Inmathematics, in particularalgebraic geometry, amoduli space is a geometric space (usually ascheme or analgebraic stack) whose points represent algebro-geometric objects of some fixed kind, orisomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smoothalgebraic curves of a fixedgenus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces isformal moduli.Bernhard Riemann first used the term "moduli" in 1857.^[1]

Every point of a moduli space corresponds to a solution of a given geometric problem. Two different solutions correspond to the same point if they are isomorphic (that is, geometrically the same). A moduli space can be thought of as giving a universal space of parameters for the problem.

For example, consider the problem of finding all circles in the Euclidean plane up to congruence. Any circle can be described uniquely by giving three points, but many different sets of three points give the same circle: the correspondence is many-to-one. However, circles are uniquely parameterized by giving their center and radius: this is two real parameters and one positive real parameter. Since we are only interested in circles "up to congruence", we identify circles having different centers but the same radius, and so the radius alone suffices to parameterize the set of interest. The moduli space is, therefore, thepositive real numbers.

Moduli spaces often carry natural geometric and topological structures as well. In the example of circles, for instance, the moduli space is not just an abstract set, but the absolute value of the difference of the radii defines ametric for determining when two circles are "close". The geometric structure of moduli spaces locally tells us when two solutions of a geometric classification problem are "close", but generally moduli spaces also have a complicated global structure as well.

For example, consider how to describe the collection of lines in${\displaystyle {\mathbb{R}}^2}$ that intersect the origin. We want to assign to each line${\displaystyle L}$ of this family a quantity that can uniquely identify it—a modulus. An example of such a quantity is the positive angle${\displaystyle \theta (L)}$ with radians. The set of lines so parametrized is known as${\displaystyle {\mathbb{P}}^1(\mathbb{R})}$ and is called thereal projective line.

We can also describe the collection of lines in${\displaystyle {\mathbb{R}}^2}$ that intersect the origin by means of a topological construction. To wit: consider the unit circle${\displaystyle S^1\subseteq {\mathbb{R}}^2}$ and notice that for every point${\displaystyle s\in S^1}$ there is a unique line joining the origin and${\displaystyle s}$. However, this is also the line we'd get from looking at${\displaystyle -s}$, so we identify opposite points via an equivalence relation${\displaystyle \sim }$ to yield${\displaystyle {\mathbb{P}}^1\cong S^1/\sim }$ , with thequotient topology.

Thus, when we consider${\displaystyle {\mathbb{P}}^1(\mathbb{R})}$ as a moduli space of lines that intersect the origin in${\displaystyle {\mathbb{R}}^2}$, we capture the ways in which the members (lines in this case) of the family can modulate by continuously varying the angle${\displaystyle \theta }$.

Real projective space${\displaystyle {\mathbb{P}}^n(\mathbb{R})}$ is the moduli space of lines through the origin in${\displaystyle {\mathbb{R}}^{n+1}}$. Similarly,complex projective space${\displaystyle {\mathbb{P}}^n(\mathbb{C})}$ is the space of all complex lines through the origin in${\displaystyle {\mathbb{C}}^{n+1}}$.

More generally, theGrassmannian${\displaystyle \mathbf{G}\mathbf{r}(k,V)}$ of a vector space${\displaystyle V}$ is the moduli space of all${\displaystyle k}$-dimensional linear subspaces ofV.

Whenever there is an embedding of a scheme${\displaystyle X}$ into the universal projective space${\displaystyle {\mathbf{P}}_{\mathbb{Z}}^n}$,^[2][3] the embedding is given by a line bundle${\displaystyle \mathcal{L}\to X}$ and${\displaystyle n+1}$ sections${\displaystyle s_0,\dots ,s_n\in \mathrm{\Gamma }(X,\mathcal{L})}$ which don't all vanish at the same time. This means, given a point

${\displaystyle x:\text{Spec}(R)\to X}$

there is an associated point

${\displaystyle \hat{x}:\text{Spec}(R)\to {\mathbf{P}}_{\mathbb{Z}}^n}$

given by the compositions

${\displaystyle [s_0:\cdots :s_n]\circ x=[s_0(x):\cdots :s_n(x)]\in {\mathbf{P}}_{\mathbb{Z}}^n(R)}$

