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

[\sqrt.]

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.

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. 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.

Moduli space

Moduli of vector bundles

Constructions

See also