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# -*- coding: utf-8 -*- 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 [FMSS1]_ 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:
.. [wp:ChowRing] :wikipedia:`Chow_ring`
.. [FMSS1] Fulton, MacPherson, Sottile, Sturmfels: *Intersection theory on spherical varieties*, J. of Alg. Geometry 4 (1995), 181-193. http://www.math.tamu.edu/~frank.sottile/research/ps/spherical.ps.gz
.. [FultonChow] Chapter 5.1 "Chow Groups" of Fulton, William: *Introduction to Toric Varieties*, Princeton University Press
EXAMPLES::
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(0,A.ngens()) ) # an element of A sage: a # long time (2s on sage.math, 2011) ( 3 | 1 mod 7 | 0 mod 2, 1 mod 2, 4, 5, 6, 7, 8 | 9 )
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 :meth:`ChowGroup_class.gens` for more details.
Cones of toric varieties can determine their own Chow cycle::
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 | 1 mod 7 | 0 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 ) sage: A_cone.degree() 1 sage: 2 * A_cone ( 0 | 2 mod 7 | 0 mod 2, 0 mod 2, 0, 0, 0, 0, 0 | 0 ) sage: A_cone + A.gen(0) ( 0 | 1 mod 7 | 0 mod 2, 1 mod 2, 0, 0, 0, 0, 0 | 0 )
Chow cycles can be of mixed degrees::
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(0,X.dimension()+1) ) == mixed True """
#***************************************************************************** # Copyright (C) 2010 Volker Braun <vbraun.name@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # # http://www.gnu.org/licenses/ #*****************************************************************************
#******************************************************************* """ The elements of the Chow group.
.. WARNING::
Do not construct :class:`ChowCycle` objects manually. Instead, use the parent :class:`ChowGroup<ChowGroup_class>` to obtain generators or Chow cycles corresponding to cones of the fan.
EXAMPLES::
sage: P2 = toric_varieties.P2() sage: A = P2.Chow_group() sage: A.gens() (( 1 | 0 | 0 ), ( 0 | 1 | 0 ), ( 0 | 0 | 1 )) sage: cone = P2.fan(1)[0] sage: A(cone) ( 0 | 1 | 0 ) sage: A( Cone([(1,0)]) ) ( 0 | 1 | 0 ) """
r""" Construct a :class:`ChowCycle`.
INPUT:
- ``parent`` -- a :class:`ChowGroup_class`.
- ``v`` -- a vector in the covering module, that is, with one entry for each cone of the toric variety.
- ``check`` -- boolean (default: ``True``). Verify that ``v`` is in the covering module. Set to ``False`` if you want to initialize from a coordinate vector.
TESTS::
sage: P2 = toric_varieties.P2() sage: A = P2.Chow_group() sage: from sage.schemes.toric.chow_group import ChowCycle sage: ChowCycle(A, (0,1,2,3,11,12,13), check=False) ( 36 | 6 | 0 ) """
r""" Return a string representation of the Chow cycle.
OUTPUT:
See the module-level documentation for details.
EXAMPLES::
sage: P2 = toric_varieties.P2() sage: A = P2.Chow_group() sage: A.degree() (Z, Z, Z) sage: A.an_element()._repr_() '( 1 | 0 | 0 )'
A more complicated example with torsion::
sage: X = toric_varieties.Cube_nonpolyhedral() sage: A = X.Chow_group() sage: A.degree() (Z, 0, C2 x Z^5, Z) sage: sum( A.gen(i) * (i+1) for i in range(0,A.ngens()) ) ( 2 || 1 mod 2, 3, 4, 5, 6, 7 | 8 ) """
r""" The degree of the Chow cycle.
OUTPUT:
Integer. The complex dimension of the subvariety representing the Chow cycle. Raises a ``ValueError`` if the Chow cycle is a sum of mixed degree cycles.
EXAMPLES::
sage: P2 = toric_varieties.P2() sage: A = P2.Chow_group() sage: [ a.degree() for a in A.gens() ] [0, 1, 2] """ return self._dim
raise ValueError('Chow cycle is not of definite degree.')
r""" 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 :class:`ChowCycle` of the same Chow group.