Then, two line bundles with sections are equivalent

${\displaystyle (\mathcal{L},(s_0,\dots ,s_n))\sim ({\mathcal{L}}^{\prime },(s_0^{\prime },\dots ,s_n^{\prime }))}$

iff there is an isomorphism${\displaystyle \varphi :\mathcal{L}\to {\mathcal{L}}^{\prime }}$ such that${\displaystyle \varphi (s_i)=s_i^{\prime }}$. This means the associated moduli functor

${\displaystyle {\mathbf{P}}_{\mathbb{Z}}^n:\text{Sch}\to \text{Sets}}$

sends a scheme${\displaystyle X}$ to the set

${\displaystyle {\mathbf{P}}_{\mathbb{Z}}^n(X)=\left\{(\mathcal{L},s_0,\dots ,s_n):\begin{array}{c}\mathcal{L}\to X\text{ is a line bundle}\\ s_0,\dots ,s_n\in \mathrm{\Gamma }(X,\mathcal{L})\\ \text{ form a basis of global sections}\end{array}\right\}/\sim }$

Showing this is true can be done by running through a series of tautologies: any projective embedding${\displaystyle i:X\to {\mathbb{P}}_{\mathbb{Z}}^n}$ gives the globally generated sheaf${\displaystyle i^{\ast }{\mathcal{O}}_{{\mathbf{P}}_{\mathbb{Z}}^n}(1)}$ with sections${\displaystyle i^{\ast }{x}_0,\dots ,i^{\ast }{x}_n}$. Conversely, given an ample line bundle${\displaystyle \mathcal{L}\to X}$ globally generated by${\displaystyle n+1}$ sections gives an embedding as above.

TheChow varietyChow(d,P³) is a projective algebraic variety which parametrizes degreed curves inP³. It is constructed as follows. LetC be a curve of degreed inP³, then consider all the lines inP³ that intersect the curveC. This is a degreeddivisorD_C inG(2, 4), the Grassmannian of lines inP³. WhenC varies, by associatingC toD_C, we obtain a parameter space of degreed curves as a subset of the space of degreed divisors of the Grassmannian:Chow(d,P³).

TheHilbert schemeHilb(X) is a moduli scheme. Every closed point ofHilb(X) corresponds to a closed subscheme of a fixed schemeX, and every closed subscheme is represented by such a point. A simple example of a Hilbert scheme is the Hilbert scheme parameterizing degree${\displaystyle d}$ hypersurfaces of projective space${\displaystyle {\mathbb{P}}^n}$. This is given by the projective bundle

${\displaystyle {\mathcal{H}\mathcal{i}\mathcal{l}\mathcal{b}}_d({\mathbb{P}}^n)=\mathbb{P}(\mathrm{\Gamma }(\mathcal{O}(d)))}$

with universal family given by

${\displaystyle \mathcal{U}=\{(V(f),f):f\in \mathrm{\Gamma }(\mathcal{O}(d))\}}$

where${\displaystyle V(f)}$ is the associated projective scheme for the degree${\displaystyle d}$ homogeneous polynomial${\displaystyle f}$.

There are several related notions of things we could call moduli spaces. Each of these definitions formalizes a different notion of what it means for the points of spaceM to represent geometric objects.

This is the standard concept. Heuristically, if we have a spaceM for which each pointm ∊M corresponds to an algebro-geometric objectU_m, then we can assemble these objects into atautological bundleU overM. (For example, the GrassmannianG(k,V) carries a rankk bundle whose fiber at any point [L] ∊G(k,V) is simply the linear subspaceL ⊂V.)M is called abase space of the familyU. We say that such a family isuniversal if any family of algebro-geometric objectsT over any base spaceB is thepullback ofU along a unique mapB →M. A fine moduli space is a spaceM which is the base of a universal family.

More precisely, suppose that we have a functorF from schemes to sets, which assigns to a schemeB the set of all suitable families of objects with baseB. A spaceM is afine moduli space for the functorF ifMrepresentsF, i.e., there is a natural isomorphismτ :F →Hom(−,M), whereHom(−,M) is the functor of points. This implies thatM carries a universal family; this family is the family onM corresponding to the identity map1_M ∊Hom(M,M).

