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