Moduli scheme: Difference between revisions

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{{short description|Moduli space in the Grothendieck category of schemes}}

{{unref|date=November 2008}}

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]]).

In [[algebraic geometry]], a '''moduli scheme''' is a [[moduli space]] that exists in the [[category of schemes]] developed by French mathematician [[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, and then apply a criterion that singles out the representable functors[[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 (algebraic geometry)|level structure]] and other 'markings') that first has tomust be addressed to get a question with a chance of having such an answer.

[[Teruhisa Matsusaka]] proved a result, now known as [[Matsusaka's big theorem]], establishing a necessary condition on a [[moduli problem]] for the existence of a coarse moduli scheme.<ref>{{cite book |first=S.J. |last=Kovács |chapter=Young person’s guide to moduli of higher dimensional varieties |chapter-url={{GBurl|gkyDAwAAQBAJ|p=711}} |title=Algebraic Geometry, Seattle 2005: 2005 Summer Research Institute, July 25-August 12, 2005, University of Washington |publisher=American Mathematical Society |date=2009 |isbn=978-0-8218-4703-9 |pages=711–743 }} p. [https://sites.math.washington.edu/~kovacs/2013/papers/Kovacs__YPG_to_moduli.pdf 13] of PDF</ref>

==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 |chapter=10.4 Coarse moduli schemes |chapter-url=https://books.google.com/books?id=ByTyBwAAQBAJ&pg=PA83|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|last1=Hauser|first1=Herwig|last2=Lipman|first2=Joseph|last3=Oort|first3=Frans|last4=Quirós|first4=Adolfo|date=2012-12-06|publisher=Birkhäuser|isbn=9783034883993|page=83|access-date=22 August 2017}}</ref> According to a recent survey by [[János Kollár]], it "has a rich and intriguing intrinsic geometry which is related to major questions in many branches of mathematics and [[theoretical physics]]."<ref>{{cite book |first=János |last=Kollár |chapter=1.1. Short History Of Moduli Problems: Theorem 1.14 |title=Families of varieties of general type |date=July 20, 2017 |pages=11 |url=https://web.math.princeton.edu/~kollar/book/modbook20170720.pdf }}</ref> Braungardt has posed the question whether [[Belyi's theorem]] can be generalised to varieties of higher dimension over the [[field of algebraic numbers]], with the formulation that they are generally birational to a finite [[étale covering]] of a moduli space of curves.<ref>{{cite book |first=W. |last=Goldring |chapter=Unifying themes suggested by Belyi’s Theorem |chapter-url= |title=Number Theory, Analysis and Geometry |publisher=Springer |date=2012 |isbn=978-1-4614-1260-1 |pages=181–214 See p. 203|doi=10.1007/978-1-4614-1260-1_10}}</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=Harris |first=Joe |title=Algebraic Geometry: Bowdoin 1985 |chapter=Curves and their moduli |chapter-url=https://books.google.com/books?id=50IECAAAQBAJ&pg=PA103 |year=1987|publisher=American Mathematical Soc.|isbn=978-0-8218-1480-2 |pages=99–143 See p. 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 |first1=W. |last1=Böhmer |first2=G. |last2=Trautman |chapter=Special Instanton bundles and Poncelet curves |pages=325–336 |doi=10.1007/BFb0078852 |editor1-last=Greuel|editor1-first=Gert-Martin|editor2-last=Trautmann|editor2-first=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|series=Lecture Notes in Mathematics |chapter-url=https://books.google.com/books?id=Ukh6CwAAQBAJ&pg=PA336 |date=2006 |volume=1273 |publisher=Springer|isbn=978-3-540-47851-5 }}</ref>

==References==

*{{eom|title = Moduli theory|id =m/m064510}}

==Notes==

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[[Category:Moduli theory]]

[[Category:Representable functors]]

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