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morphism.py
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morphism.py
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r"""
Morphisms of Toric Varieties
There are three "obvious" ways to map toric varieties to toric
varieties:
1. Polynomial maps in local coordinates, the usual morphisms in
algebraic geometry.
2. Polynomial maps in the (global) homogeneous coordinates.
3. Toric morphisms, that is, algebraic morphisms equivariant with
respect to the torus action on the toric variety.
Both 2 and 3 are special cases of 1, which is just to say that we
always remain within the realm of algebraic geometry. But apart from
that, none is included in one of the other cases. In the examples
below, we will explore some algebraic maps that can or can not be
written as a toric morphism. Often a toric morphism can be written
with polynomial maps in homogeneous coordinates, but sometimes it
cannot.
The toric morphisms are perhaps the most mysterious at the
beginning. Let us quickly review their definition (See Definition
3.3.3 of [CLS]_). Let `\Sigma_1` be a fan in `N_{1,\RR}` and `\Sigma_2` be a
fan in `N_{2,\RR}`. A morphism `\phi: X_{\Sigma_1} \to X_{\Sigma_2}`
of the associated toric varieties is toric if `\phi` maps the maximal
torus `T_{N_1} \subseteq X_{\Sigma_1}` into `T_{N_2} \subseteq
X_{\Sigma_2}` and `\phi|_{T_N}` is a group homomorphism.
The data defining a toric morphism is precisely what defines a fan
morphism (see :mod:`~sage.geometry.fan_morphism`), extending the more
familiar dictionary between toric varieties and fans. Toric geometry
is a functor from the category of fans and fan morphisms to the
category of toric varieties and toric morphisms.
.. note::
Do not create the toric morphisms (or any morphism of schemes)
directly from the the ``SchemeMorphism...`` classes. Instead, use the
:meth:`~sage.schemes.generic.scheme.hom` method common to all
algebraic schemes to create new homomorphisms.
EXAMPLES:
First, consider the following embedding of `\mathbb{P}^1` into
`\mathbb{P}^2` ::
sage: P2.<x,y,z> = toric_varieties.P2()
sage: P1.<u,v> = toric_varieties.P1()
sage: P1.hom([0,u^2+v^2,u*v], P2)
Scheme morphism:
From: 1-d CPR-Fano toric variety covered by 2 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined on coordinates by sending [u : v] to
[0 : u^2 + v^2 : u*v]
This is a well-defined morphism of algebraic varieties because
homogeneously rescaled coordinates of a point of `\mathbb{P}^1` map to the same
point in `\mathbb{P}^2` up to its homogeneous rescalings. It is not
equivariant with respect to the torus actions
.. math::
\CC^\times \times \mathbb{P}^1,
(\mu,[u:v]) \mapsto [u:\mu v]
\quad\text{and}\quad
\left(\CC^\times\right)^2 \times \mathbb{P}^2,
((\alpha,\beta),[x:y:z]) \mapsto [x:\alpha y:\beta z]
,
hence it is not a toric morphism. Clearly, the problem is that
the map in homogeneous coordinates contains summands that transform
differently under the torus action. However, this is not the only
difficulty. For example, consider ::
sage: phi = P1.hom([0,u,v], P2); phi
Scheme morphism:
From: 1-d CPR-Fano toric variety covered by 2 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined on coordinates by sending [u : v] to
[0 : u : v]
This map is actually the embedding of the
:meth:`~sage.schemes.toric.variety.ToricVariety_field.orbit_closure`
associated to one of the rays of the fan of `\mathbb{P}^2`. Now the
morphism is equivariant with respect to **some** map `\CC^\times \to
(\CC^\times)^2` of the maximal tori of `\mathbb{P}^1` and
`\mathbb{P}^2`. But this map of the maximal tori cannot be the same as
``phi`` defined above. Indeed, the image of ``phi`` completely misses
the maximal torus `T_{\mathbb{P}^2} = \{ [x:y:z] | x\not=0, y\not=0,
z\not=0 \}` of `\mathbb{P}^2`.
Consider instead the following morphism of fans::
sage: fm = FanMorphism( matrix(ZZ,[[1,0]]), P1.fan(), P2.fan() ); fm
Fan morphism defined by the matrix
[1 0]
Domain fan: Rational polyhedral fan in 1-d lattice N
Codomain fan: Rational polyhedral fan in 2-d lattice N
which also defines a morphism of toric varieties::
sage: P1.hom(fm, P2)
Scheme morphism:
From: 1-d CPR-Fano toric variety covered by 2 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined by sending Rational polyhedral fan in 1-d lattice N
to Rational polyhedral fan in 2-d lattice N.