EXAMPLES::
sage: A = toric_varieties.P2().Chow_group() sage: cycle = 10*A.gen(0) + 11*A.gen(1) + 12*A.gen(2) sage: cycle ( 10 | 11 | 12 ) sage: cycle.project_to_degree(2) ( 0 | 0 | 12 ) """
r""" 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: P2 = toric_varieties.P2() sage: A = P2.Chow_group() sage: a = 5*A.gen(0) + 7*A.gen(1); a ( 5 | 7 | 0 ) sage: 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: 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() ) 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: 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() ) 1/2 """
""" Intersect the Chow cycle with ``divisor``.
See [FultonChow]_ for a description of the toric algorithm.
INPUT:
- ``divisor`` -- a :class:`ToricDivisor <sage.schemes.toric.divisor.ToricDivisor_generic>` that can be moved away from the Chow cycle. For example, any Cartier divisor. See also :meth:`ToricDivisor.move_away_from <sage.schemes.toric.divisor.ToricDivisor_generic.move_away_from>`.
OUTPUT:
A new :class:`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: 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, 0, 1 | 0 ) sage: intersection = A(cone).intersection_with_divisor(D); intersection ( -1 | 0, 0, 0, 0 | 0 ) sage: intersection.count_points() -1
You can do the same computation over the rational Chow group since there is no torsion in this case::
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'>
TESTS:
The relations are the Chow cycles rationally equivalent to the zero cycle. Their intersection with any divisor must be the zero cycle::
sage: [ r.intersection_with_divisor(D) for r in dP6.Chow_group().relation_gens() ] [( 0 | 0, 0, 0, 0 | 0 ), ( 0 | 0, 0, 0, 0 | 0 ), ( 0 | 0, 0, 0, 0 | 0 ), ( 0 | 0, 0, 0, 0 | 0 ), ( 0 | 0, 0, 0, 0 | 0 ), ( 0 | 0, 0, 0, 0 | 0 ), ( 0 | 0, 0, 0, 0 | 0 )] sage: [ r.intersection_with_divisor(D).lift() for r in dP6.Chow_group().relation_gens() ] [(0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)] """
# full-dimensional cone = degree-0 Chow cycle # note: the relative quotients are of dimension one # print sigma._points_idx, "\t", i, D, a_i, s_i, b_gamma, gamma.A()
r""" 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 :meth:`~sage.schemes.toric.variety.ToricVariety_field.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 (:meth:`volume_class <sage.schemes.toric.variety.ToricVariety_field.volume_class>`) and :meth:`count_points` are chosen to agree.
OUTPUT:
The :class:`~sage.schemes.toric.variety.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, :meth:`cohomology_class` raises a ``ValueError``.
EXAMPLES::
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]
Here is an example of a toric variety with orbifold singularities, where we can also use the isomorphism with the rational cohomology ring::
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 """ raise ValueError
#******************************************************************* """ Factory for :class:`ChowGroup_class`. """
""" Create a key that uniquely determines the :class:`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: 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 """ raise ValueError('First argument must be a toric variety.')
raise ValueError('Base ring must be either ZZ or QQ.')
""" Create a :class:`ChowGroup_class`.
INPUT:
- ``version`` -- object version. Currently not used.
- ``key`` -- a key created by :meth:`create_key_and_extra_args`.
- ``**extra_args`` -- Currently not used.
EXAMPLES::
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 """
#******************************************************************* r""" The Chow group of a toric variety.
EXAMPLES::
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() ( 1 | 0 | 0 ) """
r""" EXAMPLES::
sage: from sage.schemes.toric.chow_group import * sage: P2=toric_varieties.P2() sage: A = ChowGroup_class(P2,ZZ,True); A Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches sage: is_ChowGroup(A) True sage: is_ChowCycle(A.an_element()) True
TESTS::
sage: A_ZZ = P2.Chow_group() sage: 2 * A_ZZ.an_element() * 3 ( 6 | 0 | 0 ) sage: 1/2 * A_ZZ.an_element() * 1/3 Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for *: 'Rational Field' and 'Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches' sage: A_ZZ.get_action(ZZ) Right scalar multiplication by Integer Ring on Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches sage: A_ZZ.get_action(QQ)
You can't multiply integer classes with fractional numbers. For that you need to go to the rational Chow group::
sage: A_QQ = P2.Chow_group(QQ) sage: 2 * A_QQ.an_element() * 3 ( 0 | 0 | 6 ) sage: 1/2 * A_QQ.an_element() * 1/3 ( 0 | 0 | 1/6 ) sage: A_QQ.get_action(ZZ) Right scalar multiplication by Integer Ring on QQ-Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches sage: A_QQ.get_action(QQ) Right scalar multiplication by Rational Field on QQ-Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches """
# cones are automatically sorted by dimension
r""" Return the underlying toric variety.
