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''Chow scheme''
 
''Chow scheme''
  
An algebraic variety whose points parametrize all algebraic subvarieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c0221801.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c0221802.png" /> and degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c0221803.png" /> of a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c0221804.png" />.
+
An algebraic variety whose points parametrize all algebraic subvarieties $  X $
 +
of dimension $  r $
 +
and degree $  d $
 +
of a projective space $  P  ^ {n} $.
  
In the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c0221805.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c0221806.png" /> is the dual of the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c0221807.png" />, parametrizing the hyperplanes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c0221808.png" />, one considers the subvariety
+
In the product $  X \times ( \widetilde{P}  {}  ^ {n} )  ^ {r+} 1 $,  
 +
where $  \widetilde{P}  {}  ^ {n} $
 +
is the dual of the projective space $  P  ^ {n} $,  
 +
parametrizing the hyperplanes $  u \subset  P  ^ {n} $,  
 +
one considers the subvariety
  
<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/c/c022/c022180/c0221809.png" /></td> </tr></table>
+
$$
 +
\Gamma  = \{ {( x , u  ^ {(} 0) \dots u  ^ {(} r) ) } : {
 +
x \in u  ^ {(} i)  \textrm{ for }  i = 0 \dots r } \}
 +
.
 +
$$
  
Its image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218010.png" /> under the projection onto the second factor is a hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218011.png" /> which is given by a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218013.png" /> systems of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218014.png" /> variables, homogeneous of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218015.png" /> in each system of variables. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218016.png" /> is called the associated form (or the Cayley form) of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218017.png" />. It completely determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218018.png" /> as a subvariety. This form was introduced by B.L. van der Waerden and W.L. Chow [[#References|[1]]]. The coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218019.png" /> are determined up to a constant factor, and are called the Chow coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218020.png" />.
+
Its image $  p _ {2} ( \Gamma ) \subset  ( \widetilde{P}  {}  ^ {n} )  ^ {r+} 1 $
 +
under the projection onto the second factor is a hypersurface in $  ( \widetilde{P}  {}  ^ {n} )  ^ {r+} 1 $
 +
which is given by a form $  F _ {X} $
 +
in $  r+ 1 $
 +
systems of $  n + 1 $
 +
variables, homogeneous of degree $  d $
 +
in each system of variables. The form $  F _ {X} $
 +
is called the associated form (or the Cayley form) of the variety $  X $.  
 +
It completely determines $  X $
 +
as a subvariety. This form was introduced by B.L. van der Waerden and W.L. Chow [[#References|[1]]]. The coefficients of $  F _ {X} $
 +
are determined up to a constant factor, and are called the Chow coordinates of $  X $.
  