The fan morphism map is equivalent to the following polynomial map::
sage: _.as_polynomial_map()
Scheme morphism:
From: 1-d CPR-Fano toric variety covered by 2 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined on coordinates by sending [u : v] to
[u : v : v]
Finally, here is an example of a fan morphism that cannot be written
using homogeneous polynomials. Consider the blowup `O_{\mathbb{P}^1}(2)
\to \CC^2/\ZZ_2`. In terms of toric data, this blowup is::
sage: A2_Z2 = toric_varieties.A2_Z2()
sage: A2_Z2.fan().rays()
N(1, 0),
N(1, 2)
in 2-d lattice N
sage: O2_P1 = A2_Z2.resolve(new_rays=[(1,1)])
sage: blowup = O2_P1.hom(identity_matrix(2), A2_Z2)
sage: blowup.as_polynomial_map()
Traceback (most recent call last):
...
TypeError: The fan morphism cannot be written in homogeneous polynomials.
If we denote the homogeneous coordinates of `O_{\mathbb{P}^1}(2)` by
`x`, `t`, `y` corresponding to the rays `(1,2)`, `(1,1)`, and `(1,0)`
then the blow-up map is [BB]_:
.. math::
f: O_{\mathbb{P}^1}(2) \to \CC^2/\ZZ_2, \quad
(x,t,y) \mapsto \left( x\sqrt{t}, y\sqrt{t} \right)
which requires square roots.
Fibrations
----------
If a toric morphism is :meth:`dominant
<SchemeMorphism_fan_toric_variety.is_dominant>`, then all fibers over
a fixed torus orbit in the base are isomorphic. Hence, studying the
fibers is again a combinatorial question and Sage implements
additional methods to study such fibrations that are not available
otherwise (however, note that you can always
:meth:`~SchemeMorphism_fan_toric_variety.factor` to pick out the part
that is dominant over the image or its closure).
For example, consider the blow-up restricted to one of the two
coordinate charts of $O_{\mathbb{P}^1}(2)$ ::
sage: O2_P1_chart = ToricVariety(Fan([O2_P1.fan().generating_cones()[1]]))
sage: single_chart = O2_P1_chart.hom(identity_matrix(2), A2_Z2)
sage: single_chart.is_dominant()
True
sage: single_chart.is_surjective()
False
sage: fiber = single_chart.fiber_generic(); fiber
(0-d affine toric variety, 1)
sage: fiber[0].embedding_morphism().as_polynomial_map()
Scheme morphism:
From: 0-d affine toric variety
To: 2-d affine toric variety
Defn: Defined on coordinates by sending [] to
[1 : 1]
The fibers are labeled by torus orbits in the base, that is, cones of
the codomain fan. In this case, the fibers over lower-dimensional
torus orbits are::
sage: A2_Z2_cones = flatten(A2_Z2.fan().cones())
sage: table([('cone', 'dim')] +
....: [(cone.ambient_ray_indices(), single_chart.fiber_dimension(cone))
....: for cone in A2_Z2_cones], header_row=True)
cone dim
+--------+-----+
() 0
(0,) 0
(1,) -1
(0, 1) 1
Lets look closer at the one-dimensional fiber. Although not the case
in this example, connected components of fibers over higher-dimensional cones
(corresponding
to lower-dimensional torus orbits) of the base are often not
irreducible. The irreducible components are labeled by the
:meth:`~sage.geometry.fan_morphism.FanMorphism.primitive_preimage_cones`,
which are certain cones of the domain fan that map to the cone in the
base that defines the torus orbit::
sage: table([('base cone', 'primitive preimage cones')] +
....: [(cone.ambient_ray_indices(),
....: single_chart.fan_morphism().primitive_preimage_cones(cone))
....: for cone in A2_Z2_cones], header_row=True)
base cone primitive preimage cones
+-----------+---------------------------------------------------------+
() (0-d cone of Rational polyhedral fan in 2-d lattice N,)
(0,) (1-d cone of Rational polyhedral fan in 2-d lattice N,)
(1,) ()
(0, 1) (1-d cone of Rational polyhedral fan in 2-d lattice N,)
The fiber over the trivial cone is the generic fiber that we have
already encountered. The interesting fiber is the one over the
2-dimensional cone, which represents the exceptional set of the
blow-up in this single coordinate chart. Lets investigate further::
sage: exceptional_cones = single_chart.fan_morphism().primitive_preimage_cones(A2_Z2.fan(2)[0])
sage: exceptional_set = single_chart.fiber_component(exceptional_cones[0])
sage: exceptional_set
1-d affine toric variety
sage: exceptional_set.embedding_morphism().as_polynomial_map()
Scheme morphism:
From: 1-d affine toric variety
To: 2-d affine toric variety
Defn: Defined on coordinates by sending [z0] to
[z0 : 0]
So we see that the fiber over this point is an affine line. Together
with another affine line in the other coordinate patch, this covers
the exceptional $\mathbb{P}^1$ of the blowup $O_{\mathbb{P}^1}(2) \to
\CC^2/\ZZ_2$.