Fine moduli spaces are desirable, but they do not always exist and are frequently difficult to construct, so mathematicians sometimes use a weaker notion, the idea of a coarse moduli space. A spaceM is acoarse moduli space for the functorF if there exists a natural transformation τ :F →Hom(−,M) and τ is universal among such natural transformations. More concretely,M is a coarse moduli space forF if any familyT over a baseB gives rise to a map φ_T :B →M and any two objectsV andW (regarded as families over a point) correspond to the same point ofM if and only ifV andW are isomorphic. Thus,M is a space which has a point for every object that could appear in a family, and whose geometry reflects the ways objects can vary in families. Note, however, that a coarse moduli space does not necessarily carry any family of appropriate objects, let alone a universal one.

In other words, a fine moduli space includesboth a base spaceM and universal familyU →M, while a coarse moduli space only has the base spaceM.

It is frequently the case that interesting geometric objects come equipped with many naturalautomorphisms. This in particular makes the existence of a fine moduli space impossible (intuitively, the idea is that ifL is some geometric object, the trivial familyL × [0,1] can be made into a twisted family on the circleS¹ by identifyingL × {0} withL × {1} via a nontrivial automorphism. Now if a fine moduli spaceX existed, the mapS¹ →X should not be constant, but would have to be constant on any proper open set by triviality), one can still sometimes obtain a coarse moduli space. However, this approach is not ideal, as such spaces are not guaranteed to exist, they are frequently singular when they do exist, and miss details about some non-trivial families of objects they classify.

A more sophisticated approach is to enrich the classification by remembering the isomorphisms. More precisely, on any baseB one can consider the category of families onB with only isomorphisms between families taken as morphisms. One then considers thefibred category which assigns to any spaceB the groupoid of families overB. The use of thesecategories fibred in groupoids to describe a moduli problem goes back to Grothendieck (1960/61). In general, they cannot be represented by schemes or evenalgebraic spaces, but in many cases, they have a natural structure of analgebraic stack.

Algebraic stacks and their use to analyze moduli problems appeared in Deligne-Mumford (1969) as a tool to prove the irreducibility of the (coarse)moduli space of curves of a given genus. The language of algebraic stacks essentially provides a systematic way to view the fibred category that constitutes the moduli problem as a "space", and themoduli stack of many moduli problems is better-behaved (such as smooth) than the corresponding coarse moduli space.

The moduli stack${\displaystyle {\mathcal{M}}_g}$ classifies families of smooth projective curves of genusg, together with their isomorphisms. Wheng > 1, this stack may be compactified by adding new "boundary" points which correspond to stable nodal curves (together with their isomorphisms). A curve is stable if it has only a finite group of automorphisms. The resulting stack is denoted${\displaystyle {\overline{\mathcal{M}}}_g}$. Both moduli stacks carry universal families of curves. One can also define coarse moduli spaces representing isomorphism classes of smooth or stable curves. These coarse moduli spaces were actually studied before the notion of moduli stack was invented. In fact, the idea of a moduli stack was invented by Deligne and Mumford in an attempt to prove the projectivity of the coarse moduli spaces. In recent years, it has become apparent that the stack of curves is actually the more fundamental object.

Both stacks above have dimension 3g−3; hence a stable nodal curve can be completely specified by choosing the values of 3g−3 parameters, wheng > 1. In lower genus, one must account for the presence of smooth families of automorphisms, by subtracting their number. There is exactly one complex curve of genus zero, the Riemann sphere, and its group of isomorphisms is PGL(2). Hence, the dimension of${\displaystyle {\mathcal{M}}_0}$ is

dim(space of genus zero curves) − dim(group of automorphisms) = 0 − dim(PGL(2)) = −3.

Likewise, in genus 1, there is a one-dimensional space of curves, but every such curve has a one-dimensional group of automorphisms. Hence, the stack${\displaystyle {\mathcal{M}}_1}$ has dimension 0. The coarse moduli spaces have dimension 3g−3 as the stacks wheng > 1 because the curves with genus g > 1 have only a finite group as its automorphism i.e. dim(a group of automorphisms) = 0. Eventually, in genus zero, the coarse moduli space has dimension zero, and in genus one, it has dimension one.

One can also enrich the problem by considering the moduli stack of genusg nodal curves withn marked points. Such marked curves are said to be stable if the subgroup of curve automorphisms which fix the marked points is finite. The resulting moduli stacks of smooth (or stable) genusg curves withn-marked points are denoted${\displaystyle {\mathcal{M}}_{g,n}}$ (or${\displaystyle {\overline{\mathcal{M}}}_{g,n}}$), and have dimension 3g − 3 + n.

