moduli problem
is to describe the parameters on which algebraic varieties depend. The use of the term modulus here for such a parameter goes back to
the same source as in modular form: a modular form is in general is some
kind of differential form (or tensor density, since the forms come
with a 'weight') on a moduli space, that is, a space whose co-ordinates are the moduli.
In the case of elliptic curves, there is one modulus, so moduli
spaces are algebraic curves. This is the quantity called k
in Jacobi's elliptic function theory, which
reduces elliptic integrals to a form involving the square root of
(1-x2)(1-k2x2). This modulus of the elliptic integral
therefore was probably the first modulus to be recognised.
The case of elliptic curves has been thoroughly studied, because of
the great interest of the modular equations in this case. The
j-invariant is a fundamental elliptic modular function. The moduli problem here is the prototype for moduli problems with
level structure, meaning in this case some 'marking' of torsion
groups of points on the curve. Each level structure gives rise to a subgroup of the modular group, and then its own modular curve. The
j-invariant is called a Hauptmodul, traditionally, meaning that the modular curve has genus 0. There are other cases of genus 0, and other
Hauptmoduls, which enter the remarkable monstrous
moonshine theory.
In general a curve of genus g has
3g-3 moduli, for g > 1. This number was known classically as the number of parameters on which a compact Riemann surface depends.
It agrees with the calculation of the dimension of the space of quadratic
differentials on a fixed such Riemann surface, which is suggested by deformation theory combined with Serre duality.
Except when g=2, this is larger than the number 2g-1 of moduli of hyperelliptic curves.
Moduli of vector bundles
There is also another major question, of determining moduli for vector
bundles V on a fixed algebraic variety X. When X has
dimension 1 and V is a line bundle, this is the theory of the Jacobian
variety of a curve.
Beginning with a paper of André Weil (who called them 'matrix divisors'),
the vector bundles on X have been studied in relation to their moduli. In applications to physics, the number of moduli of vector bundles and the closely related problem of the number of moduli of principal G-bundles has been found to be significant in gauge theory.
Constructions
Two general construction techniques for moduli spaces have been especially successful. The first is the method of geometric
invariant theory, pioneered by David Mumford. The basic strategy is
to simplify the classification problem by adding additional data in such a way that the original moduli space is the quotient of the new one by a reductive
group action. To see how this might work, consider the problem of
parametrizing curves of genus 2. Each such curve is hyperelliptic and therefore admits a unique degree 2 cover of
P1---unique, that is, up to composition with an element of the automorphism group PGL(2) of
P1. So we begin by classifying double covers X→P1 with
X of genus 2. Such a double cover is determined by its six ramification points. So now we are classifying six-element subsets of P1 (a
comparatively easy problem). We have to pay a price, though, in dividing out by the PGL(2) action at the end.
The other general approach is primarily associated with Michael Artin. Here the idea is to start with any object of the kind to be classified and study
its deformation theory. This means first constructing infinitesimal deformations, then appealing to
prorepresentability theorems to put these together into an object over a formal base. Next an appeal to Grothendieck's formal existence theorem provides an object of
the desired kind over a base which is a complete local ring. This object can be approximated via Artin's
approximation theorem by an object defined over a finitely generated ring. The spectrum of this latter ring can then be viewed as giving a kind of coordinate chart on the desired
moduli space. By gluing together enough of these charts, we can cover the space, but the map from our union of spectra to the
moduli space will in general be many to one. We therefore define an equivalence relation on the former; essentially, two points are equivalent if the objects over each
are isomorphic. This gives a scheme and an equivalence
relation, which is enough to define an algebraic space (actually an algebraic stack if we are being careful) if not always a scheme.
See also
For a physics-oriented description of moduli spaces, see moduli.
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