The Chow coordinates of a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218021.png" /> determine a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218023.png" /> is a certain function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218025.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218026.png" /> corresponding to irreducible subvarieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218027.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218028.png" /> and degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218029.png" /> form a quasi-projective subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218030.png" />, called the Chow variety. If one considers not only the irreducible subvarieties, but also positive algebraic cycles (that is, formal linear combinations of varieties with positive integer coefficients) of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218031.png" /> and degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218033.png" />, then one obtains a closed subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218034.png" />, which is also called the Chow variety. The Chow variety is the base of a universal algebraic family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218037.png" /> is the induced projection, and the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218038.png" /> above the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218039.png" /> is identified with the cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218040.png" />. The simplest examples of Chow varieties are the varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218041.png" /> of curves of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218042.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218043.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218044.png" /> is an irreducible variety of dimension 4, isomorphic to the Plücker quadric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218045.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218046.png" /> consists of two components of dimension 8, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218047.png" /> corresponds to smooth curves of order two, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218048.png" /> to pairs of lines; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218049.png" /> consists of four components of dimension 12 corresponding to triplets of lines, curves consisting of a line together with a planar quadric, planar cubics, and non-planar curves of order 3. In all these cases the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218050.png" /> is rational. However, it follows from the non-rationality of moduli schemes of curves of sufficiently high genus that for sufficiently high <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218051.png" /> the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218052.png" /> is not rational (cf. [[#References|[2]]]).
+
The Chow coordinates of a variety $  X $
 +
determine a point c ( X) \in P  ^  \nu  $,  
 +
where $  \nu $
 +
is a certain function of $  n , r $
 +
and $  d $.  
 +
The points c ( X) \in P  ^  \nu  $
 +
corresponding to irreducible subvarieties $  X \subset  P  ^ {n} $
 +
of dimension $  r $
 +
and degree $  d $
 +
form a quasi-projective subvariety $  C _ {n,r,d} $,  
 +
called the Chow variety. If one considers not only the irreducible subvarieties, but also positive algebraic cycles (that is, formal linear combinations of varieties with positive integer coefficients) of dimension $  r $
 +
and degree $  d $
 +
in $  P  ^ {n} $,  
 +
then one obtains a closed subvariety $  \overline{ {C _ {n,r,d} }}\; \subset  P  ^  \nu  $,  
 +
which is also called the Chow variety. The Chow variety is the base of a universal algebraic family $  \pi : \mathfrak X \rightarrow \overline{ {C _ {n,r,d} }}\; $,  
 +
where $  \mathfrak X \subset  \overline{ {C _ {n,r,d} }}\; \times P  ^ {n} $,  
 +
$  \pi $
 +
is the induced projection, and the fibre $  \pi  ^ {-} 1 ( c) $
 +
above the point c ( X) \in \overline{ {C _ {n,r,d} }}\; $
 +
is identified with the cycle $  X $.  
 +
The simplest examples of Chow varieties are the varieties $  C _ {3,1,d} $
 +
of curves of degree $  d $
 +
in $  P  ^ {3} $.  
 +
Thus, $  C _ {3,1,1} = \overline{ {C _ {3,1,1} }}\; $
 +
is an irreducible variety of dimension 4, isomorphic to the Plücker quadric in $  P  ^ {5} $;
 +
$  \overline{ {C _ {3,1,2} }}\; = C  ^ {(} 1) \cup C  ^ {(} 2) $
 +
consists of two components of dimension 8, where $  C  ^ {(} 1) $
 +
corresponds to smooth curves of order two, and $  C  ^ {(} 2) $
 +
to pairs of lines; $  \overline{ {C _ {3,1,3} }}\; $
 +
consists of four components of dimension 12 corresponding to triplets of lines, curves consisting of a line together with a planar quadric, planar cubics, and non-planar curves of order 3. In all these cases the variety $  C _ {3,1,d} $
 +
is rational. However, it follows from the non-rationality of moduli schemes of curves of sufficiently high genus that for sufficiently high $  d $
 +
the variety $  C _ {3,1,d} $
 +
is not rational (cf. [[#References|[2]]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218053.png" /> is an algebraic subvariety, then the cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218054.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218055.png" /> and degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218056.png" /> that lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218057.png" /> form an algebraic subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218058.png" />. This result permits one to introduce a certain algebraic structure on the set of all positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218059.png" />-dimensional cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218060.png" /> on the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022180/c02218061.png" /> (cf. [[#References|[1]]]).
+
If $  V \subset  P  ^ {n} $
 +
is an algebraic subvariety, then the cycles $  Z \subset  P  ^ {n} $
 +
of dimension $  r $
 +
and degree $  d $
 +
that lie in $  V $
 +
form an algebraic subvariety $  \overline{ {C _ {r,d} }}\; ( V) \subset  \overline{ {C _ {n,r,d} }}\; $.  
 +
This result permits one to introduce a certain algebraic structure on the set of all positive $  r $-
 +
dimensional cycles $  Z _ {r}  ^ {+} ( V) = \cup _ {d>} 0 \overline{ {C _ {r,d} }}\; ( V) $
 +
on the variety $  V $(
 +
cf. [[#References|[1]]]).
  
 
For other approaches to the problem of the classification of varieties cf. [[Hilbert scheme|Hilbert scheme]]; [[Moduli problem|Moduli problem]].
 
