View source for Moduli theory

{{TEX|done}}
A theory studying continuous families of objects in algebraic geometry.

Let $  A $ 
be a class of objects in algebraic geometry (varieties, schemes, vector bundles, etc.) on which an equivalence relation $  R $ 
has been given. The fundamental classification problem (the description of the set of classes $  A / R $ ) 
has the following two parts: 1) the description of discrete invariants, which usually allow a partition of $  A / R $ 
into a countable number of subsets, the objects of which already continuously depend on parameters; 2) the assignment and study of algebro-geometric structures on the parameter sets. The second part forms the matter of moduli theory.

Moduli theory arose in the study of elliptic functions: There is a continuous family of different fields of elliptic functions (or of their models — isomorphic elliptic curves over $  \mathbf C $ ), 
parametrized by the complex numbers. B. Riemann, who introduced the term "moduli" , showed that an algebraic function field over $  \mathbf C $ (
or their models, compact Riemann surfaces) of genus $  g \geq 2 $ 
depends on $  3 g - 3 $ 
continuous complex parameters — the moduli.

==Basic concepts in moduli theory.==
Let $  S $ 
be a [[Scheme|scheme]] (a complex or algebraic space). A family of objects parametrized by the scheme $  S $ (
or, as is often said, "scheme over Sover S" or "scheme with basis Swith basis S" ) is a set of objects$$ 
\{ {X _{s}} : {s \in S ,  X _{s} \in A} \}
,
 $$ 
equipped with an additional structure compatible with the structure of the base $  S $ . 
This structure, in each concrete case, is given explicitly. A functor of families is a contravariant functor $  {\mathcal M} $ 
from the category of the schemes (or spaces) into the category of sets defined as follows: $  {\mathcal M} (S) $ 
is the set of classes of isomorphic families over $  S $ . 
To every morphism $  f : \  T \rightarrow S $ 
is associated a mapping $  f ^ {\  *} : \  {\mathcal M} (S) \rightarrow {\mathcal M} (T) $ 
which assigns to a family over $  S $ 
the pullback, or induced, family over $  T $ .


Let $  M $ 
be an object in the category of schemes (complex or algebraic spaces) and let $  h _{M} $ 
be a functor of the points in this category, that is, $  h _{M} =  \mathop{\rm Hom}\nolimits ( S ,\  M ) $ . 
If the functor of families $  {\mathcal M} $ 
is representable, that is, $  {\mathcal M} = h _{M} $ 
for some $  M $ , 
then there exists a universal family with base $  {\mathcal M} $ , 
and $  M $ 
is called a fine moduli scheme (respectively, fine complex moduli space or fine algebraic moduli space). The functor $  {\mathcal M} $ 
is representable in very few cases. Therefore the notion of a coarse moduli scheme was introduced. $  M $ 
is called a coarse moduli scheme if there is a morphism of functors $  \phi : \  {\mathcal M} \rightarrow h _{M} $ 
with the properties: a) if $  S =  \mathop{\rm Spec}\nolimits \  K =  \mathop{\rm pt}\nolimits $ 
is one point (where $  K $ 
is an algebraically closed field), then the mapping $  \phi : \  {\mathcal M} (  \mathop{\rm pt}\nolimits ) \rightarrow h _{M} (  \mathop{\rm pt}\nolimits ) $ 
is bijective; in other words, the set of geometric points of the scheme $  M $ 
is in a natural one-to-one correspondence with the set of equivalence classes of parametrized objects; and b) for each scheme $  N $ 
and morphism of functors $  \psi : \  {\mathcal M} \rightarrow h _{N} $ 
there is a unique morphism $  \chi : \  h _{M} \rightarrow h _{N} $ 
such that $  \psi = \chi \circ \phi $ . 
Coarse schemes of complex and algebraic moduli spaces are similarly defined.

Although a coarse moduli scheme uniquely parametrizes the class of objects defined by given discrete invariants, the natural family over it (in contrast to the family over a fine moduli scheme) does not have the strong universality property. A coarse moduli scheme (space) already exists in a fairly large number of cases.

