linguistic:translation
Moduli stacks

It is frequently the case that interesting geometric objects come equipped with many natural automorphisms. 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 × can be made into a twisted family on the circle S by identifying L × {0} with L × {1} via a nontrivial automorphism. Now if a fine moduli space X existed, the map S → X should not be constant, but would have to be constant on any proper open set by triviality), one can still sometimes obtain a coarse moduli space. However, this approach is not ideal, as such spaces are not guaranteed to exist, they are frequently singular when they do exist, and miss details about some non-trivial families of objects they classify.

A more sophisticated approach is to enrich the classification by remembering the isomorphisms. More precisely, on any base B one can consider the category of families on B with only isomorphisms between families taken as morphisms. One then considers the fibred category which assigns to any space B the groupoid of families over B. The use of these categories fibred in groupoids to describe a moduli problem goes back to Grothendieck (1960/61). In general, they cannot be represented by schemes or even algebraic spaces, but in many cases, they have a natural structure of an algebraic stack.

Algebraic stacks and their use to analyze moduli problems appeared in Deligne-Mumford (1969) as a tool to prove the irreducibility of the (coarse) moduli space of curves of a given genus. The language of algebraic stacks essentially provides a systematic way to view the fibred category that constitutes the moduli problem as a "space", and the moduli stack of many moduli problems is better-behaved (such as smooth) than the corresponding coarse moduli space.