Here is an example with higher dimensional varieties involved::
sage: A3 = toric_varieties.A(3)
sage: P3 = toric_varieties.P(3)
sage: m = matrix([(2,0,0), (1,1,0), (3, 1, 0)])
sage: phi = A3.hom(m, P3)
sage: phi.as_polynomial_map()
Scheme morphism:
From: 3-d affine toric variety
To: 3-d CPR-Fano toric variety covered by 4 affine patches
Defn: Defined on coordinates by sending [z0 : z1 : z2] to
[z0^2*z1*z2^3 : z1*z2 : 1 : 1]
sage: phi.fiber_generic()
Traceback (most recent call last):
...
AttributeError: 'SchemeMorphism_fan_toric_variety' object
has no attribute 'fiber_generic'
Let's use factorization mentioned above::
sage: phi_i, phi_b, phi_s = phi.factor()
It is possible to study fibers of the last two morphisms or their composition::
sage: phi_d = phi_b * phi_s
sage: phi_d
Scheme morphism:
From: 3-d affine toric variety
To: 2-d toric variety covered by 3 affine patches
Defn: Defined by sending Rational polyhedral fan in 3-d lattice N to
Rational polyhedral fan in Sublattice <N(1, 0, 0), N(0, 1, 0)>.
sage: phi_d.as_polynomial_map()
Scheme morphism:
From: 3-d affine toric variety
To: 2-d toric variety covered by 3 affine patches
Defn: Defined on coordinates by sending [z0 : z1 : z2] to
[z0^2*z1*z2^3 : z1*z2 : 1]
sage: phi_d.codomain().fan().rays()
N( 1, 0, 0),
N( 0, 1, 0),
N(-1, -1, 0)
in Sublattice <N(1, 0, 0), N(0, 1, 0)>
sage: for c in phi_d.codomain().fan():
... c.ambient_ray_indices()
(1, 2)
(0, 2)
(0, 1)
We see that codomain fan of this morphism is a projective plane, which can be
verified by ::
sage: phi_d.codomain().fan().is_isomorphic(toric_varieties.P2().fan()) # known bug
True
(Unfortunately it cannot be verified correctly until :trac:`16012` is fixed.)
We now have access to fiber methods::
sage: fiber = phi_d.fiber_generic()
sage: fiber
(1-d affine toric variety, 2)
sage: fiber[0].embedding_morphism()
Scheme morphism:
From: 1-d affine toric variety
To: 3-d affine toric variety
Defn: Defined by sending
Rational polyhedral fan in Sublattice <N(1, 1, -1)> to
Rational polyhedral fan in 3-d lattice N.
sage: fiber[0].embedding_morphism().as_polynomial_map()
Traceback (most recent call last):
...
