Chow variety
Chow scheme
An algebraic variety whose points parametrize all algebraic subvarieties 
of dimension 
and degree 
of a projective space 
.
In the product 
, where 
is the dual of the projective space 
, parametrizing the hyperplanes 
, one considers the subvariety
 |
Its image 
under the projection onto the second factor is a hypersurface in 
which is given by a form 
in 
systems of 
variables, homogeneous of degree 
in each system of variables. The form 
is called the associated form (or the Cayley form) of the variety 
. It completely determines 
as a subvariety. This form was introduced by B.L. van der Waerden and W.L. Chow [1]. The coefficients of 
are determined up to a constant factor, and are called the Chow coordinates of 
.
The Chow coordinates of a variety 
determine a point 
, where 
is a certain function of 
and 
. The points 
corresponding to irreducible subvarieties 
of dimension 
and degree 
form a quasi-projective subvariety 
, 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 
and degree 
in 
, then one obtains a closed subvariety 
, which is also called the Chow variety. The Chow variety is the base of a universal algebraic family 
, where 
, 
is the induced projection, and the fibre 
above the point 
is identified with the cycle 
. The simplest examples of Chow varieties are the varieties 
of curves of degree 
in 
. Thus, 
is an irreducible variety of dimension 4, isomorphic to the Plücker quadric in 
; 
consists of two components of dimension 8, where 
corresponds to smooth curves of order two, and 
to pairs of lines; 
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 
is rational. However, it follows from the non-rationality of moduli schemes of curves of sufficiently high genus that for sufficiently high 
the variety 
is not rational (cf. [2]).
If 
is an algebraic subvariety, then the cycles 
of dimension 
and degree 
that lie in 
form an algebraic subvariety 
. This result permits one to introduce a certain algebraic structure on the set of all positive 
-dimensional cycles 
on the variety 
(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 |
Chow variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chow_variety&oldid=13219