Editing Moduli space (section)

==Methods for constructing moduli spaces==
The modern formulation of moduli problems and definition of moduli spaces in terms of the moduli functors (or more generally the [[fibred category|categories fibred]] in [[groupoid]]s), and spaces (almost) representing them, dates back to Grothendieck (1960/61), in which he described the general framework, approaches, and main problems using [[Teichmüller space]]s in complex analytical geometry as an example. The talks, in particular, describe the general method of constructing moduli spaces by first ''rigidifying'' the moduli problem under consideration.

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 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 the genus ''g'' > 2. A smooth curve together with a [[complete linear system]] of degree ''d'' > 2''g'' is equivalent to a closed one dimensional subscheme of the projective space '''P'''<sup>''d−g''</sup>. 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 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.

Another general approach is primarily associated with [[Michael Artin]].  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 scheme|formal]] base.  Next, an appeal to [[Alexandre Grothendieck|Grothendieck's]] [[Grothendieck existence theorem|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 a ring|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.