NotImplementedError: polynomial representations for
fans with virtual rays are not implemented yet
sage: fiber[0].fan().rays()
Empty collection
in Sublattice <N(1, 1, -1)>
We see that generic fibers of this morphism consist of 2 one-dimensional tori
each. To see what happens over boundary points we can look at fiber components
corresponding to the cones of the domain fan::
sage: fm = phi_d.fan_morphism()
sage: for c in flatten(phi_d.domain().fan().cones()):
... fc, m = phi_d.fiber_component(c, multiplicity=True)
... print "{} |-> {} ({} rays, multiplicity {}) over {}".format(
... c.ambient_ray_indices(), fc, fc.fan().nrays(),
... m, fm.image_cone(c).ambient_ray_indices())
() |-> 1-d affine toric variety (0 rays, multiplicity 2) over ()
(0,) |-> 1-d affine toric variety (0 rays, multiplicity 1) over (0,)
(1,) |-> 2-d affine toric variety (2 rays, multiplicity 1) over (0, 1)
(2,) |-> 2-d affine toric variety (2 rays, multiplicity 1) over (0, 1)
(0, 1) |-> 1-d affine toric variety (1 rays, multiplicity 1) over (0, 1)
(1, 2) |-> 1-d affine toric variety (1 rays, multiplicity 1) over (0, 1)
(0, 2) |-> 1-d affine toric variety (1 rays, multiplicity 1) over (0, 1)
(0, 1, 2) |-> 0-d affine toric variety (0 rays, multiplicity 1) over (0, 1)
Now we see that over one of the coordinate lines of the projective plane we also
have one-dimensional tori (but only one in each fiber), while over one of the
points fixed by torus action we have two affine planes intersecting along an
affine line. An alternative perspective is provided by cones of the codomain
fan::
sage: for c in flatten(phi_d.codomain().fan().cones()):
... print "{} connected components over {}, each with {} irreducible components.".format(
... fm.index(c), c.ambient_ray_indices(),
... len(fm.primitive_preimage_cones(c)))
2 connected components over (), each with 1 irreducible components.
1 connected components over (0,), each with 1 irreducible components.
None connected components over (1,), each with 0 irreducible components.
None connected components over (2,), each with 0 irreducible components.
None connected components over (1, 2), each with 0 irreducible components.
None connected components over (0, 2), each with 0 irreducible components.
1 connected components over (0, 1), each with 2 irreducible components.
REFERENCES:
.. [BB]
Gavin Brown, Jaroslaw Buczynski:
Maps of toric varieties in Cox coordinates,
http://arxiv.org/abs/1004.4924
"""
#*****************************************************************************
# Copyright (C) 2011 Volker Braun <vbraun.name@gmail.com>
# Copyright (C) 2010 Andrey Novoseltsev <novoselt@gmail.com>
# Copyright (C) 2006 William Stein <wstein@gmail.com>
# Distributed under the terms of the GNU General Public License (GPL)
# http://www.gnu.org/licenses/
#*****************************************************************************
# For now, the scheme morphism base class cannot derive from Morphism
# since this would clash with elliptic curves. So we derive only on
# the toric varieties level from Morphism. See
# https://groups.google.com/d/msg/sage-devel/qF4yU6Vdmao/wQlNrneSmWAJ
from sage.categories.morphism import Morphism
from sage.structure.sequence import Sequence
from sage.rings.all import ZZ, gcd
from sage.misc.all import cached_method
from sage.matrix.constructor import matrix, block_matrix, zero_matrix, identity_matrix
from sage.modules.free_module_element import vector
from sage.geometry.all import Cone, Fan
from sage.schemes.generic.scheme import is_Scheme
from sage.schemes.generic.morphism import (
is_SchemeMorphism,
SchemeMorphism, SchemeMorphism_point, SchemeMorphism_polynomial
)
############################################################################
# A points on a toric variety determined by homogeneous coordinates.
class SchemeMorphism_point_toric_field(SchemeMorphism_point, Morphism):
"""
A point of a toric variety determined by homogeneous coordinates
in a field.
.. WARNING::
You should not create objects of this class directly. Use the
:meth:`~sage.schemes.generic.scheme.hom` method of
:class:`toric varieties
<sage.schemes.toric.variety.ToricVariety_field>`
instead.
INPUT:
- ``X`` -- toric variety or subscheme of a toric variety.
- ``coordinates`` -- list of coordinates in the base field of ``X``.
- ``check`` -- if ``True`` (default), the input will be checked for
correctness.
OUTPUT:
A :class:`SchemeMorphism_point_toric_field`.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1(1,2,3,4)
[1 : 2 : 3 : 4]
"""
# Mimicking affine/projective classes
def __init__(self, X, coordinates, check=True):
r"""
See :class:`SchemeMorphism_point_toric_field` for documentation.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1(1,2,3,4)
[1 : 2 : 3 : 4]
"""
# Convert scheme to its set of points over the base ring
if is_Scheme(X):
X = X(X.base_ring())
super(SchemeMorphism_point_toric_field, self).__init__(X)
if check:
# Verify that there are the right number of coords
# Why is it not done in the parent?
if is_SchemeMorphism(coordinates):
coordinates = list(coordinates)
if not isinstance(coordinates, (list, tuple)):
raise TypeError("coordinates must be a scheme point, list, "
"or tuple. Got %s" % coordinates)
d = X.codomain().ambient_space().ngens()
if len(coordinates) != d:
raise ValueError("there must be %d coordinates! Got only %d: "
"%s" % (d, len(coordinates), coordinates))
# Make sure the coordinates all lie in the appropriate ring
coordinates = Sequence(coordinates, X.value_ring())
# Verify that the point satisfies the equations of X.
X.codomain()._check_satisfies_equations(coordinates)
self._coords = coordinates
############################################################################
# A morphism of toric varieties determined by homogeneous polynomials.
class SchemeMorphism_polynomial_toric_variety(SchemeMorphism_polynomial, Morphism):
"""
A morphism determined by homogeneous polynomials.
