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Encyclopedia > Moduli space

In algebraic geometry, a moduli space is a parameter space for families of algebraic objects (such as algebraic varieties, morphisms, vector bundles). The use of the term modulus here for such a parameter space goes back to the same source as in modular form: a modular form in general is some kind of differential form (or tensor density, since the forms come with a 'weight') on a moduli space, that is, a space whose co-ordinates are the moduli. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... The factual accuracy of this article is disputed. ... Modular form - Wikipedia /**/ @import /skins-1. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... Tensor field - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...


In the case of elliptic curves, there is one modulus, so moduli spaces are algebraic curves. This is the quantity called k in Jacobi's elliptic function theory, which reduces elliptic integrals to a form involving In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ...

sqrt{(1-x^2)(1-k^2x^2),}.

This modulus of the elliptic integral therefore was probably the first modulus to be recognised.


The case of elliptic curves has been thoroughly studied, because of the great interest of the modular equations in this case. The j-invariant is a fundamental elliptic modular function. The moduli problem here is the prototype for moduli problems with level structure, meaning in this case some 'marking' of torsion groups of points on the curve. Each level structure gives rise to a subgroup of the modular group, and then its own modular curve. The j-invariant is called a Hauptmodul, traditionally, meaning that the modular curve has genus 0. There are other cases of genus 0, and other Hauptmoduls, which enter the remarkable monstrous moonshine theory. In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem. ... Real part of the j-invariant as a function of the nome q on the unit disk In mathematics, Kleins j-invariant, regarded as a function of a complex variable Ï„, is a modular function defined on the upper half-plane of complex numbers. ... In mathematics, the j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half plane of complex numbers with positive imaginary part. ... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ... In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ... In mathematics, a modular curve is a Riemann surface, or corresponding algebraic curve, constructed as HΓ where H is the upper half-plane in the complex numbers, and Γ is a Fuchsian group acting on H, with Γ a subgroup of the modular group of integral 2×2 matrices. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... In mathematics, monstrous moonshine is a term devised by John Conway and Simon P. Norton in 1979, used to describe the (then totally unexpected) connection between the monster group M and modular functions (particularly, the j function). ...


In general a curve of genus g has In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...

3g − 3

moduli, for g > 1. This number was known classically as the number of parameters on which a compact Riemann surface depends. As curves of higher genus admit automorphisms, the parameter space of smooth genus g curves is a (Deligne-Mumford) stack admitting a (quasiprojective) coarse moduli scheme. A proper moduli space can be achieved by adding nodal curves satisfying a technical stability condition. A closely related moduli space parametrizes stable morphisms to a smooth projective variety. Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...


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.


Moduli of vector bundles

There is also another major question, of determining moduli for vector bundles V on a fixed algebraic variety X. When X has dimension 1 and V is a line bundle, this is the theory of the Jacobian variety of a curve. In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. ...


Beginning with a paper of André Weil (who called them 'matrix divisors'), the vector bundles on X have been studied in relation to their moduli. In applications to physics, the number of moduli of vector bundles and the closely related problem of the number of moduli of principal G-bundles has been found to be significant in gauge theory. André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world, which deals with the fundamental constituents of the universe, the forces they exert on one another, and the results of these forces. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...


Constructions

Two general construction techniques for moduli spaces have been especially successful. The first is the method of geometric invariant theory, pioneered by David Mumford. The basic strategy is to simplify the classification problem by adding additional data in such a way that the original moduli space is the quotient of the new one by a reductive group action. To see how this might work, consider the problem of parametrizing curves of genus 2. Each such curve is hyperelliptic and therefore admits a unique degree 2 cover of P1 — unique, that is, up to composition with an element of the automorphism group PGL(2) of P1. So we begin by classifying double covers In mathematics, geometric invariant theory in algebraic geometry is a (technically complex) development building on nineteenth century invariant theory. ... David Bryant Mumford (born 11 June 1937) is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. ... In mathematics, a quotient is the end result of a division problem. ... In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...

XP1

with X of genus 2. Such a double cover is determined by its six ramification points. So now we are classifying six-element subsets of P1 (a comparatively easy problem). We have to pay a price, though, in dividing out by the PGL(2) action at the end. For instance, the moduli space of vector bundles (say over a curve, for simplicity) can be constructed using G.I.T. In mathematics, ramification is a geometric term used for branching out, in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. ...


The other general approach is primarily associated with Michael Artin. Here the idea is to start with any object of the kind to be classified and study its deformation theory. This means first constructing infinitesimal deformations, then appealing to prorepresentability theorems to put these together into an object over a formal base. Next an appeal to Grothendieck's formal existence theorem provides an object of the desired kind over a base which is a complete local ring. This object can be approximated via Artin's approximation theorem by an object defined over a finitely generated ring. The spectrum 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 an equivalence 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 an algebraic space (actually an algebraic stack if we are being careful) if not always a scheme. Michael Artin is an American mathematician, known for his contributions to algebraic geometry. ... In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ... Alexander Grothendieck (born March 28, 1928, Berlin) is one of the greatest mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis. ... In mathematics, the Artin approximation theorem is a fundamental result of Michael Artin in deformation theory which implies that formal power series with coefficients in a field k are well-approximated by the algebraic functions on k. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, an algebraic space is a generalization of the affine schemes of algebraic geometry introduced by Michael Artin for use in deformation theory. ... In mathematics, an algebraic stack in algebraic geometry is a special case of the concept of a stack, which is useful for working on moduli questions. ...


See also

For a physics-oriented description of moduli spaces, see moduli. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world, which deals with the fundamental constituents of the universe, the forces they exert on one another, and the results of these forces. ... In theoretical physics, moduli are scalar fields whose different values are equally good (each one such scalar field is called a modulus). ...


  Results from FactBites:
 
Moduli space - Wikipedia, the free encyclopedia (796 words)
In algebraic geometry, a moduli space is a parameter space for families of algebraic objects (such as algebraic varieties, morphisms, vector bundles).
There is also another major question, of determining moduli for vector bundles V on a fixed algebraic variety X. When X has dimension 1 and V is a line bundle, this is the theory of the Jacobian variety of a curve.
In applications to physics, the number of moduli of vector bundles and the closely related problem of the number of moduli of principal G-bundles has been found to be significant in gauge theory.
Moduli space - definition of Moduli space in Encyclopedia (776 words)
In algebraic geometry, the moduli problem is to describe the parameters on which algebraic varieties depend.
The use of the term modulus here for such a parameter goes back to the same source as in modular form: a modular form in general is some kind of differential form (or tensor density, since the forms come with a 'weight') on a moduli space, that is, a space whose co-ordinates are the moduli.
It agrees with the calculation of the dimension of the space of quadratic differentials on a fixed such Riemann surface, which is suggested by deformation theory combined with Serre duality.
  More results at FactBites »

 
 

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