For other approaches to the problem of the classification of varieties cf. [[Hilbert scheme|Hilbert scheme]]; [[Moduli problem|Moduli problem]].
Line 17: Line 91:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.L. van der Waerden, W.L. Chow, "Zur algebraische Geometrie IX" ''Math. Ann.'' , '''113''' (1937) pp. 692–704 {{MR|}} {{ZBL|0016.04004}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Harris, D. Mumford, "On the Kodaira dimension of the moduli space of curves" ''Invent. Math.'' , '''67''' (1982) pp. 23–88 {{MR|0664324}} {{ZBL|0506.14016}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W.L.V.D. Hodge, "Methods of algebraic geometry" , '''2''' , Cambridge Univ. Press (1947–1954) {{MR|1288307}} {{MR|1288306}} {{MR|1288305}} {{MR|0061846}} {{MR|0048065}} {{MR|0028055}} {{ZBL|0796.14002}} {{ZBL|0796.14003}} {{ZBL|0796.14001}} {{ZBL|0157.27502}} {{ZBL|0157.27501}} {{ZBL|0055.38705}} {{ZBL|0048.14502}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.L. van der Waerden, W.L. Chow, "Zur algebraische Geometrie IX" ''Math. Ann.'' , '''113''' (1937) pp. 692–704 {{MR|}} {{ZBL|0016.04004}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Harris, D. Mumford, "On the Kodaira dimension of the moduli space of curves" ''Invent. Math.'' , '''67''' (1982) pp. 23–88 {{MR|0664324}} {{ZBL|0506.14016}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W.L.V.D. Hodge, "Methods of algebraic geometry" , '''2''' , Cambridge Univ. Press (1947–1954) {{MR|1288307}} {{MR|1288306}} {{MR|1288305}} {{MR|0061846}} {{MR|0048065}} {{MR|0028055}} {{ZBL|0796.14002}} {{ZBL|0796.14003}} {{ZBL|0796.14001}} {{ZBL|0157.27502}} {{ZBL|0157.27501}} {{ZBL|0055.38705}} {{ZBL|0048.14502}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Angéniol, "Familles de cycles algébriques. Schéma de Chow" , ''Lect. notes in math.'' , '''896''' , Springer (1981) {{MR|0646062}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Angéniol, "Familles de cycles algébriques. Schéma de Chow" , ''Lect. notes in math.'' , '''896''' , Springer (1981) {{MR|0646062}} {{ZBL|}} </TD></TR></table>

Latest revision as of 16:44, 4 June 2020


Chow scheme

An algebraic variety whose points parametrize all algebraic subvarieties $ X $ of dimension $ r $ and degree $ d $ of a projective space $ P ^ {n} $.

In the product $ X \times ( \widetilde{P} {} ^ {n} ) ^ {r+} 1 $, where $ \widetilde{P} {} ^ {n} $ is the dual of the projective space $ P ^ {n} $, parametrizing the hyperplanes $ u \subset P ^ {n} $, one considers the subvariety

$$ \Gamma = \{ {( x , u ^ {(} 0) \dots u ^ {(} r) ) } : { x \in u ^ {(} i) \textrm{ for } i = 0 \dots r } \} . $$

Its image $ p _ {2} ( \Gamma ) \subset ( \widetilde{P} {} ^ {n} ) ^ {r+} 1 $ under the projection onto the second factor is a hypersurface in $ ( \widetilde{P} {} ^ {n} ) ^ {r+} 1 $ which is given by a form $ F _ {X} $ in $ r+ 1 $ systems of $ n + 1 $ variables, homogeneous of degree $ d $ in each system of variables. The form $ F _ {X} $ is called the associated form (or the Cayley form) of the variety $ X $. It completely determines $ X $ as a subvariety. This form was introduced by B.L. van der Waerden and W.L. Chow [1]. The coefficients of $ F _ {X} $ are determined up to a constant factor, and are called the Chow coordinates of $ X $.

