In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly in the field of
algebraic geometry, 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 set of solutions of a system of polynomial equations over the real or complex numbers. ...
whose points correspond to effective
algebraic cycles of fixed dimension and degree on a given
projective space. More precisely, the Chow variety
is the
fine moduli variety parametrizing all effective algebraic cycles of dimension
and degree
in
.
The Chow variety
may be constructed via a Chow embedding into a sufficiently large projective space. This is a direct generalization of the construction of a
Grassmannian variety via the
Plücker embedding, as Grassmannians are the
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(周煒良), a pioneer in the study of algebraic cycles.
Background on algebraic cycles
If X is a closed
subvariety
A subvariety (Latin: ''subvarietas'') in botanical nomenclature is a taxonomic rank. They are rarely used to classify organisms.
Plant taxonomy
Subvariety is ranked:
*below that of variety (''varietas'')
*above that of form (''forma'').
Subva ...
of
of dimension
, the degree of X is the number of intersection points between X and a generic
-dimensional
projective subspace of
.
Degree is constant in families of subvarieties, except in certain degenerate limits. To see this, consider the following family parametrized by t.
:
.
Whenever
,
is a conic (an irreducible subvariety of degree 2), but
degenerates to the line
(which has degree 1). There are several approaches to reconciling this issue, but the simplest is to declare
to be a ''line of multiplicity 2'' (and more generally to attach multiplicities to subvarieties) using the language of ''algebraic cycles''.
A
-dimensional
algebraic cycle In mathematics, an algebraic cycle on an algebraic variety ''V'' is a formal linear combination of subvarieties of ''V''. These are the part of the algebraic topology of ''V'' that is directly accessible by algebraic methods. Understanding the al ...
is a finite formal linear combination
:
.
in which
s are
-dimensional irreducible closed subvarieties in
, and
s are integers. An algebraic cycle is effective if each
. The degree of an algebraic cycle is defined to be
:
.
A
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
or
homogeneous ideal in n-many variables defines an effective algebraic cycle in
, in which the multiplicity of each irreducible component is the order of vanishing at that component. In the family of algebraic cycles defined by
, the
cycle is 2 times the line
, 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
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:
:
The latter space has a distinguished system of
homogeneous coordinates, given by the
Plücker coordinates.
Dimension 0: Points
An effective algebraic cycle in
of dimension 0 and degree d is an (unordered) d-tuple of points in
, possibly with repetition. This gives an isomorphism to a
symmetric power of
:
:
.
Codimension 1: Divisors
An effective algebraic cycle in
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
denote the vector space of degree d polynomials in n-many variables, this gives an isomorphism to a
projective space:
:
.
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
parametrizes dimension 1, degree 2 cycles in
. 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
. 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
of dimension k-1 and degree d. By the definition of the degree, most
-dimensional
projective subspaces of
intersect X in d-many points. By contrast, most
-dimensional
projective subspaces of
do not intersect at X at all. This can be sharpened as follows.
Lemma. The set
parametrizing the subspaces of
which intersect X non-trivially is an irreducible hypersurface of degree d.
As a consequence, there exists a degree d form
on
which vanishes precisely on
, and this form is unique up to scaling. This construction can be extended to an algebraic cycle
by declaring that
. To each degree d algebraic cycle, this associates a degree d form
on
, called the Chow form of X, which is well-defined up to scaling.
Let
denote the vector space of degree d forms on
.
The Chow-van-der-Waerden Theorem. The map
which sends
is a closed embedding of varieties.
In particular, an effective algebraic cycle X is determined by its Chow form
.
If a basis for
has been chosen, sending
to the coefficients of
in this basis gives a system of homogeneous coordinates on the Chow variety
, called the Chow coordinates of
. However, as there is no consensus as to the ‘best’ basis for
, this term can be ambiguous.
From a foundational perspective, the above theorem is usually used as the definition of
. That is, the Chow variety is usually defined as a subvariety of
, 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
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 is the
fine moduli scheme of closed subschemes of dimension k-1 and degree d inside
.
[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 .] Each closed subscheme determines an effective algebraic cycle, and the induced map
:
.
is called the cycle map or the Hilbert-Chow morphism. This map is generically an isomorphism over the points in
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 of
stable
A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
genus-zero curves with ''n'' marked points is the Chow quotient of Grassmannian
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 ...
, 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