The real projective space is a moduli space. It is the space of lines in which pass through the origin. Similarly, complex projective space is the space of all complex lines in
More generally, the Grassmannian of a vector space V over a field F is the moduli space of all k-dimensional linear subspaces of V.
The Hilbert scheme is a moduli scheme. Every closed point of corresponds to a closed subscheme of a fixed scheme X, and every closed subscheme is represented by such a point.
There are several different related notions of what it means for a space M to be a moduli space. Each of these definitions formalizes a different notion of what it means for the points of a space to represent geometric objects.
More precisely, suppose that we have a functor F from schemes to sets, which assigns to a scheme B the set of all suitable families of objects with base B. A space M is a fine moduli space for the functor F if M represents F, i.e., the functor of points is naturally isomorphic to F. This implies that M carries a universal family; this family is the family on M corresponding to the identity map .
Fine moduli spaces are very useful, but they do not always exist and are frequently difficult to construct, so mathematicians sometimes use a weaker notion, the idea of a coarse moduli space. A space M is a coarse moduli space for the functor F if there exists a natural transformation and is universal among such natural transformations. More concretely, M is a coarse moduli space for F if any family T on a scheme B gives rise to a map and any two objects T and T' (regarded as families over a point) correspond to the same point of M if and only if T and T' are isomorphic. Thus, M is a space which has a point for every object that could appear in a family, and whose geometry reflects the ways objects can vary in families. Note, however, that a coarse moduli space does not necessarily carry any family of appropriate objects, let alone a universal one.
It is frequently the case that interesting geometric objects come equipped with lots of natural automorphisms. This in particular makes the existence of a fine moduli space impossible, one can still sometimes obtain a coarse moduli space. However, this approach is not ideal, as such spaces are not guaranteed to exist, are frequently singular when they do exist, and miss details about some non-trivial families of objects they classify. A more sophisticated approach is to enrich the classification by remembering the isomorphisms. More precisely, on any base B one can consider the category of families on B with only isomorphisms between families taken as morphisms. One then considers the fibred category which assigns to any space B the groupoid of families over B. The use of these categories fibred in groupoids to describe a moduli problem goes back to Grothendieck (1960/61). In general they cannot be represented by schemes or even algebraic spaces, but in many cases they have a natural structure of an algebraic stack. Algebraic stacks and their use to analyse moduli problems appeared in Deligne-Mumford (1969) as a tool to prove the irreducibility of the (coarse) moduli space of curves of a given genus. The language of algebraic stacks essentially provides a systematic way to view the fibred category that constitutes the moduli problem as a "space", and the moduli stack of many moduli problems is better-behaved (such as smooth) than the corresponding coarse moduli space.
The moduli stack classifies families of smooth projective curves, together with their isomorphisms. When g > 1, this stack may be compactified by adding new "boundary" points which correspond to stable nodal curves (together with their isomorphisms). A curve is stable if it has only a finite group of automorphisms. The resulting stack is denoted . Both moduli stacks carry universal families of curves. One can also define coarse moduli spaces representing isomorphism classes of smooth or stable curves. These coarse moduli spaces were actually studied before the notion of moduli stack was invented. In fact, the idea of a moduli stack was invented by Deligne and Mumford in an attempt to prove the projectivity of the coarse moduli spaces. In recent years, it has become apparent that the stack of curves is actually the more fundamental object.
Both stacks above have dimension ; hence a stable nodal curve can be completely specified by choosing the values of 3g-3 parameters, when g > 1. In lower genus, one must account for the presence of smooth families of automorphisms, by subtracting their number. There is exactly one complex curve of genus zero, the Riemann sphere, and its group of isomorphisms is PGL(2). Hence the dimension of is
Likewise, in genus 1, there is a one-dimensional space of curves, but every such curve has a one-dimensional group of automorphisms. Hence the stack has dimension 0. The coarse moduli spaces have the same dimension as the stacks when g > 1; however, in genus zero the coarse moduli space has dimension zero, and in genus one, it has dimension one.
One can also enrich the problem by considering the moduli stack of genus g nodal curves with n marked points. Such marked curves are said to be stable if the subgroup of curve automorphisms which fix the marked points is finite. The resulting moduli stacks of smooth (or stable) genus g curves with n-marked points are denoted (or ), and have dimension 3g-3 + n.
A case of particular interest is the moduli stack of genus 1 curves with one marked point. This is the stack of elliptic curves, and is the natural home of the much studied modular forms, which are meromorphic sections of bundles on this stack.
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. This is the problem underlying Siegel modular form theory. See also Shimura variety.
Another important moduli problem is to understand the geometry of (various substacks of) the moduli stack of rank n vector bundles on a fixed algebraic variety X. This stack has been most studied when X is one-dimensional, and especially when n equals one. In this case, the coarse moduli space is the Picard scheme, which like the moduli space of curves, was studied before stacks were invented. Finally, when the bundles have rank 1 and degree zero, the study of the coarse moduli space is the study of the Jacobian variety.
More precisely, the existence of non-trivial automorphisms of the objects being classified makes it impossible to have a fine moduli space. However, it is often possible to consider a modified moduli problem of classifying the original objects together with additional data, chosen in such a way that the identity is the only automorphism respecting also the additional data. With a suitable choice of the rigidifying data, the modified moduli problem will have a (fine) moduli space T, often described as a subscheme of a suitable Hilbert scheme or Quot scheme. The rigidifying data is moreover chosen so that it corresponds to a principal bundle with an algebraic structure group G. Thus one can move back from the rigidified problem to the original by taking quotient by the action of G, and the problem of constructing the moduli space becomes that of finding a scheme (or more general space) that is (in a suitably strong sense) the quotient T/G of T by the action of G. The last problem in general does not admit a solution; however, it is addressed by the groundbreaking geometric invariant theory (GIT), developed by David Mumford in Mumford (1965), which shows that under suitable conditions the quotient indeed exists.
To see how this might work, consider the problem of parametrizing smooth curves of genus g > 2. A smooth curve together with a complete linear system of degree d > 2g is equivalent to a closed one complex dimensional subscheme of the projective space . Consequently, the moduli space of smooth curves and linear systems (satisfying certain criteria) may be embedded in the Hilbert scheme of a sufficiently high-dimensional projective space. This locus H in the Hilbert scheme has an action of which mixes the elements of the linear system; consequently, the moduli space of smooth curves is then recovered as the quotient of H by the projective general linear group.
Another 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.