def find_isomorphism(fan1, fan2, check=False):
"""
Find an isomorphism of the two fans.
INPUT:
- ``fan1``, ``fan2`` -- two fans.
- ``check`` -- boolean (default: False). Passed to the fan
morphism constructor, see
:func:`~sage.geometry.fan_morphism.FanMorphism`.
OUTPUT:
A fan isomorphism. If the fans are not isomorphic, a
:class:`FanNotIsomorphicError` is raised.
EXAMPLE::
sage: rays = ((1, 1), (0, 1), (-1, -1), (3, 1))
sage: cones = [(0,1), (1,2), (2,3), (3,0)]
sage: fan1 = Fan(cones, rays)
sage: m = matrix([[-2,3],[1,-1]])
sage: m.det() == -1
True
sage: fan2 = Fan(cones, [vector(r)*m for r in rays])
sage: from sage.geometry.fan_isomorphism import find_isomorphism
sage: find_isomorphism(fan1, fan2, check=True)
Fan morphism defined by the matrix
[-2 3]
[ 1 -1]
Domain fan: Rational polyhedral fan in 2-d lattice N
Codomain fan: Rational polyhedral fan in 2-d lattice N
sage: find_isomorphism(fan1, toric_varieties.P2().fan())
Traceback (most recent call last):
...
FanNotIsomorphicError
sage: fan1 = Fan(cones=[[1,3,4,5],[0,1,2,3],[2,3,4],[0,1,5]],
... rays=[(-1,-1,0),(-1,-1,3),(-1,1,-1),(-1,3,-1),(0,2,-1),(1,-1,1)])
sage: fan2 = Fan(cones=[[0,2,3,5],[0,1,4,5],[0,1,2],[3,4,5]],
... rays=[(-1,-1,-1),(-1,-1,0),(-1,1,-1),(0,2,-1),(1,-1,1),(3,-1,-1)])
sage: fan1.is_isomorphic(fan2)
True
"""
generator = fan_isomorphism_generator(fan1, fan2)
try:
m = next(generator)
except StopIteration:
raise FanNotIsomorphicError
from sage.geometry.fan_morphism import FanMorphism
return FanMorphism(m, domain_fan=fan1, codomain=fan2, check=check)
def _element_constructor_(self, x, check=True):
"""
Construct a scheme morphism.
INPUT:
- `x` -- anything that defines a morphism of toric
varieties. A matrix, fan morphism, or a list or tuple of
homogeneous polynomials that define a morphism.
- ``check`` -- boolean (default: ``True``) passed onto
functions called by this to be more careful about input
argument type checking
OUTPUT:
The morphism of toric varieties determined by ``x``.
EXAMPLES:
First, construct from fan morphism::
sage: dP8.<t,x0,x1,x2> = toric_varieties.dP8()
sage: P2.<y0,y1,y2> = toric_varieties.P2()
sage: hom_set = dP8.Hom(P2)
sage: fm = FanMorphism(identity_matrix(2), dP8.fan(), P2.fan())
sage: hom_set(fm) # calls hom_set._element_constructor_()
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 2-d lattice N.
A matrix will automatically be converted to a fan morphism::
sage: hom_set(identity_matrix(2))
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 2-d lattice N.