.. WARNING::
You should not create objects of this class directly. Use the
:meth:`~sage.schemes.generic.scheme.hom` method of
:class:`toric varieties
<sage.schemes.toric.variety.ToricVariety_field>`
instead.
INPUT:
Same as for
:class:`~sage.schemes.toric.morphism.SchemeMorphism_polynomial`.
OUPUT:
A :class:`~sage.schemes.toric.morphism.SchemeMorphism_polynomial_toric_variety`.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1.inject_variables()
Defining z0, z1, z2, z3
sage: P1 = P1xP1.subscheme(z0-z2)
sage: H = P1xP1.Hom(P1)
sage: import sage.schemes.toric.morphism as MOR
sage: MOR.SchemeMorphism_polynomial_toric_variety(H, [z0,z1,z0,z3])
Scheme morphism:
From: 2-d toric variety covered by 4 affine patches
To: Closed subscheme of 2-d toric variety
covered by 4 affine patches defined by:
z0 - z2
Defn: Defined on coordinates by sending
[z0 : z1 : z2 : z3] to [z0 : z1 : z0 : z3]
"""
def __init__(self, parent, polynomials, check=True):
r"""
See :class:`SchemeMorphism_polynomial_toric_variety` for documentation.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1.inject_variables()
Defining z0, z1, z2, z3
sage: P1 = P1xP1.subscheme(z0-z2)
sage: H = P1xP1.Hom(P1)
sage: import sage.schemes.toric.morphism as MOR
sage: MOR.SchemeMorphism_polynomial_toric_variety(H, [z0,z1,z0,z3])
Scheme morphism:
From: 2-d toric variety covered by 4 affine patches
To: Closed subscheme of 2-d toric variety
covered by 4 affine patches defined by:
z0 - z2
Defn: Defined on coordinates by sending
[z0 : z1 : z2 : z3] to [z0 : z1 : z0 : z3]
"""
SchemeMorphism_polynomial.__init__(self, parent, polynomials, check)
if check:
# Check that defining polynomials are homogeneous (degrees can be
# different if the target uses weighted coordinates)
for p in self.defining_polynomials():
if not self.domain().ambient_space().is_homogeneous(p):
raise ValueError("%s is not homogeneous!" % p)
def as_fan_morphism(self):
"""
Express the morphism as a map defined by a fan morphism.
OUTPUT:
A :class:`SchemeMorphism_polynomial_toric_variety`. Raises a
``TypeError`` if the morphism cannot be written in such a way.
EXAMPLES::
sage: A1.<z> = toric_varieties.A1()
sage: P1 = toric_varieties.P1()
sage: patch = A1.hom([1,z], P1)
sage: patch.as_fan_morphism()
Traceback (most recent call last):
...
NotImplementedError: expressing toric morphisms as fan morphisms is
not implemented yet!
"""
raise NotImplementedError("expressing toric morphisms as fan "
"morphisms is not implemented yet!")
############################################################################
# The embedding morphism of an orbit closure
class SchemeMorphism_orbit_closure_toric_variety(SchemeMorphism, Morphism):
"""
The embedding of an orbit closure.
INPUT:
- ``parent`` -- the parent homset.
- ``defining_cone`` -- the defining cone.
- ``ray_map`` -- a dictionary ``{ambient ray generator: orbit ray
generator}``. Note that the image of the ambient ray generator
is not necessarily primitive.
.. WARNING::
You should not create objects of this class directly. Use the
:meth:`~sage.schemes.toric.variety.ToricVariety_field.orbit_closure`
method of :class:`toric varieties
<sage.schemes.toric.variety.ToricVariety_field>`
instead.
