Editing Moduli space (section)

===Fine moduli space===
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<sub>m</sub>'', then we can assemble these objects into a [[tautological bundle|tautological]] family ''U'' over ''M''.  (For example, the Grassmannian '''G'''(''k'', ''V'') carries a rank ''k'' bundle whose fiber at any point [''L''] ∊ '''G'''(''k'', ''V'') is simply the linear subspace ''L'' ⊂ ''V''.) ''M'' is called a '''base space''' of the family ''U''. We say that [[universal bundle | such a family]] is '''universal''' if any family of algebro-geometric objects ''T'' over any base space ''B'' is the [[Pullback (category theory)|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'' [[representable functor|represents]] ''F'', i.e., there is a natural isomorphism
τ : ''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'''<sub>''M''</sub> ∊ '''Hom'''(''M'', ''M'').