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Difference between revisions of "Moduli theory"

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A theory studying continuous families of objects in algebraic geometry.
 
A theory studying continuous families of objects in algebraic geometry.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m0645101.png" /> be a class of objects in algebraic geometry (varieties, schemes, vector bundles, etc.) on which an equivalence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m0645102.png" /> has been given. The fundamental classification problem (the description of the set of classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m0645103.png" />) has the following two parts: 1) the description of discrete invariants, which usually allow a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m0645104.png" /> 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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m0645105.png" />), parametrized by the complex numbers. B. Riemann, who introduced the term "moduli" , showed that an algebraic function field over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m0645106.png" /> (or their models, compact Riemann surfaces) of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m0645107.png" /> depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m0645108.png" /> continuous complex parameters — the moduli.
+
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.==
 
==Basic concepts in moduli theory.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m0645109.png" /> be a [[Scheme|scheme]] (a complex or algebraic space). A family of objects parametrized by the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451010.png" /> (or, as is often said, "scheme over Sover S" or "scheme with basis Swith basis S" ) is a set of objects
+
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 $ .
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451011.png" /></td> </tr></table>
 
  
equipped with an additional structure compatible with the structure of the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451012.png" />. This structure, in each concrete case, is given explicitly. A functor of families is a contravariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451013.png" /> from the category of the schemes (or spaces) into the category of sets defined as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451014.png" /> is the set of classes of isomorphic families over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451015.png" />. To every morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451016.png" /> is associated a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451017.png" /> which assigns to a family over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451018.png" /> the pullback, or induced, family over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451019.png" />.
+
Let $  M $
 
+
be an object in the category of schemes (complex or algebraic spaces) and let $  h _{M} $
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451020.png" /> be an object in the category of schemes (complex or algebraic spaces) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451021.png" /> be a functor of the points in this category, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451022.png" />. If the functor of families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451023.png" /> is representable, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451024.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451025.png" />, then there exists a universal family with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451026.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451027.png" /> is called a fine moduli scheme (respectively, fine complex moduli space or fine algebraic moduli space). The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451028.png" /> is representable in very few cases. Therefore the notion of a coarse moduli scheme was introduced. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451029.png" /> is called a coarse moduli scheme if there is a morphism of functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451030.png" /> with the properties: a) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451031.png" /> is one point (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451032.png" /> is an algebraically closed field), then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451033.png" /> is bijective; in other words, the set of geometric points of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451034.png" /> is in a natural one-to-one correspondence with the set of equivalence classes of parametrized objects; and b) for each scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451035.png" /> and morphism of functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451036.png" /> there is a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451037.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451038.png" />. Coarse schemes of complex and algebraic moduli spaces are similarly defined.
+
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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451039.png" /> (respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451040.png" />) be the set of classes of isomorphic projective non-singular curves (respectively, stable curves) of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451041.png" /> over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451042.png" />. A family over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451043.png" /> is a smooth (flat) proper morphism of schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451044.png" /> whose fibres are smooth (stable) curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451045.png" />. Then there is a coarse (but not a fine) moduli scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451046.png" /> (respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451047.png" />), which is a quasi-projective (projective), irreducible, normal variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451048.png" /> (see [[#References|[3]]], [[#References|[5]]], [[#References|[6]]]).
+
Examples. 1) Moduli of algebraic curves. Let $  A / R = \mathfrak M _{g} $ (
 
+
respectively $  {\overline{\mathfrak M}  _{g}} $ )  
2) Moduli of algebraic curves with level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451050.png" /> structure (with Jacobian rigidity). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451051.png" /> be a smooth family of projective curves (respectively, a flat family of stable curves) of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451052.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451053.png" /> be an integer invertible on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451054.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451055.png" /> be the first direct image of the constant sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451056.png" /> in the étale topology. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451057.png" /> is locally free, has rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451058.png" /> and is equipped with a locally non-degenerate symplectic form with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451059.png" />, up to an invertible element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451060.png" />. A Jacobian structure of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451062.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451063.png" /> is an assignment of a symplectic isomorphism
+
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 $ .  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451064.png" /></td> </tr></table>
+
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]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451065.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451066.png" />) be the functor of families of smooth (stable) curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451067.png" /> with a Jacobian rigidity of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451068.png" />. Then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451069.png" /> the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451070.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451071.png" />) is represented by a quasi-projective (projective) scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451072.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451073.png" />) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451074.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451075.png" /> is the inverse image of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451076.png" />-th root of unity, that is, there is a fine moduli scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451077.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451078.png" />) for the smooth (stable) curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451079.png" /> over a field of characteristic coprime with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451080.png" />, equipped with a Jacobian rigidity of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451081.png" />. For sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451082.png" /> the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451083.png" /> is smooth [[#References|[5]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451084.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451085.png" /> is a smooth family of varieties, i.e. a smooth proper morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451086.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451087.png" /> is the class of the relatively ample invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451088.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451089.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451090.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451091.png" /> is the relative Picard scheme and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451092.png" /> is the connected component of its zero section. In this case a functor of the polarized families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451093.png" />, with a given Hilbert polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451094.png" />, is constructed. Without additional restrictions this functor is not representable. The existence of a coarse moduli space is known (1989) only in individual cases.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451095.png" /> also exists.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451096.png" /> over an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451097.png" />. In this case a family over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451098.png" /> is a vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451099.png" />. Cf. [[#References|[7]]], [[#References|[10]]]–[[#References|[14]]] for a description of results and more detail.
+
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.==
 
