The Chow group of a toric variety#

In general, the Chow group is an algebraic version of a homology theory. That is, the objects are formal linear combinations of submanifolds modulo relations. In particular, the objects of the Chow group are formal linear combinations of algebraic subvarieties and the equivalence relation is rational equivalence. There is no relative version of the Chow group, so it is not a generalized homology theory.

The Chow groups of smooth or mildly singular toric varieties are almost the same as the homology groups:

For smooth toric varieties, \(A_{k}(X) = H_{2k}(X,\ZZ)\). While they are the same, using the cohomology ring instead of the Chow group will be much faster! The cohomology ring does not try to keep track of torsion and uses Groebner bases to encode the cup product.
For simplicial toric varieties, \(A_{k}(X)\otimes \QQ = H_{2k}(X,\QQ)\).

Note that in these cases the odd-dimensional (co)homology groups vanish. But for sufficiently singular toric varieties the Chow group differs from the homology groups (and the odd-dimensional homology groups no longer vanish). For singular varieties the Chow group is much easier to compute than the (co)homology groups.

The toric Chow group of a toric variety is the Chow group generated by the toric subvarieties, that is, closures of orbits under the torus action. These are in one-to-one correspondence with the cones of the fan and, therefore, the toric Chow group is a quotient of the free Abelian group generated by the cones. In particular, the toric Chow group has finite rank. One can show [FMSS1995] that the toric Chow groups equal the “full” Chow group of a toric variety, so there is no need to distinguish these in the following.

AUTHORS:

Volker Braun (2010-08-09): Initial version

REFERENCES:

Wikipedia article Chow_ring
[FMSS1995]
[Ful1993], Chapter 5.1, “Chow Groups”

EXAMPLES:

Sage

sage: X = toric_varieties.Cube_deformation(7)
sage: X.is_smooth()
False
sage: X.is_orbifold()
False
sage: A = X.Chow_group()
sage: A.degree()
(Z, C7, C2 x C2 x Z^5, Z)
sage: A.degree(2).ngens()
7
sage: a = sum( A.gen(i) * (i+1) for i in range(A.ngens()) )   # an element of A
sage: a  # long time (2s on sage.math, 2011)
( 9 | 1 mod 7 | 1 mod 2, 0 mod 2, 4, 5, 6, 7, 8 | 3 )

Python

>>> from sage.all import *
>>> X = toric_varieties.Cube_deformation(Integer(7))
>>> X.is_smooth()
False
>>> X.is_orbifold()
False
>>> A = X.Chow_group()
>>> A.degree()
(Z, C7, C2 x C2 x Z^5, Z)
>>> A.degree(Integer(2)).ngens()
7
>>> a = sum( A.gen(i) * (i+Integer(1)) for i in range(A.ngens()) )   # an element of A
>>> a  # long time (2s on sage.math, 2011)
( 9 | 1 mod 7 | 1 mod 2, 0 mod 2, 4, 5, 6, 7, 8 | 3 )

The Chow group elements are printed as ( a0 | a1 mod 7 | a2 mod 2, a3 mod 2, a4, a5, a6, a7, a8 | a9 ), which denotes the element of the Chow group in the same basis as A.degree(). The | separates individual degrees, so the example means:

The degree-0 part is \(3 \in \ZZ\).
The degree-1 part is \(1 \in \ZZ_7\).
The torsion of the degree-2 Chow group is \((0, 1) \in \ZZ_2\oplus\ZZ_2\).
The free part of the degree-2 Chow group is \((4, 5, 6, 7, 8) \in \ZZ^5\).
The degree-3 part is \(9 \in \ZZ\).

Note that the generators A.gens() are not sorted in any way. In fact, they may be of mixed degree. Use A.gens(degree=d) to obtain the generators in a fixed degree d. See ChowGroup_class.gens() for more details.

Cones of toric varieties can determine their own Chow cycle:

Sage

sage: A = X.Chow_group(); A
Chow group of 3-d toric variety covered by 6 affine patches
sage: cone = X.fan(dim=2)[3]; cone
2-d cone of Rational polyhedral fan in 3-d lattice N
sage: A_cone = A(cone); A_cone
( 0 | 6 mod 7 | 0 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 )
sage: A_cone.degree()
1
sage: 2 * A_cone
( 0 | 5 mod 7 | 0 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 )
sage: A_cone + A.gen(0)
( 0 | 6 mod 7 | 1 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 )

Python

>>> from sage.all import *
>>> A = X.Chow_group(); A
Chow group of 3-d toric variety covered by 6 affine patches
>>> cone = X.fan(dim=Integer(2))[Integer(3)]; cone
2-d cone of Rational polyhedral fan in 3-d lattice N
>>> A_cone = A(cone); A_cone
( 0 | 6 mod 7 | 0 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 )
>>> A_cone.degree()
1
>>> Integer(2) * A_cone
( 0 | 5 mod 7 | 0 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 )
>>> A_cone + A.gen(Integer(0))
( 0 | 6 mod 7 | 1 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 )

