Chow variety

In mathematics, particularly in the field of algebraic geometry, a Chow variety is an algebraic variety whose points correspond to effective algebraic cycles of fixed dimension and degree on a given projective space. More precisely, the Chow variety

Gr

⁡

(

k

,

d

,

n

)

{\displaystyle \operatorname {Gr} (k,d,n)}

is the fine moduli variety parametrizing all effective algebraic cycles of dimension

k

−

1

{\displaystyle k-1}

and degree

d

{\displaystyle d}

in

P

n

−

1

{\displaystyle \mathbb {P} ^{n-1}}

.

The Chow variety

Gr

⁡

(

k

,

d

,

n

)

{\displaystyle \operatorname {Gr} (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 variety via the Plücker embedding, as Grassmannians are the

d

=

1

{\displaystyle 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 (周煒良), a pioneer in the study of algebraic cycles.

Background on algebraic cycles

If X is a closed subvariety of

P

n

−

1

{\displaystyle \mathbb {P} ^{n-1}}

of dimension

k

−

1

{\displaystyle k-1}

, the degree of X is the number of intersection points between X and a generic

(

n

−

k

)

{\displaystyle (n-k)}

-dimensional projective subspace of

P

n

−

1

{\displaystyle \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

−

t

y

z

)

⊂

P

2

{\displaystyle X_{t}:=V(x^{2}-tyz)\subset \mathbb {P} ^{2}}

.