EXAMPLES::
sage: P1xP1 = toric_varieties.P1xP1()
sage: H = P1xP1.fan(1)[0]
sage: V = P1xP1.orbit_closure(H)
sage: V.embedding_morphism()
Scheme morphism:
From: 1-d toric variety covered by 2 affine patches
To: 2-d CPR-Fano toric variety covered by 4 affine patches
Defn: Defined by embedding the torus closure associated to the 1-d
cone of Rational polyhedral fan in 2-d lattice N.
TESTS::
sage: V.embedding_morphism()._reverse_ray_map()
{N(-1): 3, N(1): 2}
sage: V.embedding_morphism()._defining_cone
1-d cone of Rational polyhedral fan in 2-d lattice N
"""
def __init__(self, parent, defining_cone, ray_map):
"""
The Python constructor.
EXAMPLES::
sage: P2 = toric_varieties.P2()
sage: P1 = P2.orbit_closure(P2.fan(1)[0])
sage: P1.embedding_morphism()
Scheme morphism:
From: 1-d toric variety covered by 2 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined by embedding the torus closure associated to the 1-d cone
of Rational polyhedral fan in 2-d lattice N.
"""
SchemeMorphism.__init__(self, parent)
self._defining_cone = defining_cone
self._ray_map = ray_map
def defining_cone(self):
r"""
Return the cone corresponding to the torus orbit.
OUTPUT:
A cone of the fan of the ambient toric variety.
EXAMPLES::
sage: P2 = toric_varieties.P2()
sage: cone = P2.fan(1)[0]
sage: P1 = P2.orbit_closure(cone)
sage: P1.embedding_morphism().defining_cone()
1-d cone of Rational polyhedral fan in 2-d lattice N
sage: _ is cone
True
"""
return self._defining_cone
@cached_method
def _reverse_ray_map(self):
"""
Reverse ``self._ray_map``.
OUTPUT:
Return a dictionary `{orbit ray generator : preimage ray
index}`. Note that the orbit ray generator need not be
primitive. Also, the preimage ray is not necessarily unique.
EXAMPLES::
sage: P2_112 = toric_varieties.P2_112()
sage: P1 = P2_112.orbit_closure(Cone([(1,0)]))
sage: f = P1.embedding_morphism()
sage: f._ray_map
{N(0, 1): (1), N(1, 0): (0), N(-1, -2): (-2)}
sage: f._reverse_ray_map()
{N(-2): 2, N(1): 1}
"""
orbit = self.parent().domain()
codomain_fan = self.parent().codomain().fan()
reverse_ray_dict = dict()
defining_cone_indices = []
for n1,n2 in self._ray_map.iteritems():
ray_index = codomain_fan.rays().index(n1)
if n2.is_zero():
assert ray_index in self._defining_cone.ambient_ray_indices()
continue
n2 = orbit.fan().lattice()(n2)
n2.set_immutable()
reverse_ray_dict[n2] = ray_index
return reverse_ray_dict
def _repr_defn(self):
"""
Return a string representation of the definition of ``self``.
OUTPUT:
String.
EXAMPLES::
sage: P2 = toric_varieties.P2()
sage: V = P2.orbit_closure(P2.fan(1)[0]); V
1-d toric variety covered by 2 affine patches
sage: V.embedding_morphism()._repr_defn()
'Defined by embedding the torus closure associated to the 1-d cone of
Rational polyhedral fan in 2-d lattice N.'
"""
s = 'Defined by embedding the torus closure associated to the '
s += str(self._defining_cone)
s += '.'
return s
def as_polynomial_map(self):
"""
Express the morphism via homogeneous polynomials.
OUTPUT:
A :class:`SchemeMorphism_polynomial_toric_variety`. Raises a
``TypeError`` if the morphism cannot be written in terms of
homogeneous polynomials.
The defining polynomials are not necessarily unique. There are
choices if multiple ambient space ray generators project to
the same orbit ray generator, and one such choice is made
implicitly. The orbit embedding can be written as a polynomial
map if and only if each primitive orbit ray generator is the
image of at least one primitive ray generator of the ambient
toric variety.
EXAMPLES::
sage: P2 = toric_varieties.P2()
sage: V = P2.orbit_closure(P2.fan(1)[0]); V
1-d toric variety covered by 2 affine patches
sage: V.embedding_morphism().as_polynomial_map()
Scheme morphism:
From: 1-d toric variety covered by 2 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined on coordinates by sending [z0 : z1] to
[0 : z1 : z0]
If the toric variety is singular, then some orbit closure
embeddings cannot be written with homogeneous polynomials::
sage: P2_112 = toric_varieties.P2_112()
sage: P1 = P2_112.orbit_closure(Cone([(1,0)]))
sage: P1.embedding_morphism().as_polynomial_map()
Traceback (most recent call last):
...
