def _element_constructor_(self, x, check=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 )
"""
fan = self._variety.fan()
if is_Cone(x):
cone = fan.embed(x)
return self.element_class(self, self._cone_to_V(cone), False)
if is_ToricDivisor(x):
v = sum(x.coefficient(i)*self._cone_to_V(onecone)
for i,onecone in enumerate(fan(1)))
return self.element_class(self, v, False)
return super(ChowGroup_class,self)._element_constructor_(x, check)
def intersection_with_divisor(self, divisor):
"""
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())
<type 'sage.rings.rational.Rational'>
sage: type(intersection.count_points())
<type '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)]
"""
assert is_ToricDivisor(divisor), str(divisor)+' is not a toric divisor.'
A = self.parent() # the Chow group
X = A._variety # the toric variety
intersection = A(0)
coefficients = self.lift()
for sigma_idx, sigma in enumerate(A._cones):
if sigma.dim()==X.dimension():
# full-dimensional cone = degree-0 Chow cycle
continue
coefficient = coefficients[sigma_idx]
if coefficient==0:
continue
D = divisor.move_away_from(sigma)
for gamma in sigma.facet_of():
# note: the relative quotients are of dimension one
n = gamma.relative_quotient(sigma).gen(0).lift()
perp = sigma.relative_orthogonal_quotient(gamma).gen(0).lift()
I_gamma = set(gamma.ambient_ray_indices()) - set(sigma.ambient_ray_indices())
i = I_gamma.pop() # index of a ray in gamma but not sigma
v_i = X.fan().ray(i)
a_i = D.coefficient(i)
s_i = (v_i*perp)/(n*perp)
b_gamma = a_i/s_i
# print sigma._points_idx, "\t", i, D, a_i, s_i, b_gamma, gamma.A()
intersection += self.base_ring()(coefficient*b_gamma) * A(gamma)
return intersection