OUTPUT:
A :class:`ToricVariety <sage.schemes.toric.variety.ToricVariety_field>`.
EXAMPLES::
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 """
r""" Construct a :class:`ChowCycle`.
INPUT:
- ``x`` -- a cone of the fan, a toric divisor, or a valid input for :class:`sage.modules.fg_pid.fgp_module.FGP_Module_class`.
- ``check`` -- bool (default: ``True``). See :class:`sage.modules.fg_pid.fgp_module.FGP_Module_class`.
EXAMPLES::
sage: dP6 = toric_varieties.dP6() sage: A = dP6.Chow_group() sage: cone = dP6.fan(dim=1)[4] sage: A(cone) ( 0 | 0, 1, 0, 0 | 0 ) sage: A(Cone(cone)) # isomorphic but not identical to a cone of the fan! ( 0 | 0, 1, 0, 0 | 0 ) sage: A( dP6.K() ) ( 0 | -1, -2, -2, -1 | 0 ) """ for i,onecone in enumerate(fan(1)))
""" Return true if S canonically coerces to self.
EXAMPLES::
sage: A = toric_varieties.P2().Chow_group() sage: A._coerce_map_from_(ZZ) # private method False sage: A.has_coerce_map_from(ZZ) # recommended usage False """ # We might want to coerce Cone_of_fans into ChowCycles # but cones don't have parents at the moment.
r""" Return the rational equivalence relations between the cones of the fan.
See :meth:`relation_gens` for details.
EXAMPLES::
sage: points_mod = lambda k: 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: points = lambda k: 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=points(k)) ) sage: from sage.schemes.toric.chow_group import ChowGroup sage: A = ChowGroup( X_Delta(2) ) sage: rel = A._rational_equivalence_relations(A.cover()).basis() sage: matrix(rel).submatrix(col=0, ncols=1).elementary_divisors() [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] sage: matrix(rel).submatrix(col=1, ncols=8).elementary_divisors() [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] sage: matrix(rel).submatrix(col=9, ncols=12).elementary_divisors() [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0] sage: matrix(rel).submatrix(col=21, ncols=6).elementary_divisors() [1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] """
r""" Return the quotient of the Chow group by a subgroup.
OUTPUT:
Currently not implemented.
EXAMPLES::
sage: A = toric_varieties.dP6().Chow_group() sage: Asub = A.submodule([ A.gen(0), A.gen(3) ]) sage: A/Asub Traceback (most recent call last): ... NotImplementedError: Quotients of the Chow group are not implemented. """
""" Return a string representation.
EXAMPLES::
sage: P2=toric_varieties.P2() sage: from sage.schemes.toric.chow_group import ChowGroup sage: ChowGroup(P2,ZZ)._repr_() 'Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches' sage: ChowGroup(P2,QQ)._repr_() 'QQ-Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches' """ else: raise ValueError
r""" Comparison of two Chow groups.
INPUT:
- ``other`` -- anything.
OUTPUT:
``True`` or ``False``.
EXAMPLES::
sage: P2 = toric_varieties.P2() sage: P2.Chow_group() == P2.Chow_group() True sage: P2.Chow_group(ZZ) == P2.Chow_group(QQ) False """
r""" Convert a cone into the corresponding vector in ``self._V``
INPUT:
- ``cone`` -- a :class:`sage.geometry.cone.ConvexRationalPolyhedralCone`.
OUTPUT:
The corresponding element of ``self.V()``.