The Chow coordinates of a variety $ X $ determine a point $ c ( X) \in P ^ \nu $, where $ \nu $ is a certain function of $ n , r $ and $ d $. The points $ c ( X) \in P ^ \nu $ corresponding to irreducible subvarieties $ X \subset P ^ {n} $ of dimension $ r $ and degree $ d $ form a quasi-projective subvariety $ C _ {n,r,d} $, called the Chow variety. If one considers not only the irreducible subvarieties, but also positive algebraic cycles (that is, formal linear combinations of varieties with positive integer coefficients) of dimension $ r $ and degree $ d $ in $ P ^ {n} $, then one obtains a closed subvariety $ \overline{ {C _ {n,r,d} }}\; \subset P ^ \nu $, which is also called the Chow variety. The Chow variety is the base of a universal algebraic family $ \pi : \mathfrak X \rightarrow \overline{ {C _ {n,r,d} }}\; $, where $ \mathfrak X \subset \overline{ {C _ {n,r,d} }}\; \times P ^ {n} $, $ \pi $ is the induced projection, and the fibre $ \pi ^ {-} 1 ( c) $ above the point $ c ( X) \in \overline{ {C _ {n,r,d} }}\; $ is identified with the cycle $ X $. The simplest examples of Chow varieties are the varieties $ C _ {3,1,d} $ of curves of degree $ d $ in $ P ^ {3} $. Thus, $ C _ {3,1,1} = \overline{ {C _ {3,1,1} }}\; $ is an irreducible variety of dimension 4, isomorphic to the Plücker quadric in $ P ^ {5} $; $ \overline{ {C _ {3,1,2} }}\; = C ^ {(} 1) \cup C ^ {(} 2) $ consists of two components of dimension 8, where $ C ^ {(} 1) $ corresponds to smooth curves of order two, and $ C ^ {(} 2) $ to pairs of lines; $ \overline{ {C _ {3,1,3} }}\; $ consists of four components of dimension 12 corresponding to triplets of lines, curves consisting of a line together with a planar quadric, planar cubics, and non-planar curves of order 3. In all these cases the variety $ C _ {3,1,d} $ is rational. However, it follows from the non-rationality of moduli schemes of curves of sufficiently high genus that for sufficiently high $ d $ the variety $ C _ {3,1,d} $ is not rational (cf. [2]).

If $ V \subset P ^ {n} $ is an algebraic subvariety, then the cycles $ Z \subset P ^ {n} $ of dimension $ r $ and degree $ d $ that lie in $ V $ form an algebraic subvariety $ \overline{ {C _ {r,d} }}\; ( V) \subset \overline{ {C _ {n,r,d} }}\; $. This result permits one to introduce a certain algebraic structure on the set of all positive $ r $- dimensional cycles $ Z _ {r} ^ {+} ( V) = \cup _ {d>} 0 \overline{ {C _ {r,d} }}\; ( V) $ on the variety $ V $( cf. [1]).

For other approaches to the problem of the classification of varieties cf. Hilbert scheme; Moduli problem.

References

[1] B.L. van der Waerden, W.L. Chow, "Zur algebraische Geometrie IX" Math. Ann. , 113 (1937) pp. 692–704 Zbl 0016.04004
[2] J. Harris, D. Mumford, "On the Kodaira dimension of the moduli space of curves" Invent. Math. , 67 (1982) pp. 23–88 MR0664324 Zbl 0506.14016
[3] W.L.V.D. Hodge, "Methods of algebraic geometry" , 2 , Cambridge Univ. Press (1947–1954) MR1288307 MR1288306 MR1288305 MR0061846 MR0048065 MR0028055 Zbl 0796.14002 Zbl 0796.14003 Zbl 0796.14001 Zbl 0157.27502 Zbl 0157.27501 Zbl 0055.38705 Zbl 0048.14502
[4] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001

Comments

References

[a1] B. Angéniol, "Familles de cycles algébriques. Schéma de Chow" , Lect. notes in math. , 896 , Springer (1981) MR0646062
How to Cite This Entry:
Chow variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chow_variety&oldid=46342
This article was adapted from an original article by Val.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article