Chow cycles can be of mixed degrees:

Sage

sage: mixed = sum(A.gens()); mixed
( 1 | 4 mod 7 | 1 mod 2, 1 mod 2, 1, 1, 1, 1, 1 | 1 )
sage: mixed.project_to_degree(1)
( 0 | 4 mod 7 | 0 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 )
sage: sum( mixed.project_to_degree(i) for i in range(X.dimension()+1) ) == mixed
True

Python

>>> from sage.all import *
>>> mixed = sum(A.gens()); mixed
( 1 | 4 mod 7 | 1 mod 2, 1 mod 2, 1, 1, 1, 1, 1 | 1 )
>>> mixed.project_to_degree(Integer(1))
( 0 | 4 mod 7 | 0 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 )
>>> sum( mixed.project_to_degree(i) for i in range(X.dimension()+Integer(1)) ) == mixed
True

class sage.schemes.toric.chow_group.ChowCycle(parent, v, check=True)[source]#

Bases: FGP_Element

The elements of the Chow group.

Warning

Do not construct ChowCycle objects manually. Instead, use the parent ChowGroup to obtain generators or Chow cycles corresponding to cones of the fan.

EXAMPLES:

Sage

sage: P2 = toric_varieties.P2()
sage: A = P2.Chow_group()
sage: A.gens()
(( 0 | 0 | 1 ), ( 0 | 1 | 0 ), ( 1 | 0 | 0 ))
sage: cone = P2.fan(1)[0]
sage: A(cone)
( 0 | 1 | 0 )
sage: A( Cone([(1,0)]) )
( 0 | 1 | 0 )

Python

>>> from sage.all import *
>>> P2 = toric_varieties.P2()
>>> A = P2.Chow_group()
>>> A.gens()
(( 0 | 0 | 1 ), ( 0 | 1 | 0 ), ( 1 | 0 | 0 ))
>>> cone = P2.fan(Integer(1))[Integer(0)]
>>> A(cone)
( 0 | 1 | 0 )
>>> A( Cone([(Integer(1),Integer(0))]) )
( 0 | 1 | 0 )

cohomology_class()[source]#

Return the (Poincaré-dual) cohomology class.

Consider a simplicial cone of the fan, that is, a \(d\)-dimensional cone spanned by \(d\) rays. Take the product of the corresponding \(d\) homogeneous coordinates. This monomial represents a cohomology classes of the toric variety \(X\), see cohomology_ring(). Its cohomological degree is \(2d\), which is the same degree as the Poincaré-dual of the (real) \(\dim(X)-2d\)-dimensional torus orbit associated to the simplicial cone. By linearity, we can associate a cohomology class to each Chow cycle of a simplicial toric variety.

If the toric variety is compact and smooth, the associated cohomology class actually is the Poincaré dual (over the integers) of the Chow cycle. In particular, integrals of dual cohomology classes perform intersection computations.

If the toric variety is compact and has at most orbifold singularities, the torsion parts in cohomology and the Chow group can differ. But they are still isomorphic as rings over the rationals. Moreover, the normalization of integration (volume_class) and count_points() are chosen to agree.

OUTPUT:

The CohomologyClass which is associated to the Chow cycle.

If the toric variety is not simplicial, that is, has worse than orbifold singularities, there is no way to associate a cohomology class of the correct degree. In this case, cohomology_class() raises a ValueError.

EXAMPLES:

Sage

sage: # needs sage.libs.singular
sage: dP6 = toric_varieties.dP6()
sage: cone = dP6.fan().cone_containing(2,3)
sage: HH = dP6.cohomology_ring()
sage: A = dP6.Chow_group()
sage: HH(cone)
[-w^2]
sage: A(cone)
( 1 | 0, 0, 0, 0 | 0 )
sage: A(cone).cohomology_class()
[-w^2]

Python

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> dP6 = toric_varieties.dP6()
>>> cone = dP6.fan().cone_containing(Integer(2),Integer(3))
>>> HH = dP6.cohomology_ring()
>>> A = dP6.Chow_group()
>>> HH(cone)
[-w^2]
>>> A(cone)
( 1 | 0, 0, 0, 0 | 0 )
>>> A(cone).cohomology_class()
[-w^2]

Here is an example of a toric variety with orbifold singularities, where we can also use the isomorphism with the rational cohomology ring:

Sage

sage: # needs sage.libs.singular
sage: WP4 = toric_varieties.P4_11169()
sage: A = WP4.Chow_group()
sage: HH = WP4.cohomology_ring()
sage: cone3d = Cone([(0,0,1,0), (0,0,0,1), (-9,-6,-1,-1)])
sage: A(cone3d)
( 0 | -1 | 0 | 0 | 0 )
sage: HH(cone3d)
[3*z4^3]
sage: D = -WP4.K()  # the anticanonical divisor
sage: A(D)
( 0 | 0 | 0 | -18 | 0 )
sage: HH(D)
[18*z4]
sage: WP4.integrate( A(cone3d).cohomology_class() * D.cohomology_class() )
1
sage: WP4.integrate( HH(cone3d) * D.cohomology_class() )
1
sage: A(cone3d).intersection_with_divisor(D).count_points()
1

Python

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> WP4 = toric_varieties.P4_11169()
>>> A = WP4.Chow_group()
>>> HH = WP4.cohomology_ring()
>>> cone3d = Cone([(Integer(0),Integer(0),Integer(1),Integer(0)), (Integer(0),Integer(0),Integer(0),Integer(1)), (-Integer(9),-Integer(6),-Integer(1),-Integer(1))])
>>> A(cone3d)
( 0 | -1 | 0 | 0 | 0 )
>>> HH(cone3d)
[3*z4^3]
>>> D = -WP4.K()  # the anticanonical divisor
>>> A(D)
( 0 | 0 | 0 | -18 | 0 )
>>> HH(D)
[18*z4]
>>> WP4.integrate( A(cone3d).cohomology_class() * D.cohomology_class() )
1
>>> WP4.integrate( HH(cone3d) * D.cohomology_class() )
1
>>> A(cone3d).intersection_with_divisor(D).count_points()
1

count_points()[source]#

Return the number of points in the Chow cycle.

OUTPUT:

An element of self.base_ring(), which is usually \(\ZZ\). The number of points in the Chow cycle.

EXAMPLES:

Sage

sage: P2 = toric_varieties.P2()
sage: A = P2.Chow_group()
sage: a = 5*A.gen(2) + 7*A.gen(1); a
( 5 | 7 | 0 )
sage: a.count_points()
5

Python

>>> from sage.all import *
>>> P2 = toric_varieties.P2()
>>> A = P2.Chow_group()
>>> a = Integer(5)*A.gen(Integer(2)) + Integer(7)*A.gen(Integer(1)); a
( 5 | 7 | 0 )
>>> a.count_points()
5

In the case of a smooth complete toric variety, the Chow (homology) groups are Poincaré dual to the integral cohomology groups. Here is such a smooth example:

Sage

sage: D = P2.divisor(1)
sage: a = D.Chow_cycle()
sage: aD = a.intersection_with_divisor(D)
sage: aD.count_points()
1
sage: P2.integrate(aD.cohomology_class())                                   # needs sage.libs.singular
1

Python

>>> from sage.all import *
>>> D = P2.divisor(Integer(1))
>>> a = D.Chow_cycle()
>>> aD = a.intersection_with_divisor(D)
>>> aD.count_points()
1
>>> P2.integrate(aD.cohomology_class())                                   # needs sage.libs.singular
1

For toric varieties with at most orbifold singularities, the isomorphism only holds over \(\QQ\). But the normalization of the integral is still chosen such that the intersection numbers (which are potentially rational) computed both ways agree:

Sage

sage: P1xP1_Z2 = toric_varieties.P1xP1_Z2()
sage: Dt = P1xP1_Z2.divisor(1);  Dt
V(t)
sage: Dy = P1xP1_Z2.divisor(3);  Dy
V(y)
sage: Dt.Chow_cycle(QQ).intersection_with_divisor(Dy).count_points()
1/2
sage: P1xP1_Z2.integrate(Dt.cohomology_class() * Dy.cohomology_class())     # needs sage.libs.singular
1/2

Python

>>> from sage.all import *
>>> P1xP1_Z2 = toric_varieties.P1xP1_Z2()
>>> Dt = P1xP1_Z2.divisor(Integer(1));  Dt
V(t)
>>> Dy = P1xP1_Z2.divisor(Integer(3));  Dy
V(y)
>>> Dt.Chow_cycle(QQ).intersection_with_divisor(Dy).count_points()
1/2
>>> P1xP1_Z2.integrate(Dt.cohomology_class() * Dy.cohomology_class())     # needs sage.libs.singular
1/2

degree()[source]#

Return the degree of the Chow cycle.

OUTPUT:

Integer. The complex dimension of the subvariety representing the Chow cycle.

This raises a ValueError if the Chow cycle is a sum of mixed degree cycles.

EXAMPLES:

Sage

sage: P2 = toric_varieties.P2()
sage: A = P2.Chow_group()
sage: [ a.degree() for a in A.gens() ]
[2, 1, 0]

Python

>>> from sage.all import *
>>> P2 = toric_varieties.P2()
>>> A = P2.Chow_group()
>>> [ a.degree() for a in A.gens() ]
[2, 1, 0]

intersection_with_divisor(divisor)[source]#

Intersect the Chow cycle with divisor.

See Chapter 5.1 of [Ful1993] for a description of the toric algorithm.