EXAMPLES::
sage: P2 = toric_varieties.P2() sage: A = P2.Chow_group() sage: cone = P2.fan(dim=1)[0] sage: A._cone_to_V(cone) (0, 1, 0, 0, 0, 0, 0) """
r""" 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 [FultonP65]_. First, an example with `A_1(X)=\ZZ\oplus\ZZ/3\ZZ`::
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
Second, an example with `A_2(X)=\ZZ^2`::
sage: points = [[1,0,0],[0,1,0],[0,0,1],[1,-1,1],[-1,0,-1]] sage: l = LatticePolytope(points) sage: l.show3d() sage: X = ToricVariety(FaceFan(l)) sage: A = X.Chow_group() sage: A.degree(2) Z^2
Third, an example with `A_2(X)=\ZZ^5`::
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
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: 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
Finally, Example 1.3 of [FS]_::
sage: points_mod = lambda k: 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: rays = lambda k: 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) """
for d in range(0,self._variety.dimension()+1))
r""" Return the coordinate vector of the ``chow_cycle``.
INPUT:
- ``chow_cycle`` -- a :class:`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: 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) """
r""" 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: A = toric_varieties.P2().Chow_group() sage: A.gens() (( 1 | 0 | 0 ), ( 0 | 1 | 0 ), ( 0 | 0 | 1 )) sage: A.gens(degree=1) (( 0 | 1 | 0 ),) """ else:
r""" 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
.. MATH::
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 [CLS]_.
See also :meth:`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: P2 = toric_varieties.P2() sage: A = P2.Chow_group() sage: first = A.relation_gens()[0] sage: first ( 0 | 0 | 0 ) sage: first.is_zero() True sage: first.lift() (0, 1, 0, -1, 0, 0, 0) """
#******************************************************************* r""" A fixed-degree subgroup of the Chow group of a toric variety.
.. WARNING::
Use :meth:`~sage.schemes.toric.chow_group.ChowGroup_class.degree` to construct :class:`ChowGroup_degree_class` instances.
EXAMPLES::
sage: P2 = toric_varieties.P2() sage: A = P2.Chow_group() sage: 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'> """
r""" Construct a :class:`ChowGroup_degree_class`.
INPUT:
- ``A`` -- A :class:`ChowGroup_class`.
- ``d`` -- integer. The degree of the Chow group.
EXAMPLES::
sage: P2 = toric_varieties.P2() sage: A = P2.Chow_group() sage: from sage.schemes.toric.chow_group import ChowGroup_degree_class sage: A2 = ChowGroup_degree_class(A,2) sage: A2 Z """
# Some generators
# The minimal set of generators for a in self._module.gens() ])
""" Return a string representation.
OUTPUT:
String.
EXAMPLES::
sage: projective_plane = toric_varieties.P2() sage: A2 = projective_plane.Chow_group().degree(2) sage: A2._repr_() 'Z' sage: A2_QQ = projective_plane.Chow_group(base_ring=QQ).degree(2) sage: A2_QQ._repr_() 'Q' """
else: raise NotImplementedError('Base ring must be ZZ or QQ.')
""" Return the submodule of the toric Chow group generated.
OUTPUT:
A :class:`sage.modules.fg_pid.fgp_module.FGP_Module_class`
EXAMPLES::
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) """
""" Return the number of generators.
OUTPUT:
An integer.
EXAMPLES::
sage: projective_plane = toric_varieties.P2() sage: A2 = projective_plane.Chow_group().degree(2) sage: A2.ngens() 1 """
""" 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 tuple of Chow cycles of fixed degree generating :meth:`module`.
EXAMPLES::
sage: projective_plane = toric_varieties.P2() sage: A2 = projective_plane.Chow_group().degree(2) sage: A2.gen(0) ( 0 | 0 | 1 ) """
""" Return the generators of the Chow group of fixed degree.
OUTPUT:
A tuple of Chow cycles of fixed degree generating :meth:`module`.
EXAMPLES::
sage: projective_plane = toric_varieties.P2() sage: A2 = projective_plane.Chow_group().degree(2) sage: A2.gens() (( 0 | 0 | 1 ),) """
#******************************************************************* r""" Return whether ``x`` is a :class:`ChowGroup_class`
INPUT:
- ``x`` -- anything.
OUTPUT:
``True`` or ``False``.
EXAMPLES::
sage: P2=toric_varieties.P2() sage: A = P2.Chow_group() sage: from sage.schemes.toric.chow_group import is_ChowGroup sage: is_ChowGroup(A) True sage: is_ChowGroup('Victoria') False """
#******************************************************************* r""" Return whether ``x`` is a :class:`ChowGroup_class`
INPUT:
- ``x`` -- anything.
OUTPUT:
``True`` or ``False``.
EXAMPLES::
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 """ |