Examples. 1) Moduli of algebraic curves. Let $  A / R = \mathfrak M _{g} $ (
respectively $  {\overline{\mathfrak M}  _{g}} $ ) 
be the set of classes of isomorphic projective non-singular curves (respectively, stable curves) of genus $  g \geq 2 $ 
over an algebraically closed field $  K $ . 
A family over $  S $ 
is a smooth (flat) proper morphism of schemes $  f : \  X \rightarrow S $ 
whose fibres are smooth (stable) curves of genus $  g $ . 
Then there is a coarse (but not a fine) moduli scheme $  M _{g} $ (
respectively $  \overline{M}  _{g} $ ), 
which is a quasi-projective (projective), irreducible, normal variety over $  K $ (
see [[#References|[3]]], [[#References|[5]]], [[#References|[6]]]).

2) Moduli of algebraic curves with level $  n $ 
structure (with Jacobian rigidity). Let $  f : \  X \rightarrow S $ 
be a smooth family of projective curves (respectively, a flat family of stable curves) of genus $  g \geq 1 $ , 
let $  n $ 
be an integer invertible on $  S $ , 
and let $  R ^{1} f _{*} ( \mathbf Z / n \mathbf Z ) $ 
be the first direct image of the constant sheaf $  \mathbf Z / n \mathbf Z $ 
in the étale topology. Then $  R ^{1} f _{*} ( \mathbf Z / n \mathbf Z ) $ 
is locally free, has rank $  2 g $ 
and is equipped with a locally non-degenerate symplectic form with values in $  \mathbf Z / n \mathbf Z $ , 
up to an invertible element in $  \mathbf Z / n \mathbf Z $ . 
A Jacobian structure of level $  n $ 
on $  X $ 
is an assignment of a symplectic isomorphism$$ 
R ^{1} f _{*} ( \mathbf Z / n \mathbf Z )    \stackrel{ \sim } \rightarrow    
( \mathbf Z / n \mathbf Z ) ^{2g} .
 $$ 
Let $  {\mathcal M} _{g},n $ (
respectively, $  \overline{ {\mathcal M} }  _{g},n $ ) 
be the functor of families of smooth (stable) curves of genus $  g \geq 2 $ 
with a Jacobian rigidity of level $  n $ . 
Then for $  n \geq 3 $ 
the functor $  {\mathcal M} _{g},n $ (
respectively, $  \overline{ {\mathcal M} }  _{g},n $ ) 
is represented by a quasi-projective (projective) scheme $  M _{g},n $ (
respectively, $  \overline{M}  _{g},n $ ) 
over $   \mathop{\rm Spec}\nolimits \  \mathbf Z [ 1 / n ,\  \xi _{n} ] $ , 
where $  \xi _{n} $ 
is the inverse image of an $  n $ -
th root of unity, that is, there is a fine moduli scheme $  M _{g},n $ (
respectively, $  \overline{M}  _{g},n $ ) 
for the smooth (stable) curves of genus $  g \geq 2 $ 
over a field of characteristic coprime with $  n $ , 
equipped with a Jacobian rigidity of level $  n $ . 
For sufficiently large $  n $ 
the scheme $  M _{g},n $ 
is smooth [[#References|[5]]].

3) Polarized algebraic varieties. A polarized family is a pair $  ( X / S ,\  \mathfrak P / S ) $ , 
where $  X / S $ 
is a smooth family of varieties, i.e. a smooth proper morphism $  f : \  X \rightarrow S $ , 
and $  \mathfrak P / S $ 
is the class of the relatively ample invertible sheaf $  {\mathcal L} _{X}/S $ 
in $   \mathop{\rm Hom}\nolimits ( S ,\   \mathop{\rm Pic}\nolimits \  X / S ) $ 
modulo $   \mathop{\rm Hom}\nolimits ( S ,\   \mathop{\rm Pic}\nolimits ^{0} \  X / S ) $ , 
where $   \mathop{\rm Pic}\nolimits \  X / S $ 
is the relative Picard scheme and $   \mathop{\rm Pic}\nolimits ^{0} \  X / S $ 
is the connected component of its zero section. In this case a functor of the polarized families $  {\mathcal M} _{h} $ , 
with a given Hilbert polynomial $  h $ , 
is constructed. Without additional restrictions this functor is not representable. The existence of a coarse moduli space is known (1989) only in individual cases.