TypeError: The embedding cannot be written with homogeneous polynomials.
"""
orbit = self.domain()
codomain_fan = self.codomain().fan()
R = orbit.coordinate_ring()
polys = [ R.one() ] * codomain_fan.nrays()
for i in self._defining_cone.ambient_ray_indices():
polys[i] = R.zero()
ray_index_map = self._reverse_ray_map()
for i, ray in enumerate(orbit.fan().rays()):
try:
ray_index = ray_index_map[ray]
except KeyError:
raise TypeError('The embedding cannot be written with homogeneous polynomials.')
polys[ray_index] = R.gen(i)
return SchemeMorphism_polynomial_toric_variety(self.parent(), polys)
def pullback_divisor(self, divisor):
r"""
Pull back a toric divisor.
INPUT:
- ``divisor`` -- a torus-invariant QQ-Cartier divisor on the
codomain of the embedding map.
OUTPUT:
A divisor on the domain of the embedding map (the orbit
closure) that is isomorphic to the pull-back divisor `f^*(D)`
but with possibly different linearization.
EXAMPLES::
sage: P2 = toric_varieties.P2()
sage: P1 = P2.orbit_closure(P2.fan(1)[0])
sage: f = P1.embedding_morphism()
sage: D = P2.divisor([1,2,3]); D
V(x) + 2*V(y) + 3*V(z)
sage: f.pullback_divisor(D)
4*V(z0) + 2*V(z1)
"""
from sage.schemes.toric.divisor import is_ToricDivisor
if not (is_ToricDivisor(divisor) and divisor.is_QQ_Cartier()):
raise ValueError('The divisor must be torus-invariant and QQ-Cartier.')
m = divisor.m(self._defining_cone)
values = []
codomain_rays = self.codomain().fan().rays()
for ray in self.domain().fan().rays():
ray = codomain_rays[self._reverse_ray_map()[ray]]
value = divisor.function_value(ray) - m*ray
values.append(value)
return self.domain().divisor(values)
############################################################################
# A morphism of toric varieties determined by a fan morphism
class SchemeMorphism_fan_toric_variety(SchemeMorphism, Morphism):
"""
Construct a morphism determined by a fan morphism
.. WARNING::
You should not create objects of this class directly. Use the
:meth:`~sage.schemes.generic.scheme.hom` method of
:class:`toric varieties
<sage.schemes.toric.variety.ToricVariety_field>`
instead.
INPUT:
- ``parent`` -- Hom-set whose domain and codomain are toric varieties.
- ``fan_morphism`` -- A morphism of fans whose domain and codomain
fans equal the fans of the domain and codomain in the ``parent``
Hom-set.
- ``check`` -- boolean (optional, default:``True``). Whether to
check the input for consistency.
.. WARNING::
A fibration is a dominant morphism; if you are interested in
these then you have to make sure that your fan morphism is
dominant. For example, this can be achieved by
:meth:`factoring the morphism
<sage.schemes.toric.morphism.SchemeMorphism_fan_toric_variety.factor>`. See
:class:`SchemeMorphism_fan_toric_variety_dominant` for
additional functionality for fibrations.
OUPUT:
A :class:`~sage.schemes.toric.morphism.SchemeMorphism_fan_toric_variety`.
EXAMPLES::
sage: P1xP1 = toric_varieties.P1xP1()
sage: P1 = toric_varieties.P1()
sage: f = P1.hom(matrix([[1,0]]), P1xP1); f
Scheme morphism:
From: 1-d CPR-Fano toric variety covered by 2 affine patches
To: 2-d CPR-Fano toric variety covered by 4 affine patches
Defn: Defined by sending Rational polyhedral fan in 1-d lattice N
to Rational polyhedral fan in 2-d lattice N.
sage: type(f)
<class 'sage.schemes.toric.morphism.SchemeMorphism_fan_toric_variety'>
Slightly more explicit construction::
sage: P1xP1 = toric_varieties.P1xP1()
sage: P1 = toric_varieties.P1()
sage: hom_set = P1xP1.Hom(P1)
sage: fm = FanMorphism( matrix(ZZ,[[1],[0]]), P1xP1.fan(), P1.fan() )
sage: hom_set(fm)
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 1-d lattice N.
sage: P1xP1.hom(fm, P1)
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 1-d lattice N.