INPUT:

divisor – a ToricDivisor that can be moved away from the Chow cycle. For example, any Cartier divisor. See also ToricDivisor.move_away_from.

OUTPUT:

A new ChowCycle. If the divisor is not Cartier then this method potentially raises a ValueError, indicating that the divisor cannot be made transversal to the Chow cycle.

EXAMPLES:

Sage

sage: dP6 = toric_varieties.dP6()
sage: cone = dP6.fan().cone_containing(2); cone
1-d cone of Rational polyhedral fan in 2-d lattice N
sage: D = dP6.divisor(cone); D
V(y)
sage: A = dP6.Chow_group()
sage: A(cone)
( 0 | 0, 0, 1, 0 | 0 )
sage: intersection = A(cone).intersection_with_divisor(D); intersection
( -1 | 0, 0, 0, 0 | 0 )
sage: intersection.count_points()
-1

Python

>>> from sage.all import *
>>> dP6 = toric_varieties.dP6()
>>> cone = dP6.fan().cone_containing(Integer(2)); cone
1-d cone of Rational polyhedral fan in 2-d lattice N
>>> D = dP6.divisor(cone); D
V(y)
>>> A = dP6.Chow_group()
>>> A(cone)
( 0 | 0, 0, 1, 0 | 0 )
>>> intersection = A(cone).intersection_with_divisor(D); intersection
( -1 | 0, 0, 0, 0 | 0 )
>>> intersection.count_points()
-1

You can do the same computation over the rational Chow group since there is no torsion in this case:

Sage

sage: A_QQ = dP6.Chow_group(base_ring=QQ)
sage: A_QQ(cone)
( 0 | 0, 0, 0, 1 | 0 )
sage: intersection_QQ = A_QQ(cone).intersection_with_divisor(D); intersection
( -1 | 0, 0, 0, 0 | 0 )
sage: intersection_QQ.count_points()
-1
sage: type(intersection_QQ.count_points())
<... 'sage.rings.rational.Rational'>
sage: type(intersection.count_points())
<... 'sage.rings.integer.Integer'>

Python

>>> from sage.all import *
>>> A_QQ = dP6.Chow_group(base_ring=QQ)
>>> A_QQ(cone)
( 0 | 0, 0, 0, 1 | 0 )
>>> intersection_QQ = A_QQ(cone).intersection_with_divisor(D); intersection
( -1 | 0, 0, 0, 0 | 0 )
>>> intersection_QQ.count_points()
-1
>>> type(intersection_QQ.count_points())
<... 'sage.rings.rational.Rational'>
>>> type(intersection.count_points())
<... 'sage.rings.integer.Integer'>

project_to_degree(degree)[source]#

Project a (mixed-degree) Chow cycle to the given degree.

INPUT:

degree – integer. The degree to project to.

OUTPUT:

The projection of the Chow class to the given degree as a new ChowCycle of the same Chow group.

EXAMPLES:

Sage

sage: A = toric_varieties.P2().Chow_group()
sage: cycle = 10*A.gen(0) + 11*A.gen(1) + 12*A.gen(2); cycle
( 12 | 11 | 10 )
sage: cycle.project_to_degree(2)
( 0 | 0 | 10 )

Python

>>> from sage.all import *
>>> A = toric_varieties.P2().Chow_group()
>>> cycle = Integer(10)*A.gen(Integer(0)) + Integer(11)*A.gen(Integer(1)) + Integer(12)*A.gen(Integer(2)); cycle
( 12 | 11 | 10 )
>>> cycle.project_to_degree(Integer(2))
( 0 | 0 | 10 )

class sage.schemes.toric.chow_group.ChowGroupFactory[source]#

Bases: UniqueFactory

Factory for ChowGroup_class.

create_key_and_extra_args(toric_variety, base_ring=Integer Ring, check=True)[source]#

Create a key that uniquely determines the ChowGroup_class.

INPUT:

toric_variety – a toric variety.
base_ring – either \(\ZZ\) (default) or \(\QQ\). The coefficient ring of the Chow group.
check – boolean (default: True).

EXAMPLES:

Sage

sage: from sage.schemes.toric.chow_group import *
sage: P2 = toric_varieties.P2()
sage: ChowGroup(P2, ZZ, check=True) == ChowGroup(P2, ZZ, check=False)   # indirect doctest
True

Python

>>> from sage.all import *
>>> from sage.schemes.toric.chow_group import *
>>> P2 = toric_varieties.P2()
>>> ChowGroup(P2, ZZ, check=True) == ChowGroup(P2, ZZ, check=False)   # indirect doctest
True

create_object(version, key, **extra_args)[source]#

Create a ChowGroup_class.

INPUT:

version – object version. Currently not used.
key – a key created by create_key_and_extra_args().
**extra_args – Currently not used.

