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''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 [[#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  $  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  $  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]].

====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>

====Comments====

====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>