On this blog we’ve extensively looked at lots of things from deformation theory over the past few years. Deformation theory is in some sense a local examination of a more global object called a moduli space. Today we’ll start a brief series on moduli spaces.
Roughly speaking a moduli space is a “space” whose “points” are objects you are trying to classify. You could make the moduli space of elliptic curves in which the points are elliptic curves. You could make the moduli space of rank vector bundles over some . Each point of this space would correspond to a vector bundle of rank on (up to isomorphism). You could make a moduli space of morphisms between and .
In theory, any type of mathematical thing you think up you could try to make a space of them. Algebraic geometers do this a lot, but I see no reason why you couldn’t try to study Borel measures on some metric space by trying to make a space whose points correspond to Borel measures (maybe up to mutual absolute continuity or something).
We’ve talked about the deformation functor of an object. Roughly speaking you should expect something along the following lines. Fix an object . This corresponds to a point on your moduli space . The tangent space of at should correlated to the first order infinitesimal deformation of . Nearby points to on should correspond to more similar objects (whatever that means) and far away objects should correspond to quite different objects.
The reason I want to keep this series brief is that the subject turns incredibly technical quickly because there are lots of conditions that people impose on their objects to get the moduli space to be small enough and nice enough to study. We’ll restrict ourselves to some fairly straightforward examples.
The general idea behind constructing a moduli space is that by specifying what the type of object you want to make a space out of, you’ve told me what the functor of points of the space is. In order for it to be some sort of “space” all that we need to do is figure out what space represents this functor.
Let’s start with making some of this more precise. Unfortunately, even the easiest cases have some strange technical points that can’t be avoided. Fix an algebraically closed field (unnecessary in general), . The moduli functor will be a functor from schemes over to sets. The set is the set of equivalence classes of our objects “over” (which will have a meaning depending on the type of object).
The coarse moduli space (if it exists) for this functor is a scheme (a highly restrictive condition we’ll remove if this series goes very far) over with the property that there is a natural transformation such that is bijective and satisfies a universal property: given any other natural transformation where there is a unique map of schemes so that the original map factors .
If our moduli functor has a coarse moduli space , then we define a universal family for the moduli problem to be a family of objects over (i.e. an element of ) with the property that for each closed point , the object over is the one corresponding to under the bijection .
How should we think of this? Well, our coarse moduli space is just a scheme whose closed points are the objects we are considering. This was our motivating definition. (Un)fortunately, schemes have a ton more structure than their closed points. This is what is meant by “coarse”. Other than being the universal space in some sense and actually having as its points the objects we want the rest of the scheme structure is basically irrelevant for a coarse moduli space. The universal family is essentially geometrically designating to each point of the object that it corresponds to.
We’ll end on an extremely simple example that will become somewhat annoying next time when we want a better notion of moduli space. Let be the functor that classifies smooth, projective, genus curves over up to isomorphism. We need to make precise our notion of a relative curve over some if we want a well-defined functor on all of .
Define to be the set of which are smooth and projective with geometric fibers curves of genus . This is exactly what one would expect a family of genus curves to be. The functor’s value on is just a single point because all genus curves (over an algebraically closed field) are isomorphic. This easily shows that is the coarse moduli space and there is a universal family which is to just put the one object over , i.e. is the universal family.