EXAMPLES:

Sage

sage: from sage.schemes.toric.chow_group import *
sage: P2 = toric_varieties.P2()
sage: ChowGroup(P2)    # indirect doctest
Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches

Python

>>> from sage.all import *
>>> from sage.schemes.toric.chow_group import *
>>> P2 = toric_varieties.P2()
>>> ChowGroup(P2)    # indirect doctest
Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches

class sage.schemes.toric.chow_group.ChowGroup_class(toric_variety, base_ring, check)[source]#

Bases: FGP_Module_class, WithEqualityById

The Chow group of a toric variety.

EXAMPLES:

Sage

sage: P2 = toric_varieties.P2()
sage: from sage.schemes.toric.chow_group import ChowGroup_class
sage: A = ChowGroup_class(P2,ZZ,True);  A
Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches
sage: A.an_element()
( 0 | 0 | 1 )

Python

>>> from sage.all import *
>>> P2 = toric_varieties.P2()
>>> from sage.schemes.toric.chow_group import ChowGroup_class
>>> A = ChowGroup_class(P2,ZZ,True);  A
Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches
>>> A.an_element()
( 0 | 0 | 1 )

Element[source]#: alias of ChowCycle

coordinate_vector(chow_cycle, degree=None, reduce=True)[source]#

Return the coordinate vector of the chow_cycle.

INPUT:

chow_cycle – a ChowCycle.
degree – None (default) or an integer.
reduce – boolean (default: True). Whether to reduce modulo the invariants.

OUTPUT:

If degree is None (default), the coordinate vector relative to the basis self.gens() is returned.
If some integer degree=d is specified, the chow cycle is projected to the given degree and the coordinate vector relative to the basis self.gens(degree=d) is returned.

EXAMPLES:

Sage

sage: A = toric_varieties.P2().Chow_group()
sage: a = A.gen(0) + 2*A.gen(1) + 3*A.gen(2)
sage: A.coordinate_vector(a)
(1, 2, 3)
sage: A.coordinate_vector(a, degree=1)
(2)

Python

>>> from sage.all import *
>>> A = toric_varieties.P2().Chow_group()
>>> a = A.gen(Integer(0)) + Integer(2)*A.gen(Integer(1)) + Integer(3)*A.gen(Integer(2))
>>> A.coordinate_vector(a)
(1, 2, 3)
>>> A.coordinate_vector(a, degree=Integer(1))
(2)

degree(k=None)[source]#

Return the degree-\(k\) Chow group.

INPUT:

k – an integer or None (default). The degree of the Chow group.

OUTPUT:

if \(k\) was specified, the Chow group \(A_k\) as an Abelian group.
if \(k\) was not specified, a tuple containing the Chow groups in all degrees.

Note

For a smooth toric variety, this is the same as the Poincaré-dual cohomology group \(H^{d-2k}(X,\ZZ)\).
For a simplicial toric variety (“orbifold”), \(A_k(X)\otimes \QQ = H^{d-2k}(X,\QQ)\).

EXAMPLES:

Four exercises from page 65 of [Ful1993]. First, an example with \(A_1(X)=\ZZ\oplus\ZZ/3\ZZ\):

Sage

sage: X = ToricVariety(Fan(cones=[[0,1], [1,2], [2,0]],
....:                      rays=[[2,-1], [-1,2], [-1,-1]]))
sage: A = X.Chow_group()
sage: A.degree(1)
C3 x Z

Python

>>> from sage.all import *
>>> X = ToricVariety(Fan(cones=[[Integer(0),Integer(1)], [Integer(1),Integer(2)], [Integer(2),Integer(0)]],
...                      rays=[[Integer(2),-Integer(1)], [-Integer(1),Integer(2)], [-Integer(1),-Integer(1)]]))
>>> A = X.Chow_group()
>>> A.degree(Integer(1))
C3 x Z

Second, an example with \(A_2(X)=\ZZ^2\):

Sage

sage: points = [[1,0,0], [0,1,0], [0,0,1], [1,-1,1], [-1,0,-1]]
sage: l = LatticePolytope(points)
sage: l.show3d()                                                            # needs sage.plot
sage: X = ToricVariety(FaceFan(l))
sage: A = X.Chow_group()
sage: A.degree(2)
Z^2

Python

>>> from sage.all import *
>>> points = [[Integer(1),Integer(0),Integer(0)], [Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)], [Integer(1),-Integer(1),Integer(1)], [-Integer(1),Integer(0),-Integer(1)]]
>>> l = LatticePolytope(points)
>>> l.show3d()                                                            # needs sage.plot
>>> X = ToricVariety(FaceFan(l))
>>> A = X.Chow_group()
>>> A.degree(Integer(2))
Z^2

Third, an example with \(A_2(X)=\ZZ^5\):

