Moduli space

In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects.

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.

We can also describe the collection of lines in R2 that intersect the origin by means of a topological construction.

To wit: consider the unit circle S1 ⊂ R2 and notice that every point s ∈ S1 gives a line L(s) in the collection (which joins the origin and s).

Each of these definitions formalizes a different notion of what it means for the points of space M to represent geometric objects.

Heuristically, if we have a space M for which each point m ∊ M corresponds to an algebro-geometric object Um, then we can assemble these objects into a tautological family U over M. (For example, the Grassmannian G(k, V) carries a rank k bundle whose fiber at any point [L] ∊ G(k, V) is simply the linear subspace L ⊂ V.) M is called a base space of the family U.

Note, however, that a coarse moduli space does not necessarily carry any family of appropriate objects, let alone a universal one.

This in particular makes the existence of a fine moduli space impossible (intuitively, the idea is that if L is some geometric object, the trivial family L × [0,1] can be made into a twisted family on the circle S1 by identifying L × {0} with L × {1} via a nontrivial automorphism.

The use of these categories fibred in groupoids to describe a moduli problem goes back to Grothendieck (1960/61).

When g > 1, this stack may be compactified by adding new "boundary" points which correspond to stable nodal curves (together with their isomorphisms).

One can also define coarse moduli spaces representing isomorphism classes of smooth or stable curves.

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(2).

One can also enrich the problem by considering the moduli stack of genus g nodal curves with n marked points.

The resulting moduli stacks of smooth (or stable) genus g curves with n-marked points are denoted

In higher dimensions, moduli of algebraic varieties are more difficult to construct and study.

Using techniques arising out of the minimal model program, moduli spaces of varieties of general type were constructed by János Kollár and Nicholas Shepherd-Barron, now known as KSB moduli spaces.

[4] Using techniques arising out of differential geometry and birational geometry simultaneously, the construction of moduli spaces of Fano varieties has been achieved by restricting to a special class of K-stable varieties.

In this setting important results about boundedness of Fano varieties proven by Caucher Birkar are used, for which he was awarded the 2018 Fields medal.

[citation needed] Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces.

The modern formulation of moduli problems and definition of moduli spaces in terms of the moduli functors (or more generally the categories fibred in groupoids), and spaces (almost) representing them, dates back to Grothendieck (1960/61), in which he described the general framework, approaches, and main problems using Teichmüller spaces in complex analytical geometry as an example.

More precisely, the existence of non-trivial automorphisms of the objects being classified makes it impossible to have a fine moduli space.

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 1965, which shows that under suitable conditions the quotient indeed exists.

A smooth curve together with a complete linear system of degree d > 2g is equivalent to a closed one dimensional subscheme of the projective space Pd−g.

This locus H in the Hilbert scheme has an action of PGL(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.

Here the idea is to start with an 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.

The spectrum of this latter ring can then be viewed as giving a kind of coordinate chart on the desired moduli space.