A case of particular interest is the moduli stack${\displaystyle {\overline{\mathcal{M}}}_{1,1}}$ of genus 1 curves with one marked point. This is the stack ofelliptic curves, and is the natural home of the much studiedmodular forms, which are meromorphic sections of bundles on this stack.

In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. For instance, the higher-dimensional analogue of the moduli space of elliptic curves discussed above is the moduli space of abelian varieties, such as theSiegel modular variety. This is the problem underlyingSiegel modular form theory. See alsoShimura variety.

Using techniques arising out of the minimal model program, moduli spaces of varieties of general type were constructed byJános Kollár andNicholas Shepherd-Barron, now known as KSB moduli spaces.^[4]

It has been known that a well-behaved moduli theory can not be established for all Fano varieties. Led byChenyang Xu, the construction of moduli spaces ofFano varieties has been achieved by restricting to a special class ofK-stable varieties. More precisely, there exists a projective scheme which parametrizes K-polystable Fano varieties, which is the good moduli space of the moduli stack parametrizing K-semistable Fano varieties.

The construction of moduli spaces of Calabi-Yau varieties is an important open problem, and only special cases such as moduli spaces ofK3 surfaces orAbelian varieties are understood.^[5]

Another important moduli problem is to understand the geometry of (various substacks of) the moduli stack Vect_n(X) of ranknvector bundles on a fixedalgebraic varietyX.^[6] This stack has been most studied whenX is one-dimensional, and especially when n equals one. In this case, the coarse moduli space is thePicard scheme, which like the moduli space of curves, was studied before stacks were invented. When the bundles have rank 1 and degree zero, the study of coarse moduli space is the study of theJacobian variety.

In applications tophysics, the number of moduli of vector bundles and the closely related problem of the number of moduli ofprincipal G-bundles has been found to be significant ingauge theory.^{[citation needed]}

Simple geodesics and Weil-Peterssonvolumes of moduli spaces of bordered Riemann surfaces.

The modern formulation of moduli problems and definition of moduli spaces in terms of the moduli functors (or more generally thecategories fibred ingroupoids), and spaces (almost) representing them, dates back to Grothendieck (1960/61), in which he described the general framework, approaches, and main problems usingTeichmüller spaces in complex analytical geometry as an example. The talks, in particular, describe the general method of constructing moduli spaces by firstrigidifying the moduli problem under consideration.

More precisely, the existence of non-trivial automorphisms of the objects being classified makes it impossible to have a fine moduli space. However, it is often possible to consider a modified moduli problem of classifying the original objects together with additional data, chosen in such a way that the identity is the only automorphism respecting also the additional data. With a suitable choice of the rigidifying data, the modified moduli problem will have a (fine) moduli spaceT, often described as a subscheme of a suitableHilbert scheme orQuot scheme. The rigidifying data is moreover chosen so that it corresponds to a principal bundle with an algebraic structure groupG. Thus one can move back from the rigidified problem to the original by taking quotient by the action ofG, and the problem of constructing the moduli space becomes that of finding a scheme (or more general space) that is (in a suitably strong sense) the quotientT/G ofT by the action ofG. The last problem, in general, does not admit a solution; however, it is addressed by the groundbreakinggeometric invariant theory (GIT), developed byDavid Mumford in 1965, which shows that under suitable conditions the quotient indeed exists.

To see how this might work, consider the problem of parametrizing smooth curves of the genusg > 2. A smooth curve together with acomplete linear system of degreed > 2g is equivalent to a closed one dimensional subscheme of the projective spaceP^d−g. Consequently, the moduli space of smooth curves and linear systems (satisfying certain criteria) may be embedded in the Hilbert scheme of a sufficiently high-dimensional projective space. This locusH in the Hilbert scheme has an action of PGL(n) which mixes the elements of the linear system; consequently, the moduli space of smooth curves is then recovered as the quotient ofH by the projective general linear group.

