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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.
==Examples==
Mumford proved that if ''g'' > 1, there exists a coarse moduli scheme of smooth curves of genus ''g'', which is [[quasi-projective]].<ref>{{cite book|last1=Hauser|first1=Herwig|last2=Lipman|first2=Joseph|last3=Oort|first3=Frans|coauthors=Adolfo Quirós|title=Resolution of Singularities: A research textbook in tribute to Oscar Zariski Based on the courses given at the Working Week in Obergurgl, Austria, September 7–14, 1997|url=http://books.google.com/books?id=ByTyBwAAQBAJ&pg=PA83|accessdate=22 August 2017|date=2012-12-06|publisher=Birkhäuser|isbn=9783034883993|page=83}}</ref> Using the notion of [[stable vector bundle]], coarse moduli schemes for the vector bundles on any smooth [[complex variety]] have been shown to exist, and to be quasi-projective: the statement uses the concept of [[semistable vector bundle|semistability]].<ref>{{cite book|last=Bloch|first=Spencer|title=Algebraic Geometry: Bowdoin 1985|url=http://books.google.com/books?id=50IECAAAQBAJ&pg=PA103|accessdate=22 August 2017|year=1987|publisher=American Mathematical Soc.|isbn=9780821814802|page=103}}</ref> It is possible to identify the coarse moduli space of special [[instanton bundle]]s, in mathematical physics, with objects in the classical geometry of conics, in certain cases.<ref>{{cite book|last1=Greuel|first1=Gert-Martin|last2=Trautmann|first2=Günther|title=Singularities, Representation of Algebras, and Vector Bundles: Proceedings of a Symposium held in Lambrecht/Pfalz, Fed.Rep. of Germany, Dec. 13-17, 1985|url=http://books.google.com/books?id=Ukh6CwAAQBAJ&pg=PA336|accessdate=22 August 2017|date=2006-11-15|publisher=Springer|isbn=9783540478515|page=336}}</ref>
==References==
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