Alternatively, one can use homogeneous polynomials to define morphisms::
sage: P2.inject_variables()
Defining y0, y1, y2
sage: dP8.inject_variables()
Defining t, x0, x1, x2
sage: hom_set([x0,x1,x2])
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined on coordinates by sending [t : x0 : x1 : x2] to
[x0 : x1 : x2]
A morphism of the coordinate ring will also work::
sage: ring_hom = P2.coordinate_ring().hom([x0,x1,x2], dP8.coordinate_ring())
sage: ring_hom
Ring morphism:
From: Multivariate Polynomial Ring in y0, y1, y2 over Rational Field
To: Multivariate Polynomial Ring in t, x0, x1, x2 over Rational Field
Defn: y0 |--> x0
y1 |--> x1
y2 |--> x2
sage: hom_set(ring_hom)
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined on coordinates by sending [t : x0 : x1 : x2] to
[x0 : x1 : x2]
"""
from sage.schemes.toric.morphism import SchemeMorphism_polynomial_toric_variety
if isinstance(x, (list, tuple)):
return SchemeMorphism_polynomial_toric_variety(self,
x,
check=check)
from sage.categories.map import Map
from sage.categories.all import Rings
if isinstance(x, Map) and x.category_for().is_subcategory(Rings()):
# x is a morphism of Rings
assert x.domain() is self.codomain().coordinate_ring()
assert x.codomain() is self.domain().coordinate_ring()
return SchemeMorphism_polynomial_toric_variety(self,
x.im_gens(),
check=check)
if is_Matrix(x):
x = FanMorphism(x, self.domain().fan(), self.codomain().fan())
if isinstance(x, FanMorphism):
if x.is_dominant():
from sage.schemes.toric.morphism import SchemeMorphism_fan_toric_variety_dominant
return SchemeMorphism_fan_toric_variety_dominant(self,
x,
check=check)
else:
from sage.schemes.toric.morphism import SchemeMorphism_fan_toric_variety
return SchemeMorphism_fan_toric_variety(self, x, check=check)
raise TypeError(
"x must be a fan morphism or a list/tuple of polynomials")
def _element_constructor_(self, x, check=True):
"""
Construct a scheme morphism.
INPUT:
- `x` -- anything that defines a morphism of toric
varieties. A matrix, fan morphism, or a list or tuple of
homogeneous polynomials that define a morphism.
- ``check`` -- boolean (default: ``True``) passed onto
functions called by this to be more careful about input
argument type checking
OUTPUT:
The morphism of toric varieties determined by ``x``.
EXAMPLES:
First, construct from fan morphism::
sage: dP8.<t,x0,x1,x2> = toric_varieties.dP8()
sage: P2.<y0,y1,y2> = toric_varieties.P2()
sage: hom_set = dP8.Hom(P2)
sage: fm = FanMorphism(identity_matrix(2), dP8.fan(), P2.fan())
sage: hom_set(fm) # calls hom_set._element_constructor_()
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 2-d lattice N.
A matrix will automatically be converted to a fan morphism::
sage: hom_set(identity_matrix(2))
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 2-d lattice N.
Alternatively, one can use homogeneous polynomials to define morphisms::
sage: P2.inject_variables()
Defining y0, y1, y2
sage: dP8.inject_variables()
Defining t, x0, x1, x2
sage: hom_set([x0,x1,x2])
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined on coordinates by sending [t : x0 : x1 : x2] to
[x0 : x1 : x2]
A morphism of the coordinate ring will also work::
sage: ring_hom = P2.coordinate_ring().hom([x0,x1,x2], dP8.coordinate_ring())
sage: ring_hom
Ring morphism:
From: Multivariate Polynomial Ring in y0, y1, y2 over Rational Field
To: Multivariate Polynomial Ring in t, x0, x1, x2 over Rational Field
Defn: y0 |--> x0
y1 |--> x1
y2 |--> x2
sage: hom_set(ring_hom)
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined on coordinates by sending [t : x0 : x1 : x2] to
[x0 : x1 : x2]
"""
from sage.schemes.toric.morphism import SchemeMorphism_polynomial_toric_variety
if isinstance(x, (list, tuple)):
return SchemeMorphism_polynomial_toric_variety(self, x, check=check)
if is_RingHomomorphism(x):
assert x.domain() is self.codomain().coordinate_ring()
assert x.codomain() is self.domain().coordinate_ring()
return SchemeMorphism_polynomial_toric_variety(self, x.im_gens(), check=check)
if is_Matrix(x):
x = FanMorphism(x, self.domain().fan(), self.codomain().fan())
if isinstance(x, FanMorphism):
if x.is_dominant():
from sage.schemes.toric.morphism import SchemeMorphism_fan_toric_variety_dominant
return SchemeMorphism_fan_toric_variety_dominant(self, x, check=check)
else:
from sage.schemes.toric.morphism import SchemeMorphism_fan_toric_variety
return SchemeMorphism_fan_toric_variety(self, x, check=check)
raise TypeError("x must be a fan morphism or a list/tuple of polynomials")