Another general approach is primarily associated withMichael Artin. Here the idea is to start with an object of the kind to be classified and study itsdeformation theory. This means first constructinginfinitesimal deformations, then appealing toprorepresentability theorems to put these together into an object over aformal base. Next, an appeal toGrothendieck'sformal existence theorem provides an object of the desired kind over a base which is a complete local ring. This object can be approximated viaArtin's approximation theorem by an object defined over a finitely generated ring. Thespectrum of this latter ring can then be viewed as giving a kind of coordinate chart on the desired moduli space. By gluing together enough of these charts, we can cover the space, but the map from our union of spectra to the moduli space will, in general, be many to one. We, therefore, define anequivalence relation on the former; essentially, two points are equivalent if the objects over each are isomorphic. This gives a scheme and an equivalence relation, which is enough to define analgebraic space (actually analgebraic stack if we are being careful) if not always a scheme.

The term moduli space is sometimes used inphysics to refer specifically to the moduli space ofvacuum expectation values of a set ofscalar fields, or to the moduli space of possiblestring backgrounds.

Moduli spaces also appear in physics intopological field theory, where one can useFeynman path integrals to compute theintersection numbers of various algebraic moduli spaces.

• Hilbert scheme
• Quot scheme
• Deformation theory
• GIT quotient
• Artin's criterion, general criterion for constructing moduli spaces as algebraic stacks from moduli functors

• Moduli of algebraic curves
• Moduli stack of elliptic curves
• Moduli spaces of K-stable Fano varieties
• Modular curve
• Picard functor
• Moduli of semistable sheaves on a curve
• Seiberg–Witten moduli space
• Kontsevich moduli space
• Moduli of semistable sheaves

• Moduli theory
• Moduli stacks in P-adic modular forms and Langlands program

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• Mumford, David,Geometric invariant theory.Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34 Springer-Verlag, Berlin-New York 1965 vi+145 ppMR 0214602
• Mumford, David; Fogarty, J.; Kirwan, F.Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994. xiv+292 pp.MR 1304906ISBN 3-540-56963-4

• Deligne, Pierre; Mumford, David (1969)."The irreducibility of the space of curves of given genus"(PDF).Publications Mathématiques de l'IHÉS.36 (1):75–109.CiteSeerX 10.1.1.589.288.doi:10.1007/bf02684599.
• Faltings, Gerd; Chai, Ching-Li (1990).Degeneration of Abelian Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 22. With an appendix by David Mumford. Berlin: Springer-Verlag.doi:10.1007/978-3-662-02632-8.ISBN 978-3-540-52015-3.MR 1083353.
• Katz, Nicholas M;Mazur, Barry (1985).Arithmetic Moduli of Elliptic Curves. Annals of Mathematics Studies. Vol. 108.Princeton University Press.ISBN 978-0-691-08352-0.MR 0772569.

• Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I, IRMA Lectures in Mathematics and Theoretical Physics, 11, European Mathematical Society (EMS), Zürich,doi:10.4171/029,ISBN 978-3-03719-029-6,MR 2284826
• Papadopoulos, Athanase, ed. (2009), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematics and Theoretical Physics, 13, European Mathematical Society (EMS), Zürich,doi:10.4171/055,ISBN 978-3-03719-055-5,MR 2524085
• Papadopoulos, Athanase, ed. (2012), Handbook of Teichmüller theory. Vol. III, IRMA Lectures in Mathematics and Theoretical Physics, 17, European Mathematical Society (EMS), Zürich,doi:10.4171/103,ISBN 978-3-03719-103-3.

• Harris, Joe; Morrison, Ian (1998).Moduli of Curves. Graduate Texts in Mathematics. Vol. 187. New York:Springer Verlag.doi:10.1007/b98867.ISBN 978-0-387-98429-2.MR 1631825.

• Viehweg, Eckart (1995).Quasi-Projective Moduli for Polarized Manifolds(PDF). Springer Verlag.ISBN 978-3-540-59255-6.

• Simpson, Carlos (1994)."Moduli of representations of the fundamental group of a smooth projective variety I"(PDF).Publications Mathématiques de l'IHÉS.79 (1):47–129.doi:10.1007/bf02698887.
• Maryam Mirzakhani (2007)"Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces"Inventiones Mathematicae

• Lurie, J. (2011). "Moduli Problems for Ring Spectra".Proceedings of the International Congress of Mathematicians 2010 (ICM 2010). pp. 1099–1125.doi:10.1142/9789814324359_0088.ISBN 978-981-4324-30-4.

• Moduli theory
• Invariant theory

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