For polarized algebraic varieties the idea of rigidity of level $  n $ 
also exists.

4) Vector bundles. There are also results on moduli spaces for vector bundles of rank $  n $ 
over an algebraic variety $  X $ . 
In this case a family over $  S $ 
is a vector bundle over $  X \times S $ . 
Cf. [[#References|[7]]], [[#References|[10]]]–[[#References|[14]]] for a description of results and more detail.

==Local and global theory.==
The local theory arose as the theory of deformations of complex structures (see [[Deformation|Deformation]] 1) and 2)). The fundamental methods of the global theory are those of the theory of representable functors and geometric invariant theory, the theory of algebraic stacks, and the algebraization of formal moduli.

The method of construction of a global moduli space goes back to the classical theory of invariants (cf. [[Invariants, theory of|Invariants, theory of]]). It is as follows. A sufficiently large family $  X \rightarrow H $ 
is constructed which contains representatives of all equivalence classes of the objects in questions, and so that the equivalence relation on $  H $ 
reduces to the action of an algebraic group $  G $ . 
Then the theory of actions of algebraic groups on algebraic varieties (schemes, spaces) is exploited with the aim of clarifying conditions for the existence of the quotient $  H / G $ 
in the corresponding category. The basic tool in the construction of the family $  X \rightarrow H $ 
is the theory of Hilbert schemes (cf. [[Hilbert scheme|Hilbert scheme]]). In such an approach the difficulty in constructing the family $  X \rightarrow H $ 
is reduced to the problem on a simultaneous immersion of the objects in question into a projective space. An important result on the possibility of such a simultaneous immersion is Matsusaka's theorem. Then the difficult problem remains of the existence of the quotient $  H / G $ . 
Here one has the notions of categorical and geometric quotients. The construction of a coarse moduli space reduces to the problem of the existence of geometric quotients; here the idea of stability of points, corresponding to the idea of orbits in general position, is used. Results concerning actions of reductive groups on algebraic varieties over fields of characteristic $  0 $ 
have been extended to the case of fields of characteristic $  p > 0 $ .


Another approach to global moduli theory is the method of algebraic stacks, that is, a method of globalization of local deformation theory. The first step in the investigation of the representability of a global functor of families in this approach is the establishment of the algebraizability of a formal versal deformation for each object $  X _{0} $ . 
The difficulty in the construction of a global moduli space is that not every factorization of the base of the family with respect to an equivalence relation is a separated space. In such cases the object representing the functor $  {\mathcal M} $ 
is replaced by an algebraic stack, the study of the properties of which gives some information on the moduli space.

One of the approaches to global moduli theory over $  \mathbf C $ 
is the theory of period mappings (cf. [[Period mapping|Period mapping]]). The fundamental object here is the classifying space $  D $ 
of polarized Hodge structures (cf. [[Hodge structure|Hodge structure]]) of weight $  k $ 
for given Hodge numbers. For a family $  X \rightarrow S $ 
of polarized algebraic varieties over $  \mathbf C $ 
the periods define a mapping of $  S $ 
onto the corresponding classifying space $  D $ 
of Hodge structures. The moduli problem reduces to the study of conditions for the period mapping to be bijective. The presence of (global) injectivity for the period mapping is the so-called local-global Torelli problem. Along this route the existence of coarse moduli spaces has been proved for curves, Abelian varieties and $  K 3 $ -
surfaces.

