In mathematics, a moduli scheme is a moduli space that exists in the category of schemes developed by Alexander Grothendieck. Some important

moduli problem In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such s ...

s of algebraic geometry can be satisfactorily solved by means of

scheme theory In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...

alone, while others require some extension of the 'geometric object' concept (

algebraic space In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, wh ...

s, algebraic stacks of

Michael Artin Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.David Mumford David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded t ...

(see

geometric invariant theory In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in clas ...

) 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 In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets a ...

question, then apply a criterion that singles out the representable

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...

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 This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...

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 In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

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 In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in clas ...

'', one might want to have the fine version, but there is a technical issue (

level structure In the mathematical subfield of graph theory a level structure of an undirected graph is a partition of the vertices into subsets that have the same distance from a given root vertex.. Definition and construction Given a connected graph ''G'' ...

and other 'markings') that must be addressed to get a question with a chance of having such an answer.

Teruhisa Matsusaka (1926–2006) was a Japanese-born American mathematician, who specialized in algebraic geometry. Matsusaka received his Ph.D. in 1952 at Kyoto University; he was a member of the Brandeis Mathematics Department from 1961 until his retirement in 1 ...

proved a result, now known as

Matsusaka's big theorem In algebraic geometry, given an ample line bundle ''L'' on a compact complex manifold ''X'', Matsusaka's big theorem gives an integer ''m'', depending only on the Hilbert polynomial of ''L'', such that the tensor power ''L'n'' is very ample for ...

, establishing a necessary condition on a

for the existence of a coarse moduli scheme.

Examples

Mumford proved that if ''g'' > 1, there exists a coarse moduli scheme of smooth curves of genus ''g'', which is

quasi-projective In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in ...

. According to a recent survey by

János Kollár János Kollár (born 7 June 1956) is a Hungarian mathematician, specializing in algebraic geometry. Professional career Kollár began his studies at the Eötvös University in Budapest and later received his PhD at Brandeis University in 1984 ...

, it "has a rich and intriguing intrinsic geometry which is related to major questions in many branches of mathematics and theoretical physics." Braungardt has posed the question whether

Belyi's theorem In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve ''C'', defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified a ...

can be generalised to varieties of higher dimension over the

field of algebraic numbers An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the p ...

, with the formulation that they are generally birational to a finite

étale covering In mathematics, more specifically in algebra, the adjective étale refers to several closely related concepts: * Étale morphism ** Formally étale morphism * Étale cohomology * Étale topology * Étale fundamental group * Étale group scheme ...

of a moduli space of curves. Using the notion of

stable vector bundle In mathematics, a stable vector bundle is a ( holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable ...

, coarse moduli schemes for the vector bundles on any smooth

complex variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

have been shown to exist, and to be quasi-projective: the statement uses the concept of semistability. It is possible to identify the coarse moduli space of special

instanton bundle An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. M ...

s, in mathematical physics, with objects in the classical geometry of conics, in certain cases.

References

Notes

Moduli theory Representable functors {{algebraic-geometry-stub