==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 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510100.png" /> is constructed which contains representatives of all equivalence classes of the objects in questions, and so that the equivalence relation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510101.png" /> reduces to the action of an algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510102.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510103.png" /> in the corresponding category. The basic tool in the construction of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510104.png" /> is the theory of Hilbert schemes (cf. [[Hilbert scheme|Hilbert scheme]]). In such an approach the difficulty in constructing the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510105.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510106.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510107.png" /> have been extended to the case of fields of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510108.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510109.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510110.png" /> is replaced by an algebraic stack, the study of the properties of which gives some information on the moduli space.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510111.png" /> is the theory of period mappings (cf. [[Period mapping|Period mapping]]). The fundamental object here is the classifying space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510112.png" /> of polarized Hodge structures (cf. [[Hodge structure|Hodge structure]]) of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510113.png" /> for given Hodge numbers. For a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510114.png" /> of polarized algebraic varieties over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510115.png" /> the periods define a mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510116.png" /> onto the corresponding classifying space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510117.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510118.png" />-surfaces.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510119.png" /> is that of finding a natural and complete (projective or compact, in the theory over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510120.png" />) variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510121.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510122.png" /> as a dense open subset, and also the description and geometric interpretation of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510123.png" />. In example 1) the natural compactification of the coarse moduli variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510124.png" /> of curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510125.png" /> is the projective moduli variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510126.png" /> of stable curves. For polarized Abelian varieties over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510127.png" /> several means for compactifying moduli varieties are known.
+
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====
 
====References====
Line 47: Line 171:
  
 
====Comments====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510128.png" /> (see the appendix to Chapt. 5 in [[#References|[a1]]], which is an enlarged edition of [[#References|[5]]]) and the proof that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510129.png" /> is of general type for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510130.png" /> [[#References|[a2]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510131.png" /> of principally-polarized Abelian varieties over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m064510132.png" />, see [[#References|[a3]]].
+
G. Faltings has constructed a compactification of the moduli space $  A _{g} $
 +
of principally-polarized Abelian varieties over $  \mathbf Z $ ,  
 +
see [[#References|[a3]]].
  
 
====References====
 
====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>
 
<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>

Latest revision as of 11:49, 16 December 2019

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 (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 [3], [5], [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 [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. [7], [10][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 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). 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). 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). The fundamental object here is the classifying space $ D $ of polarized Hodge structures (cf. 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

[1] M. Artin, "Algebraization of formal moduli 1" , Global analysis , Univ. Tokyo Press (1969) pp. 21–71 MR260746
[2] M. Artin, "Versal deformations and algebraic stacks" Invent. Math. , 27 (1974) pp. 165–189 MR0399094 Zbl 0317.14001
[3] P. Deligne, D. Mumford, "The irreducibility of the space of curves of given genus" Publ. Math. IHES , 36 (1969) pp. 75–109 MR0262240 Zbl 0181.48803
[4] J. Dieudonné, J.B. Carrell, "Invariant theory, old and new" , Acad. Press (1971) MR0279102 Zbl 0258.14011
[5] D. Mumford, "Geometric invariant theory" , Springer (1965) MR0214602 Zbl 0147.39304
[6] D. Mumford, "Stability of projective varieties" l'Enseign. Math. (2) , 23 : 1–2 (1977) pp. 39–110 MR0450273 MR0450272 Zbl 0497.14004 Zbl 0376.14007 Zbl 0363.14003
[7] D. Gieseker, "Global moduli for varieties of general type" Invent. Math. , 43 (1977) pp. 233–282
[8] D. Gieseker, "On the moduli of vector bundles on an algebraic surface" Ann. of Math. , 106 (1977) pp. 45–60 MR0466475 Zbl 0381.14003
[9] T. Matsusaka, "Polarized varieties with a given Hilbert polynomial" Amer. J. Math. , 94 : 4 (1972) pp. 1027–1077 MR0337960 Zbl 0256.14004
[10] P.E. Newstead, "Lectures on introduction to moduli problems and orbit spaces" , Springer (1978) MR546290 Zbl 0411.14003
[11] C. Okonek, M. Schneider, H. Spindler, "Vector bundles on complex projective spaces" , Birkhäuser (1980) MR0561910 Zbl 0438.32016
[12] H. Popp, "Moduli theory and classification theory of algebraic varieties" , Springer (1977) MR0466143 Zbl 0359.14005
[13] C.S. Seshadri, "Spaces of unitary vector bundles on a compact Riemann surface" Ann. of Math. , 85 (1967) pp. 302–336 MR0233371
[14] 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 Zbl 0325.14016


Comments

Much progress has been made recently in the study of the moduli spaces of algebraic curves and Abelian varieties; see the appendices of [a2]. Among the most important ones are the question of the compactification of $ M _{g} $ ( see the appendix to Chapt. 5 in [a1], which is an enlarged edition of [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 [a3].

References

[a1] D. Mumford, J. Fogarty, "Geometric invariant theory" , Springer (1982) MR0719371 Zbl 0504.14008
[a2] 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 MR0927953 Zbl 0646.14019
[a3] 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 MR0797429 Zbl 0597.14036
How to Cite This Entry:
Moduli theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moduli_theory&oldid=21899
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article