Sage

sage: cube = [[ 1,0,0], [0, 1,0], [0,0, 1], [-1, 1, 1],
....:         [-1,0,0], [0,-1,0], [0,0,-1], [ 1,-1,-1]]
sage: lat_cube = LatticePolytope(cube)
sage: X = ToricVariety(FaceFan((LatticePolytope(lat_cube))))
sage: X.Chow_group().degree(2)
Z^5

Python

>>> from sage.all import *
>>> cube = [[ Integer(1),Integer(0),Integer(0)], [Integer(0), Integer(1),Integer(0)], [Integer(0),Integer(0), Integer(1)], [-Integer(1), Integer(1), Integer(1)],
...         [-Integer(1),Integer(0),Integer(0)], [Integer(0),-Integer(1),Integer(0)], [Integer(0),Integer(0),-Integer(1)], [ Integer(1),-Integer(1),-Integer(1)]]
>>> lat_cube = LatticePolytope(cube)
>>> X = ToricVariety(FaceFan((LatticePolytope(lat_cube))))
>>> X.Chow_group().degree(Integer(2))
Z^5

Fourth, a fan that is not the fan over a polytope. Combinatorially, the fan is the same in the third example, only the coordinates of the first point are different. But the resulting fan is not the face fan of a cube, so the variety is “more singular”. Its Chow group has torsion, \(A_2(X)=\ZZ^5 \oplus \ZZ/2\):

Sage

sage: rays = [[ 1, 2, 3], [ 1,-1, 1], [-1, 1, 1], [-1,-1, 1],
....:         [-1,-1,-1], [-1, 1,-1], [ 1,-1,-1], [ 1, 1,-1]]
sage: cones = [[0,1,2,3], [4,5,6,7], [0,1,7,6],
....:          [4,5,3,2], [0,2,5,7], [4,6,1,3]]
sage: X = ToricVariety(Fan(cones, rays))
sage: X.Chow_group().degree(2)  # long time (2s on sage.math, 2011)
C2 x Z^5

Python

>>> from sage.all import *
>>> rays = [[ Integer(1), Integer(2), Integer(3)], [ Integer(1),-Integer(1), Integer(1)], [-Integer(1), Integer(1), Integer(1)], [-Integer(1),-Integer(1), Integer(1)],
...         [-Integer(1),-Integer(1),-Integer(1)], [-Integer(1), Integer(1),-Integer(1)], [ Integer(1),-Integer(1),-Integer(1)], [ Integer(1), Integer(1),-Integer(1)]]
>>> cones = [[Integer(0),Integer(1),Integer(2),Integer(3)], [Integer(4),Integer(5),Integer(6),Integer(7)], [Integer(0),Integer(1),Integer(7),Integer(6)],
...          [Integer(4),Integer(5),Integer(3),Integer(2)], [Integer(0),Integer(2),Integer(5),Integer(7)], [Integer(4),Integer(6),Integer(1),Integer(3)]]
>>> X = ToricVariety(Fan(cones, rays))
>>> X.Chow_group().degree(Integer(2))  # long time (2s on sage.math, 2011)
C2 x Z^5

Finally, Example 1.3 of [FS1994]:

Sage

sage: def points_mod(k):
....:     return matrix([[ 1, 1, 2*k+1], [ 1,-1, 1],
....:                    [-1, 1, 1], [-1,-1, 1], [-1,-1,-1],
....:                    [-1, 1,-1], [ 1,-1,-1], [ 1, 1,-1]])
sage: def rays(k):
....:     return matrix([[ 1,  1,  1],
....:                    [ 1, -1,  1],
....:                    [-1,  1,  1]]).solve_left(points_mod(k)).rows()
sage: cones = [[0,1,2,3], [4,5,6,7], [0,1,7,6], [4,5,3,2], [0,2,5,7], [4,6,1,3]]
sage: X_Delta = lambda k: ToricVariety(Fan(cones=cones, rays=rays(k)))
sage: X_Delta(0).Chow_group().degree()  # long time (3s on sage.math, 2011)
(Z, Z, Z^5, Z)
sage: X_Delta(1).Chow_group().degree()  # long time (3s on sage.math, 2011)
(Z, 0, Z^5, Z)
sage: X_Delta(2).Chow_group().degree()  # long time (3s on sage.math, 2011)
(Z, C2, Z^5, Z)
sage: X_Delta(2).Chow_group(base_ring=QQ).degree()  # long time (4s on sage.math, 2011)
(Q, 0, Q^5, Q)

