Moduli space

In mathematics, in particular algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is formal moduli. Bernhard Riemann first used the term "moduli" in 1857.

Motivation

Every point of a moduli space corresponds to a solution of a given geometric problem. Two different solutions correspond to the same point if they are isomorphic. A moduli space can be thought of as giving a universal space of parameters for the problem.
For example, consider the problem of finding all circles in the Euclidean plane up to congruence. Any circle can be described uniquely by giving three points, but many different sets of three points give the same circle: the correspondence is many-to-one. However, circles are uniquely parameterized by giving their center and radius: this is two real parameters and one positive real parameter. Since we are only interested in circles "up to congruence", we identify circles having different centers but the same radius, and so the radius alone suffices to parameterize the set of interest. The moduli space is, therefore, the positive real numbers.
Moduli spaces often carry natural geometric and topological structures as well. In the example of circles, for instance, the moduli space is not just an abstract set, but the absolute value of the difference of the radii defines a metric for determining when two circles are "close". The geometric structure of moduli spaces locally tells us when two solutions of a geometric classification problem are "close", but generally moduli spaces also have a complicated global structure as well.
For example, consider how to describe the collection of lines in that intersect the origin. We want to assign to each line of this family a quantity that can uniquely identify it—a modulus. An example of such a quantity is the positive angle with radians. The set of lines so parametrized is known as and is called the real projective line.
We can also describe the collection of lines in that intersect the origin by means of a topological construction. To wit: consider the unit circle and notice that for every point there is a unique line joining the origin and. However, this is also the line we'd get from looking at, so we identify opposite points via an equivalence relation to yield , with the quotient topology.
Thus, when we consider as a moduli space of lines that intersect the origin in, we capture the ways in which the members of the family can modulate by continuously varying the angle.

Basic examples

Projective space and Grassmannians

is the moduli space of lines through the origin in. Similarly, complex projective space is the space of all complex lines through the origin in.
More generally, the Grassmannian of a vector space is the moduli space of all -dimensional linear subspaces of V.

Projective space as moduli of very ample line bundles generated by global sections

Whenever there is an embedding of a scheme into the universal projective space, the embedding is given by a line bundle and sections which don't all vanish at the same time. This means, given a point

there is an associated point

given by the compositions

Then, two line bundles with sections are equivalent

iff there is an isomorphism such that. This means the associated moduli functor

sends a scheme to the set

Showing this is true can be done by running through a series of tautologies: any projective embedding gives the globally generated sheaf with sections. Conversely, given an ample line bundle globally generated by sections gives an embedding as above.

Chow variety

The Chow variety Chow is a projective algebraic variety which parametrizes degree d curves in P³. It is constructed as follows. Let C be a curve of degree d in P³, then consider all the lines in P³ that intersect the curve C. This is a degree d divisor D_C in G, the Grassmannian of lines in P³. When C varies, by associating C to D_C, we obtain a parameter space of degree d curves as a subset of the space of degree d divisors of the Grassmannian: Chow.

Hilbert scheme

The Hilbert scheme Hilb is a moduli scheme. Every closed point of Hilb corresponds to a closed subscheme of a fixed scheme X, and every closed subscheme is represented by such a point. A simple example of a Hilbert scheme is the Hilbert scheme parameterizing degree hypersurfaces of projective space. This is given by the projective bundle

with universal family given by

where is the associated projective scheme for the degree homogeneous polynomial.

Definitions

There are several related notions of things we could call moduli spaces. Each of these definitions formalizes a different notion of what it means for the points of space M to represent geometric objects.

Fine moduli space

This is the standard concept. Heuristically, if we have a space M for which each point m ∊ M corresponds to an algebro-geometric object U_m, then we can assemble these objects into a tautological bundle U over M. carries a rank k bundle whose fiber at any point ∊ G M is called a base space of the family U. We say that such a family is universal if any family of algebro-geometric objects T over any base space B is the pullback of U along a unique map B → M. A fine moduli space is a space M which is the base of a universal family.
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., there is a natural isomorphism
τ : F → Hom, where Hom is the functor of points. This implies that M carries a universal family; this family is the family on M corresponding to the identity map 1_M ∊ Hom.

Coarse moduli space

Fine moduli spaces are desirable, 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 τ : F → Hom and τ is universal among such natural transformations. More concretely, M is a coarse moduli space for F if any family T over a base B gives rise to a map φ_T : B → M and any two objects V and W correspond to the same point of M if and only if V and W 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.
In other words, a fine moduli space includes both a base space M and universal family U → M, while a coarse moduli space only has the base space M.

Moduli stack

It is frequently the case that interesting geometric objects come equipped with many 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, they 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. 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 analyze moduli problems appeared in Deligne-Mumford as a tool to prove the irreducibility of the 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 than the corresponding coarse moduli space.

Further examples

Moduli of curves

The moduli stack classifies families of smooth projective curves of genus g, together with their isomorphisms. When g > 1, this stack may be compactified by adding new "boundary" points which correspond to stable nodal curves. 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 3g−3; 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. 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 dimension 3g−3 as the stacks when g > 1 because the curves with genus g > 1 have only a finite group as its automorphism i.e. dim = 0. Eventually, 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 genus g curves with n-marked points are denoted , 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.