Chow variety

If X is a closed subvariety of \mathbb{P}^{n-1} of dimension k-1, the degree of X is the number of intersection points between X and a generic (n-k)-dimensional projective subspace of \mathbb{P}^{n-1}. Degree is constant in families of subvarieties, except in certain degenerate limits. To see this, consider the following family parametrized by t. :X_t := V(x^2-tyz) \subset \mathbb{P}^{2}. Whenever t\neq0, X_t is a conic (an irreducible subvariety of degree 2), but X_0 degenerates to the line x=0 (which has degree 1). There are several approaches to reconciling this issue, but the simplest is to declare X_0 to be a line of multiplicity 2 (and more generally to attach multiplicities to subvarieties) using the language of algebraic cycles. A (k-1)-dimensional algebraic cycle is a finite formal linear combination :X=\sum_{i} m_{i}X_{i}. in which X_{i}s are (k-1)-dimensional irreducible closed subvarieties in \mathbb{P}^{n-1}, and m_{i}s are integers. An algebraic cycle is effective if each m_i\geq0. The degree of an algebraic cycle is defined to be :\deg(X):=\sum_{i} m_{i}\deg(X_{i}). A homogeneous polynomial or homogeneous ideal in n-many variables defines an effective algebraic cycle in \mathbb{P}^{n-1}, in which the multiplicity of each irreducible component is the order of vanishing at that component. In the family of algebraic cycles defined by x^2-tyz, the t=0 cycle is 2 times the line x=0, which has degree 2. More generally, the degree of an algebraic cycle is constant in families, and so it makes sense to consider the moduli problem of effective algebraic cycles of fixed dimension and degree. ==Examples of Chow varieties==

There are three special classes of Chow varieties with particularly simple constructions. Degree 1: Subspaces An effective algebraic cycle in \mathbb{P}^{n-1} of dimension k-1 and degree 1 is the projectivization of a k-dimensional subspace of n-dimensional affine space. This gives an isomorphism to a Grassmannian variety: :\operatorname{Gr}(k,1,n) \simeq \operatorname{Gr}(k,n) The latter space has a distinguished system of homogeneous coordinates, given by the Plücker coordinates. Dimension 0: Points An effective algebraic cycle in \mathbb{P}^{n-1} of dimension 0 and degree d is an (unordered) d-tuple of points in \mathbb{P}^{n-1}, possibly with repetition. This gives an isomorphism to a symmetric power of \mathbb{P}^{n-1}: :\operatorname{Gr}(1,d,n) \simeq \operatorname{Sym}_d\mathbb{P}^{n-1}. Codimension 1: Divisors An effective algebraic cycle in \mathbb{P}^{n-1} of codimension 1 and degree d can be defined by the vanishing of a single degree d polynomial in n-many variables, and this polynomial is unique up to rescaling. Letting V_{d,n} denote the vector space of degree d polynomials in n-many variables, this gives an isomorphism to a projective space: :\operatorname{Gr}(n-1,d,n) \simeq \mathbb{P}V_{d,n}. Note that the latter space has a distinguished system of homogeneous coordinates, which send a polynomial to the coefficient of a fixed monomial. A non-trivial example The Chow variety \operatorname{Gr}(2,2,4) parametrizes dimension 1, degree 2 cycles in \mathbb{P}^{3}. This Chow variety has two irreducible components. These two 8-dimensional components intersect in the moduli of coplanar pairs of lines, which is the singular locus in \operatorname{Gr}(2,2,4). This shows that, in contrast with the special cases above, Chow varieties need not be smooth or irreducible. ==The Chow embedding==

Let X be an irreducible subvariety in \mathbb{P}^{n-1} of dimension k-1 and degree d. By the definition of the degree, most (n-k)-dimensional projective subspaces of \mathbb{P}^{n-1} intersect X in d-many points. By contrast, most (n-k-1)-dimensional projective subspaces of \mathbb{P}^{n-1} do not intersect at X at all. This can be sharpened as follows. Lemma. The set Z(X) \subset \operatorname{Gr}(n-k,n) parametrizing the subspaces of \mathbb{P}^{n-1} which intersect X non-trivially is an irreducible hypersurface of degree d. As a consequence, there exists a degree d form R_X on \operatorname{Gr}(n-k,n) which vanishes precisely on Z(X), and this form is unique up to scaling. This construction can be extended to an algebraic cycle X=\sum_{i} m_{i}X_{i} by declaring that R_X:= \prod_{i} R_{X_i}^{m_i}. To each degree d algebraic cycle, this associates a degree d form R_X on \operatorname{Gr}(n-k,n), called the Chow form of X, which is well-defined up to scaling. Let V_{k,d,n} denote the vector space of degree d forms on \operatorname{Gr}(n-k,n). The Chow-van-der-Waerden Theorem. The map \operatorname{Gr}(k,d,n) \hookrightarrow \mathbb{P}V_{k,d,n} which sends X\mapsto R_X is a closed embedding of varieties. In particular, an effective algebraic cycle X is determined by its Chow form R_X. If a basis for V_{k,d,n} has been chosen, sending X to the coefficients of R_X in this basis gives a system of homogeneous coordinates on the Chow variety \operatorname{Gr}(k,d,n), called the Chow coordinates of X. However, as there is no consensus as to the ‘best’ basis for V_{k,d,n}, this term can be ambiguous. From a foundational perspective, the above theorem is usually used as the definition of \operatorname{Gr}(k,d,n). That is, the Chow variety is usually defined as a subvariety of \mathbb{P}V_{k,d,n}, and only then shown to be a fine moduli space for the moduli problem in question. ==Relation to the Hilbert scheme==

A more sophisticated solution to the problem of 'correctly' counting the degree of a degenerate subvariety is to work with subschemes of \mathbb{P}^{n-1} rather than subvarieties. Schemes can keep track of infinitesimal information that varieties and algebraic cycles cannot. For example, if two points in a variety approach each other in an algebraic family, the limiting subvariety is a single point, the limiting algebraic cycle is a point with multiplicity 2, and the limiting subscheme is a 'fat point' which contains the tangent direction along which the two points collided. The Hilbert scheme \operatorname{Hilb}(k,d,n) is the fine moduli scheme of closed subschemes of dimension k-1 and degree d inside \mathbb{P}^{n-1}. Each closed subscheme determines an effective algebraic cycle, and the induced map :\operatorname{Hilb}(k,d,n) \longrightarrow \operatorname{Gr}(k,d,n). is called the cycle map or the Hilbert-Chow morphism. This map is generically an isomorphism over the points in \operatorname{Gr}(k,d,n) corresponding to irreducible subvarieties of degree d, but the fibers over non-simple algebraic cycles can be more interesting. ==Chow quotient ==

A Chow quotient parametrizes closures of generic orbits. It is constructed as a closed subvariety of a Chow variety. Kapranov's theorem says that the moduli space \overline{M}_{0, n} of stable genus-zero curves with n marked points is the Chow quotient of Grassmannian \operatorname{Gr}(2, \C^n) by the standard maximal torus. ==See also==