The compactification problem for a moduli variety $  M $ 
is that of finding a natural and complete (projective or compact, in the theory over the field $  \mathbf C $ ) 
variety $  \overline{M}  $ 
containing $  M $ 
as a dense open subset, and also the description and geometric interpretation of the boundary $  \overline{M}  \setminus M $ . 
In example 1) the natural compactification of the coarse moduli variety $  M _{g} $ 
of curves of genus $  g \geq 2 $ 
is the projective moduli variety $  \overline{M}  _{g} $ 
of stable curves. For polarized Abelian varieties over $  \mathbf C $ 
several means for compactifying moduli varieties are known.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M. Artin, "Algebraization of formal moduli 1" , ''Global analysis'' , Univ. Tokyo Press (1969) pp. 21–71 {{MR|260746}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Artin, "Versal deformations and algebraic stacks" ''Invent. Math.'' , '''27''' (1974) pp. 165–189 {{MR|0399094}} {{ZBL|0317.14001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P. Deligne, D. Mumford, "The irreducibility of the space of curves of given genus" ''Publ. Math. IHES'' , '''36''' (1969) pp. 75–109 {{MR|0262240}} {{ZBL|0181.48803}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J. Dieudonné, J.B. Carrell, "Invariant theory, old and new" , Acad. Press (1971) {{MR|0279102}} {{ZBL|0258.14011}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> D. Mumford, "Geometric invariant theory" , Springer (1965) {{MR|0214602}} {{ZBL|0147.39304}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> D. Mumford, "Stability of projective varieties" ''l'Enseign. Math. (2)'' , '''23''' : 1–2 (1977) pp. 39–110 {{MR|0450273}} {{MR|0450272}} {{ZBL|0497.14004}} {{ZBL|0376.14007}} {{ZBL|0363.14003}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> D. Gieseker, "Global moduli for varieties of general type" ''Invent. Math.'' , '''43''' (1977) pp. 233–282 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> D. Gieseker, "On the moduli of vector bundles on an algebraic surface" ''Ann. of Math.'' , '''106''' (1977) pp. 45–60 {{MR|0466475}} {{ZBL|0381.14003}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> T. Matsusaka, "Polarized varieties with a given Hilbert polynomial" ''Amer. J. Math.'' , '''94''' : 4 (1972) pp. 1027–1077 {{MR|0337960}} {{ZBL|0256.14004}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> P.E. Newstead, "Lectures on introduction to moduli problems and orbit spaces" , Springer (1978) {{MR|546290}} {{ZBL|0411.14003}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> C. Okonek, M. Schneider, H. Spindler, "Vector bundles on complex projective spaces" , Birkhäuser (1980) {{MR|0561910}} {{ZBL|0438.32016}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> H. Popp, "Moduli theory and classification theory of algebraic varieties" , Springer (1977) {{MR|0466143}} {{ZBL|0359.14005}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> C.S. Seshadri, "Spaces of unitary vector bundles on a compact Riemann surface" ''Ann. of Math.'' , '''85''' (1967) pp. 302–336 {{MR|0233371}} {{ZBL|}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> A.N. Tyurin, "The geometry of moduli of vector bundles" ''Russian Math. Surveys'' , '''29''' : 6 (1974) pp. 57–88 ''Uspekhi Mat. Nauk'' , '''29''' : 6 (1974) pp. 59–88 {{MR|}} {{ZBL|0325.14016}} </TD></TR></table>



====Comments====
Much progress has been made recently in the study of the moduli spaces of algebraic curves and Abelian varieties; see the appendices of [[#References|[a2]]]. Among the most important ones are the question of the compactification of $  M _{g} $ (
see the appendix to Chapt. 5 in [[#References|[a1]]], which is an enlarged edition of [[#References|[5]]]) and the proof that $  M _{g} $ 
is of general type for $  g \geq 24 $ [[#
References|[a2]]].

G. Faltings has constructed a compactification of the moduli space $  A _{g} $ 
of principally-polarized Abelian varieties over $  \mathbf Z $ , 
see [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Mumford, J. Fogarty, "Geometric invariant theory" , Springer (1982) {{MR|0719371}} {{ZBL|0504.14008}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Harris, "Curves and their moduli" S.J. Bloch (ed.) , ''Algebraic geometry'' , ''Proc. Symp. Pure Math.'' , '''46.1''' , Amer. Math. Soc. (1985) pp. 99–143 {{MR|0927953}} {{ZBL|0646.14019}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> G. Faltings, "Arithmetische Kompaktifizierung des Modulraums der abelschen Varietäten" F. Hirzebruch (ed.) J. Schwermer (ed.) S. Suter (ed.) , ''Arbeitstagung Bonn 1984'' , ''Lect. notes in math.'' , '''1111''' , Springer (1985) pp. 321–383 {{MR|0797429}} {{ZBL|0597.14036}} </TD></TR></table>