"""
def __init__(self, parent, fan_morphism, check=True):
r"""
See :class:`SchemeMorphism_polynomial_toric_variety` for documentation.
TESTS::
sage: P1xP1 = toric_varieties.P1xP1()
sage: P1 = toric_varieties.P1()
sage: hom_set = P1xP1.Hom(P1)
sage: fan_morphism = FanMorphism( matrix(ZZ,[[1],[0]]), P1xP1.fan(), P1.fan() )
sage: from sage.schemes.toric.morphism import SchemeMorphism_fan_toric_variety
sage: SchemeMorphism_fan_toric_variety(hom_set, fan_morphism)
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 1-d lattice N.
"""
SchemeMorphism.__init__(self, parent)
if check and self.domain().fan()!=fan_morphism.domain_fan():
raise ValueError('The fan morphism domain must be the fan of the domain.')
if check and self.codomain().fan()!=fan_morphism.codomain_fan():
raise ValueError('The fan morphism codomain must be the fan of the codomain.')
self._fan_morphism = fan_morphism
def __cmp__(self, right):
r"""
Compare ``self`` and ``right``.
INPUT:
- ``right`` -- anything.
OUTPUT:
- 0 if ``right`` is also a toric morphism between the same domain and
codomain, given by an equal fan morphism. 1 or -1 otherwise.
TESTS::
sage: A2 = toric_varieties.A2()
sage: P3 = toric_varieties.P(3)
sage: m = matrix([(2,0,0), (1,1,0)])
sage: phi = A2.hom(m, P3)
sage: cmp(phi, phi)
0
sage: cmp(phi, prod(phi.factor()))
0
sage: abs(cmp(phi, phi.factor()[0]))
1
sage: cmp(phi, 1) * cmp(1, phi)
-1
"""
if isinstance(right, SchemeMorphism_fan_toric_variety):
return cmp(
[self.domain(), self.codomain(), self.fan_morphism()],
[right.domain(), right.codomain(), right.fan_morphism()])
else:
return cmp(type(self), type(right))
def _composition_(self, right, homset):
"""
Return the composition of ``self`` and ``right``.
INPUT:
- ``right`` -- a toric morphism defined by a fan morphism.
OUTPUT:
- a toric morphism.
EXAMPLES::
sage: A2 = toric_varieties.A2()
sage: P3 = toric_varieties.P(3)
sage: m = matrix([(2,0,0), (1,1,0)])
sage: phi = A2.hom(m, P3)
sage: phi1, phi2, phi3 = phi.factor()
sage: phi1 * phi2
Scheme morphism:
From: 2-d affine toric variety
To: 3-d CPR-Fano toric variety covered by 4 affine patches
Defn: Defined by sending Rational polyhedral fan in Sublattice
<N(1, 0, 0), N(0, 1, 0)> to Rational polyhedral fan in 3-d lattice N.
sage: phi1 * phi2 * phi3 == phi
True
"""
f = self.fan_morphism() * right.fan_morphism()
return homset(f, self.codomain())
def _repr_defn(self):
"""
Return a string representation of the definition of ``self``.
OUTPUT:
String.
EXAMPLES::
sage: P1xP1 = toric_varieties.P1xP1()
sage: P1 = toric_varieties.P1()
sage: f = P1xP1.hom(matrix([[1],[0]]), P1)
sage: f._repr_defn()
'Defined by sending Rational polyhedral fan in 2-d lattice N to Rational polyhedral fan in 1-d lattice N.'
"""
s = 'Defined by sending '
s += str(self.domain().fan())
s += ' to '
s += str(self.codomain().fan())
s += '.'
return s
def factor(self):
r"""
Factor ``self`` into injective * birational * surjective morphisms.
OUTPUT:
- a triple of toric morphisms `(\phi_i, \phi_b, \phi_s)`, such that
`\phi_s` is surjective, `\phi_b` is birational, `\phi_i` is injective,
and ``self`` is equal to `\phi_i \circ \phi_b \circ \phi_s`.
The intermediate varieties are universal in the following sense. Let
``self`` map `X` to `X'` and let `X_s`, `X_i` sit in between, that is,
.. math::
X
\twoheadrightarrow
X_s
\to
X_i
\hookrightarrow
X'.
Then any toric morphism from `X` coinciding with ``self`` on the maximal
torus factors through `X_s` and any toric morphism into `X'` coinciding
with ``self`` on the maximal torus factors through `X_i`. In particular,
`X_i` is the closure of the image of ``self`` in `X'`.