Chow variety

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

References

[a1]	B. Angéniol, "Familles de cycles algébriques. Schéma de Chow" , Lect. notes in math. , 896 , Springer (1981) MR0646062