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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, particularly in the field of
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, a Chow variety is an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
whose points correspond to effective algebraic cycles of fixed dimension and degree on a given
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
. More precisely, the Chow variety \operatorname(k,d,n) is the fine moduli variety parametrizing all effective algebraic cycles of dimension k-1 and degree d in \mathbb^. The Chow variety \operatorname(k,d,n) may be constructed via a Chow embedding into a sufficiently large projective space. This is a direct generalization of the construction of a
Grassmannian In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a ...
variety via the Plücker embedding, as Grassmannians are the d=1 case of Chow varieties. Chow varieties are distinct from Chow groups, which are the abelian group of all algebraic cycles on a variety (not necessarily projective space) up to rational equivalence. Both are named for
Wei-Liang Chow Chow Wei-Liang (; October 1, 1911, Shanghai – August 10, 1995, Baltimore) was a Chinese-American mathematician and stamp collector. He was well known for his work in algebraic geometry. Biography Chow was a student in the US, graduating from ...
(周煒良), a pioneer in the study of algebraic cycles.


Background on algebraic cycles

If X is a closed subvariety of \mathbb^ 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^. 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^. 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_ m_X_. in which X_s are (k-1)-dimensional irreducible closed subvarieties in \mathbb^, and m_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_ m_\deg(X_). A homogeneous polynomial or homogeneous ideal in n-many variables defines an effective algebraic cycle in \mathbb^, 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^ 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 In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a ...
variety: :\operatorname(k,1,n) \simeq \operatorname(k,n) The latter space has a distinguished system of
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
, given by the Plücker coordinates.


Dimension 0: Points

An effective algebraic cycle in \mathbb^ of dimension 0 and degree d is an (unordered) d-tuple of points in \mathbb^, possibly with repetition. This gives an isomorphism to a symmetric power of \mathbb^: :\operatorname(1,d,n) \simeq \operatorname_d\mathbb^.


Codimension 1: Divisors

An effective algebraic cycle in \mathbb^ 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_ denote the vector space of degree d polynomials in n-many variables, this gives an isomorphism to a
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
: :\operatorname(n-1,d,n) \simeq \mathbbV_. Note that the latter space has a distinguished system of
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
, which send a polynomial to the coefficient of a fixed monomial.


A non-trivial example

The Chow variety \operatorname(2,2,4) parametrizes dimension 1, degree 2 cycles in \mathbb^. 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(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^ of dimension k-1 and degree d. By the definition of the degree, most (n-k)-dimensional projective subspaces of \mathbb^ intersect X in d-many points. By contrast, most (n-k-1)-dimensional projective subspaces of \mathbb^ do not intersect at X at all. This can be sharpened as follows. Lemma. The set Z(X) \subset \operatorname(n-k,n) parametrizing the subspaces of \mathbb^ 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(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_ m_X_ by declaring that R_X:= \prod_ R_^. To each degree d algebraic cycle, this associates a degree d form R_X on \operatorname(n-k,n), called the Chow form of X, which is well-defined up to scaling. Let V_ denote the vector space of degree d forms on \operatorname(n-k,n). The Chow-van-der-Waerden Theorem. The map \operatorname(k,d,n) \hookrightarrow \mathbbV_ 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_ 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(k,d,n), called the Chow coordinates of X. However, as there is no consensus as to the ‘best’ basis for V_, this term can be ambiguous. From a foundational perspective, the above theorem is usually used as the definition of \operatorname(k,d,n). That is, the Chow variety is usually defined as a subvariety of \mathbbV_, 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^ 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(k,d,n) is the fine moduli scheme of closed subschemes of dimension k-1 and degree d inside \mathbb^.There is considerable variance in how the term 'Hilbert scheme' is used. Some authors don't subdivide by dimension or degree, others assume the dimension is 0 (i.e. a Hilbert scheme of points), and still others consider more general schemes than \mathbb^. Each closed subscheme determines an effective algebraic cycle, and the induced map :\operatorname(k,d,n) \longrightarrow \operatorname(k,d,n). is called the cycle map or the Hilbert-Chow morphism. This map is generically an isomorphism over the points in \operatorname(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 In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
\overline_ of
stable A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use tod ...
genus-zero curves with ''n'' marked points is the Chow quotient of Grassmannian \operatorname(2, \C^n) by the standard maximal torus.


See also

* Picard variety * GIT quotient


References

* * * * * Mikhail Kapranov, Chow quotients of Grassmannian, I.M. Gelfand Seminar Collection, 29–110, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, RI, 1993. * * * *{{Cite book, last1=Mumford , first1=David , author1-link=David Mumford , last2=Fogarty , first2=John , last3=Kirwan , first3=Frances , author3-link=Frances Kirwan , title=Geometric invariant theory , publisher=
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, location=Berlin, New York , edition=3rd , series=Ergebnisse der Mathematik und ihrer Grenzgebiete (2) esults in Mathematics and Related Areas (2), isbn=978-3-540-56963-3 , mr=1304906 , year=1994 , volume=34 Algebraic geometry