A fine moduli space is a scheme representing this functor. That is to say, there is a universal family over and, for every family of objects over a scheme , there is a unique map which induces it by pulling back the universal family. This is usually impossible (for example, when there are extra automorphisms for some objects). If we are willing to enlarge the category in which we work, we can sometimes obtain a fine moduli space by working with stacks instead of schemes. Otherwise, the coarse moduli space is the scheme which best approximates the fine moduli space. By this, we mean that given a family of objects over a parameter space , there will be a unique map from to the coarse space , and is ``universal'' with respect to this property, which is to say that if there is also with this property, then the map factors uniquely as . Note that not every map to the coarse moduli space gives rise to a family. In particular there may not be a universal family of objects over itself.

Another requirement for a coarse moduli space is that its points should be in bijection with the objects being parametrized. I.e. algebraically-closed-field-valued points should be in bijection with the objects defined over that algebraically closed field.

Jeffrey Herschel Giansiracusa 2005-06-27