Moduli space
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In algebraic geometry, a moduli space is a parameter space for families of algebraic objects (such as algebraic varieties, morphisms, vector bundles). The use of the term modulus here for such a parameter space goes back to the same source as in modular form: a modular form 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
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. As curves of higher genus admit automorphisms, the parameter space of smooth genus g curves is a (Deligne-Mumford) stack admitting a (quasiprojective) coarse moduli scheme. A proper moduli space can be achieved by adding nodal curves satisfying a technical stability condition. A closely related moduli space parametrizes stable morphisms to a smooth projective variety.
In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. For instance, the higher dimensional analogue of the moduli space of elliptic curves discussed above is the moduli space of abelian varieties.
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[edit] 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.
[edit] 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. By Hurwitz theorem, 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. For instance, the moduli space of vector bundles (say over a curve, for simplicity) can be constructed using G.I.T.
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.
[edit] Fine versus Coarse Moduli Spaces
Often, the distinction is made between a fine moduli space and a coarse moduli space. The difference between these two concepts is subtle and demonstrates the inherent ambiguity in trying to define a 'parameter space for families of objects'. In the following, the words 'object' and 'family' are used to mean the type of algebraic object and family being considered.
A fine moduli space is an object X, together with a family 
called the universal family, with the property that given any family g:A → B, there is a unique map φ(g):B → X such that the pullback of f along φ(g) is g. Conceptually, this means that it is possible to put a family over the moduli space with the property that any other family can be gotten as a pullback from it in exactly one way; hence, the 'universal family'.
Another way of saying this: Let F be the contravariant functor from objects to Set which takes an object A to the set of families with A as the base. Then X is a fine moduli space if and only if F is a representable functor which is represented by X.
Unfortunately, most practical moduli spaces fail to be fine, and one must settle for the notion of a coarse moduli space. First, here are a couple of properties one might hope for in an object X claiming to be any sort of moduli space:
1) Given a family, g:A → B, there is a unique map φ(g):B → X, such that φ is natural.
2) Let {p} be a point (depending on the objects at hand, it could be Spec(
), Spec(
) or an actual point), and let f:C → {p} and g:D → {p} be families. Then φ(f) and φ(g) take {p} to the same point in X if and only if C and D are isomorphic objects.
Roughly speaking, 2) says that there is exactly one point in X for each allowable fiber of a family, and 1) says that the base of any family can be mapped into X such that the fibers above each point correspond to the point in X they are taken to.
With this established, a coarse moduli space is defined as an object that is universal with respect to all objects that satisfy 1) and 2). This is a bit of a messy definition, but at least it has a point for every possible fiber, a topology that captures some of the information about how those fibers can vary in families, and is as simple as possible (by universality).
Coarse moduli spaces are easier to define using the language of category theory. As above, one considers the functor F from bases to sets of families. However, instead of requiring that F be naturally equivalent to Hom(-,X), as in the case of representability, one only requires that there is a natural transformation from F to Hom(-,X), and that X be universal amongst objects that satisfy this condition.
[edit] Why do most moduli spaces fail to be fine?
The simple answer is that a family can have only one kind of fiber but still not be trivial, which can happen any time a fiber has a non-trivial automorphism. For example, consider families of elliptic curves, and take an arbitrary elliptic curve E. It has a hyperelliptic involution h, which is an order 2 non-trivial automorphism. Therefore, one can create a family over the circle S1, where every fiber is isomorphic to E, but travelling around the circle induces the map h in the fiber. The map from the circle into moduli space must then go to a single point, since all fibers are identical. This immediately prevents a universal family from existing, since it is impossible to pullback a non-trivial family from a point. A family with all fibers identical is called an isotrivial family.
[edit] Turning coarse moduli into fine moduli
In many cases, coarse moduli spaces are just not sufficient. Therefore, a number of techniques have been developed to turn the moduli problem at hand into one with a fine moduli space.
The simplest technique is to add structure to the types of families being considered. For example, instead of families of elliptic curves, one can consider families of elliptic curves with n marked points, which yield a fine moduli space if n is large enough.
Another technique that is very common is to enlarge the category of objects being considered, until the functor F is representable. This is the primary motivation behind the theory of stacks.
[edit] See also
For a physics-oriented description of moduli spaces, see moduli.