Python

>>> from sage.all import *
>>> def points_mod(k):
...     return matrix([[ Integer(1), Integer(1), Integer(2)*k+Integer(1)], [ Integer(1),-Integer(1), Integer(1)],
...                    [-Integer(1), Integer(1), Integer(1)], [-Integer(1),-Integer(1), Integer(1)], [-Integer(1),-Integer(1),-Integer(1)],
...                    [-Integer(1), Integer(1),-Integer(1)], [ Integer(1),-Integer(1),-Integer(1)], [ Integer(1), Integer(1),-Integer(1)]])
>>> def rays(k):
...     return matrix([[ Integer(1),  Integer(1),  Integer(1)],
...                    [ Integer(1), -Integer(1),  Integer(1)],
...                    [-Integer(1),  Integer(1),  Integer(1)]]).solve_left(points_mod(k)).rows()
>>> cones = [[Integer(0),Integer(1),Integer(2),Integer(3)], [Integer(4),Integer(5),Integer(6),Integer(7)], [Integer(0),Integer(1),Integer(7),Integer(6)], [Integer(4),Integer(5),Integer(3),Integer(2)], [Integer(0),Integer(2),Integer(5),Integer(7)], [Integer(4),Integer(6),Integer(1),Integer(3)]]
>>> X_Delta = lambda k: ToricVariety(Fan(cones=cones, rays=rays(k)))
>>> X_Delta(Integer(0)).Chow_group().degree()  # long time (3s on sage.math, 2011)
(Z, Z, Z^5, Z)
>>> X_Delta(Integer(1)).Chow_group().degree()  # long time (3s on sage.math, 2011)
(Z, 0, Z^5, Z)
>>> X_Delta(Integer(2)).Chow_group().degree()  # long time (3s on sage.math, 2011)
(Z, C2, Z^5, Z)
>>> X_Delta(Integer(2)).Chow_group(base_ring=QQ).degree()  # long time (4s on sage.math, 2011)
(Q, 0, Q^5, Q)

gens(degree=None)[source]#

Return the generators of the Chow group.

INPUT:

degree – integer (optional). The degree of the Chow group.

OUTPUT:

if no degree is specified, the generators of the whole Chow group. The chosen generators may be of mixed degree.
if degree= \(k\) was specified, the generators of the degree-\(k\) part \(A_k\) of the Chow group.

EXAMPLES:

Sage

sage: A = toric_varieties.P2().Chow_group()
sage: A.gens()
(( 0 | 0 | 1 ), ( 0 | 1 | 0 ), ( 1 | 0 | 0 ))
sage: A.gens(degree=1)
(( 0 | 1 | 0 ),)

Python

>>> from sage.all import *
>>> A = toric_varieties.P2().Chow_group()
>>> A.gens()
(( 0 | 0 | 1 ), ( 0 | 1 | 0 ), ( 1 | 0 | 0 ))
>>> A.gens(degree=Integer(1))
(( 0 | 1 | 0 ),)

relation_gens()[source]#

Return the Chow cycles equivalent to zero.

For each \(d-k-1\)-dimensional cone \(\rho \in \Sigma^{(d-k-1)}\), the relations in \(A_k(X)\), that is the cycles equivalent to zero, are generated by

\[0 \stackrel{!}{=} \mathop{\mathrm{div}}(u) = \sum_{\rho < \sigma \in \Sigma^{(n-p)} } \big< u, n_{\rho,\sigma} \big> V(\sigma) ,\qquad u \in M(\rho)\]

where \(n_{\rho,\sigma}\) is a (randomly chosen) lift of the generator of \(N_\sigma/N_\rho \simeq \ZZ\). See also Exercise 12.5.7 of [CLS2011].

See also relations() to obtain the relations as submodule of the free module generated by the cones. Or use self.relations().gens() to list the relations in the free module.

OUTPUT:

A tuple of Chow cycles, each rationally equivalent to zero, that generates the rational equivalence.

EXAMPLES:

Sage

sage: P2 = toric_varieties.P2()
sage: A = P2.Chow_group()
sage: first = A.relation_gens()[0]; first
( 0 | 0 | 0 )
sage: first.is_zero()
True
sage: first.lift()
(0, 1, 0, -1, 0, 0, 0)

Python

>>> from sage.all import *
>>> P2 = toric_varieties.P2()
>>> A = P2.Chow_group()
>>> first = A.relation_gens()[Integer(0)]; first
( 0 | 0 | 0 )
>>> first.is_zero()
True
>>> first.lift()
(0, 1, 0, -1, 0, 0, 0)

scheme()[source]#

Return the underlying toric variety.

OUTPUT: A ToricVariety.

EXAMPLES:

Sage

sage: P2 = toric_varieties.P2()
sage: A = P2.Chow_group()
sage: A.scheme()
2-d CPR-Fano toric variety covered by 3 affine patches
sage: A.scheme() is P2
True

Python

>>> from sage.all import *
>>> P2 = toric_varieties.P2()
>>> A = P2.Chow_group()
>>> A.scheme()
2-d CPR-Fano toric variety covered by 3 affine patches
>>> A.scheme() is P2
True

class sage.schemes.toric.chow_group.ChowGroup_degree_class(A, d)[source]#

Bases: SageObject

A fixed-degree subgroup of the Chow group of a toric variety.

Warning

Use degree() to construct ChowGroup_degree_class instances.

