structural:expansion
Fine moduli spaces

This is the standard concept. Heuristically, if we have a space M for which each point m ∊ M corresponds to an algebro-geometric object U, 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 ∊ G (k, V) is simply the linear subspace L ⊂ V.) M is called a base space of the family U. We say that such a family is universal if any family of algebro-geometric objects T over any base space B is the pullback of U along a unique map B → M. A fine moduli space is a space M which is the base of a universal family.

More precisely, suppose that we have a functor F from schemes to sets, which assigns to a scheme B the set of all suitable families of objects with base B. A space M is a fine moduli space for the functor F if M represents F, i.e., there is a natural isomorphismτ : F → Hom (−, M), where Hom (−, M) is the functor of points. This implies that M carries a universal family; this family is the family on M corresponding to the identity map 1 ∊ Hom (M, M).