Moduli space

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In algebraic geometry, a moduli space is a geometric space (usually a scheme or a stack) 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 (e.g., the smooth algebraic curves of a fixed genus) 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.

1 Basic Examples
- 1.1 Projective Space and Grassmanians
- 1.2 Hilbert Scheme
2 Definitions
3 Further Examples
4 Methods for constructing moduli spaces
5 In Physics

[edit] Basic Examples

[edit] Projective Space and Grassmanians

The real projective space $\mathbb{R}P^n$ is a moduli space. It is the space of lines in $\mathbb{R}^{n+1}$ which pass through the origin. Similarly, complex projective space is the space of all complex lines in $\mathbb{C}^{n+1}.$

More generally, the Grassmannian $G k (V)$ of a vector space V over a field F is the moduli space of all k-dimensional linear subspaces of V.

[edit] Hilbert Scheme

The Hilbert scheme $H i l b (X)$ is a moduli scheme. Every closed point of $H i l b (X)$ corresponds to a closed subscheme of a fixed scheme X, and every closed subscheme is represented by such a point.

[edit] Definitions

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.

[edit] Fine Moduli Spaces

This is the most important notion. Heuristically, if we have a space M for which each point $m \in M$ corresponds to an algebro-geometric object $T m$ , then we can assemble these objects into a tautological family T over M. (For example, the Grassmanian $G k (V)$ carries a rank k bundle whose fiber at any point $[L] \in G_k(V)$ is simply the linear subspace $L \subset V$ .) 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 T along a unique map $B \to M$ . 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., the functor of points $H o m ( - , M)$ 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 $id_M \in Hom(M,M)$ .

[edit] Coarse Moduli Spaces

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 $\tau: F \to Hom(-,M)$ and $τ$ 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 $\phi_T: B \to M$ 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 coarse moduli space do not necessarily carry any family of appropriate objects, let alone a universal one.

[edit] Moduli stacks

It is frequently the case that interesting geometric objects come equipped with lots of natural isomorphisms, i.e., that the collection of all such objects naturally forms a groupoid rather than a merely a set. In this situation, one can sometimes obtain a moduli space by looking for a space which coarsely represents the functor assigning to a space B the set of isomorphism classes of families over B. However, this approach is not ideal, as such spaces are not guaranteed to exist, are frequently singular when they do exist, and are never fine moduli spaces. A more modern approach is to enrich the classification by remembering the isomorphisms; roughly speaking, one considers the higher functor which assigns to any space B the groupoid of families over B. Such groupoid-valued functors are not representable by schemes, but in many cases they can be interpreted as algebraic stacks. In this situation, one speaks of moduli stacks instead of moduli spaces.

[edit] Further Examples

[edit] Moduli of curves

For more details on this topic, see Deligne-Mumford moduli space of curves.

The moduli stack $\mathcal{M}_{g}$ 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 $\overline{\mathcal{M}}_{g}$ . 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 $3 g - 3$ ; 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(2). Hence the dimension of $\mathcal{M}_0$ is

dim(space of genus zero curves) - dim(group of automorphisms) = 0 - dim(PGL(2)) = -3.

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 $\mathcal{M}_1$ 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 $\mathcal{M}_{g,n}$ (or $\overline{\mathcal{M}}_{g,n}$ ), and have dimension 3g-3 + n.

A case of particular interest is the moduli stack $\overline{\mathcal{M}}_{1,1}$ 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.

[edit] Moduli of varieties

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.

[edit] Moduli of vector bundles

Another important moduli problem is to understand the geometry of (various substacks of) the moduli stack $V e c t n (X)$ 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.

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] Methods for constructing moduli spaces

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 desired moduli space is the quotient by a reductive group action of the moduli space parametrizing objects and additional data.

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 $P d - g$ . 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 $P G L n$ 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.

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] In Physics

For more details on this topic, see moduli (physics).

The term moduli space is sometimes used in physics to refer specifically the moduli space of vacuum expectation values of a set of scalar fields, or to the moduli space of possible string backgrounds.

Moduli spaces also appear in physics in cohomological field theory, where one can use Feynamn path integrals to compute the intersection numbers of various algebraic moduli spaces.

Categories: Moduli theory | Invariant theory