EXAMPLES:

Sage

sage: P2 = toric_varieties.P2()
sage: A = P2.Chow_group(); A
Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches
sage: A.degree()
(Z, Z, Z)
sage: A.degree(2)
Z
sage: type(_)
<class 'sage.schemes.toric.chow_group.ChowGroup_degree_class'>

Python

>>> from sage.all import *
>>> P2 = toric_varieties.P2()
>>> A = P2.Chow_group(); A
Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches
>>> A.degree()
(Z, Z, Z)
>>> A.degree(Integer(2))
Z
>>> type(_)
<class 'sage.schemes.toric.chow_group.ChowGroup_degree_class'>

gen(i)[source]#

Return the i-th generator of the Chow group of fixed degree.

INPUT:

i – integer. The index of the generator to be returned.

OUTPUT: A Chow cycle.

EXAMPLES:

Sage

sage: projective_plane = toric_varieties.P2()
sage: A2 = projective_plane.Chow_group().degree(2)
sage: A2.gen(0)
( 0 | 0 | 1 )

Python

>>> from sage.all import *
>>> projective_plane = toric_varieties.P2()
>>> A2 = projective_plane.Chow_group().degree(Integer(2))
>>> A2.gen(Integer(0))
( 0 | 0 | 1 )

gens()[source]#

Return the generators of the Chow group of fixed degree.

OUTPUT: A tuple of Chow cycles of fixed degree generating module().

EXAMPLES:

Sage

sage: projective_plane = toric_varieties.P2()
sage: A2 = projective_plane.Chow_group().degree(2)
sage: A2.gens()
(( 0 | 0 | 1 ),)

Python

>>> from sage.all import *
>>> projective_plane = toric_varieties.P2()
>>> A2 = projective_plane.Chow_group().degree(Integer(2))
>>> A2.gens()
(( 0 | 0 | 1 ),)

module()[source]#

Return the submodule of the toric Chow group generated.

OUTPUT: A sage.modules.fg_pid.fgp_module.FGP_Module_class.

EXAMPLES:

Sage

sage: projective_plane = toric_varieties.P2()
sage: A2 = projective_plane.Chow_group().degree(2)
sage: A2.module()
Finitely generated module V/W over Integer Ring with invariants (0)

Python

>>> from sage.all import *
>>> projective_plane = toric_varieties.P2()
>>> A2 = projective_plane.Chow_group().degree(Integer(2))
>>> A2.module()
Finitely generated module V/W over Integer Ring with invariants (0)

ngens()[source]#

Return the number of generators.

OUTPUT: An integer.

EXAMPLES:

Sage

sage: projective_plane = toric_varieties.P2()
sage: A2 = projective_plane.Chow_group().degree(2)
sage: A2.ngens()
1

Python

>>> from sage.all import *
>>> projective_plane = toric_varieties.P2()
>>> A2 = projective_plane.Chow_group().degree(Integer(2))
>>> A2.ngens()
1

sage.schemes.toric.chow_group.is_ChowCycle(x)[source]#

Return whether x is a ChowCycle.

INPUT:

x – anything.

OUTPUT: True or False.

EXAMPLES:

Sage

sage: P2 = toric_varieties.P2()
sage: A = P2.Chow_group()
sage: from sage.schemes.toric.chow_group import *
sage: is_ChowCycle(A)
False
sage: is_ChowCycle(A.an_element())
True
sage: is_ChowCycle('Victoria')
False

Python

>>> from sage.all import *
>>> P2 = toric_varieties.P2()
>>> A = P2.Chow_group()
>>> from sage.schemes.toric.chow_group import *
>>> is_ChowCycle(A)
False
>>> is_ChowCycle(A.an_element())
True
>>> is_ChowCycle('Victoria')
False

sage.schemes.toric.chow_group.is_ChowGroup(x)[source]#

Return whether x is a ChowGroup_class.

INPUT:

x – anything.

OUTPUT: True or False.

EXAMPLES:

Sage

sage: P2 = toric_varieties.P2()
sage: A = P2.Chow_group()
sage: from sage.schemes.toric.chow_group import is_ChowGroup
sage: is_ChowGroup(A)
doctest:warning...
DeprecationWarning: The function is_ChowGroup is deprecated; use 'isinstance(..., ChowGroup_class)' instead.
See https://github.com/sagemath/sage/issues/38022 for details.
True
sage: is_ChowGroup('Victoria')
False

Python

>>> from sage.all import *
>>> P2 = toric_varieties.P2()
>>> A = P2.Chow_group()
>>> from sage.schemes.toric.chow_group import is_ChowGroup
>>> is_ChowGroup(A)
doctest:warning...
DeprecationWarning: The function is_ChowGroup is deprecated; use 'isinstance(..., ChowGroup_class)' instead.
See https://github.com/sagemath/sage/issues/38022 for details.
True
>>> is_ChowGroup('Victoria')
False