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In [[mathematics]], a '''moduli scheme''' is a [[moduli space]] that exists in the [[category of schemes]] developed by [[Alexander Grothendieck]]. Some important [[moduli problem]]s of [[algebraic geometry]] can be satisfactorily solved by means of [[scheme theory]] alone, while others require some extension of the 'geometric object' concept ([[algebraic space]]s, [[algebraic stack]]s of [[Michael Artin]]).
==History==
Work of Grothendieck and [[David Mumford]] (see [[geometric invariant theory]]) opened up this area in the early 1960s. The more algebraic and abstract approach to moduli problems is to set them up as a [[representable functor]] question, then apply a criterion that singles out the representable [[functor]]s for schemes. When this programmatic approach works, the result is a ''fine moduli scheme''. Under the influence of more geometric ideas, it suffices to find a scheme that gives the correct [[geometric point]]s. This is more like the classical idea that the moduli problem is to express the algebraic structure naturally coming with a set (say of isomorphism classes of [[elliptic curve]]s). The result is then a ''coarse moduli scheme''. Its lack of refinement is, roughly speaking, that it doesn't guarantee for families of objects what is inherent in the fine moduli scheme. As Mumford pointed out in his book ''Geometric Invariant Theory'', one might want to have the fine version, but there is a technical issue (level structure and other 'markings') that must be addressed to get a question with a chance of having such an answer.
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