Morphisms between toric lattices compatible with fans#
This module is a part of the framework for toric varieties
(variety
,
fano_variety
). Its main purpose is to
provide support for working with lattice morphisms compatible with fans via
FanMorphism
class.
AUTHORS:
Andrey Novoseltsev (2010-10-17): initial version.
- Andrey Novoseltsev (2011-04-11): added tests for injectivity/surjectivity,
fibration, bundle, as well as some related methods.
EXAMPLES:
Let’s consider the face and normal fans of the “diamond” and the projection to the \(x\)-axis:
sage: diamond = lattice_polytope.cross_polytope(2)
sage: face = FaceFan(diamond, lattice=ToricLattice(2))
sage: normal = NormalFan(diamond)
sage: N = face.lattice()
sage: H = End(N)
sage: phi = H([N.0, 0])
sage: phi
Free module morphism defined by the matrix
[1 0]
[0 0]
Domain: 2-d lattice N
Codomain: 2-d lattice N
sage: FanMorphism(phi, normal, face)
Traceback (most recent call last):
...
ValueError: the image of generating cone #1 of the domain fan
is not contained in a single cone of the codomain fan!
>>> from sage.all import *
>>> diamond = lattice_polytope.cross_polytope(Integer(2))
>>> face = FaceFan(diamond, lattice=ToricLattice(Integer(2)))
>>> normal = NormalFan(diamond)
>>> N = face.lattice()
>>> H = End(N)
>>> phi = H([N.gen(0), Integer(0)])
>>> phi
Free module morphism defined by the matrix
[1 0]
[0 0]
Domain: 2-d lattice N
Codomain: 2-d lattice N
>>> FanMorphism(phi, normal, face)
Traceback (most recent call last):
...
ValueError: the image of generating cone #1 of the domain fan
is not contained in a single cone of the codomain fan!
Some of the cones of the normal fan fail to be mapped to a single cone of the face fan. We can rectify the situation in the following way:
sage: fm = FanMorphism(phi, normal, face, subdivide=True)
sage: fm
Fan morphism defined by the matrix
[1 0]
[0 0]
Domain fan: Rational polyhedral fan in 2-d lattice N
Codomain fan: Rational polyhedral fan in 2-d lattice N
sage: fm.domain_fan().rays()
N( 1, 1),
N( 1, -1),
N(-1, -1),
N(-1, 1),
N( 0, -1),
N( 0, 1)
in 2-d lattice N
sage: normal.rays()
N( 1, 1),
N( 1, -1),
N(-1, -1),
N(-1, 1)
in 2-d lattice N
>>> from sage.all import *
>>> fm = FanMorphism(phi, normal, face, subdivide=True)
>>> fm
Fan morphism defined by the matrix
[1 0]
[0 0]
Domain fan: Rational polyhedral fan in 2-d lattice N
Codomain fan: Rational polyhedral fan in 2-d lattice N
>>> fm.domain_fan().rays()
N( 1, 1),
N( 1, -1),
N(-1, -1),
N(-1, 1),
N( 0, -1),
N( 0, 1)
in 2-d lattice N
>>> normal.rays()
N( 1, 1),
N( 1, -1),
N(-1, -1),
N(-1, 1)
in 2-d lattice N
As you see, it was necessary to insert two new rays (to prevent “upper” and “lower” cones of the normal fan from being mapped to the whole \(x\)-axis).
- class sage.geometry.fan_morphism.FanMorphism(morphism, domain_fan, codomain=None, subdivide=False, check=True, verbose=False)[source]#
Bases:
FreeModuleMorphism
Create a fan morphism.
Let \(\Sigma_1\) and \(\Sigma_2\) be two fans in lattices \(N_1\) and \(N_2\) respectively. Let \(\phi\) be a morphism (i.e. a linear map) from \(N_1\) to \(N_2\). We say that \(\phi\) is compatible with \(\Sigma_1\) and \(\Sigma_2\) if every cone \(\sigma_1\in\Sigma_1\) is mapped by \(\phi\) into a single cone \(\sigma_2\in\Sigma_2\), i.e. \(\phi(\sigma_1)\subset\sigma_2\) (\(\sigma_2\) may be different for different \(\sigma_1\)).
By a fan morphism we understand a morphism between two lattices compatible with specified fans in these lattices. Such morphisms behave in exactly the same way as “regular” morphisms between lattices, but:
fan morphisms have a special constructor allowing some automatic adjustments to the initial fans (see below);
fan morphisms are aware of the associated fans and they can be accessed via
codomain_fan()
anddomain_fan()
;fan morphisms can efficiently compute
image_cone()
of a given cone of the domain fan andpreimage_cones()
of a given cone of the codomain fan.
INPUT:
morphism
– either a morphism between domain and codomain, or an integral matrix defining such a morphism;domain_fan
– afan
in the domain;codomain
– (default:None
) either a codomain lattice or a fan in the codomain. If the codomain fan is not given, the image fan (fan generated by images of generating cones) ofdomain_fan
will be used, if possible;subdivide
– (default:False
) ifTrue
anddomain_fan
is not compatible with the codomain fan because it is too coarse, it will be automatically refined to become compatible (the minimal refinement is canonical, so there are no choices involved);check
– (default:True
) ifFalse
, given fans and morphism will be assumed to be compatible. Be careful when using this option, since wrong assumptions can lead to wrong and hard-to-detect errors. On the other hand, this option may save you some time;verbose
– (default:False
) ifTrue
, some information may be printed during construction of the fan morphism.
OUTPUT:
a fan morphism.
EXAMPLES:
Here we consider the face and normal fans of the “diamond” and the projection to the \(x\)-axis:
sage: diamond = lattice_polytope.cross_polytope(2) sage: face = FaceFan(diamond, lattice=ToricLattice(2)) sage: normal = NormalFan(diamond) sage: N = face.lattice() sage: H = End(N) sage: phi = H([N.0, 0]) sage: phi Free module morphism defined by the matrix [1 0] [0 0] Domain: 2-d lattice N Codomain: 2-d lattice N sage: fm = FanMorphism(phi, face, normal) sage: fm.domain_fan() is face True
>>> from sage.all import * >>> diamond = lattice_polytope.cross_polytope(Integer(2)) >>> face = FaceFan(diamond, lattice=ToricLattice(Integer(2))) >>> normal = NormalFan(diamond) >>> N = face.lattice() >>> H = End(N) >>> phi = H([N.gen(0), Integer(0)]) >>> phi Free module morphism defined by the matrix [1 0] [0 0] Domain: 2-d lattice N Codomain: 2-d lattice N >>> fm = FanMorphism(phi, face, normal) >>> fm.domain_fan() is face True
Note, that since
phi
is compatible with these fans, the returned fan is exactly the same object as the initialdomain_fan
.sage: FanMorphism(phi, normal, face) Traceback (most recent call last): ... ValueError: the image of generating cone #1 of the domain fan is not contained in a single cone of the codomain fan! sage: fm = FanMorphism(phi, normal, face, subdivide=True) sage: fm.domain_fan() is normal False sage: fm.domain_fan().ngenerating_cones() 6
>>> from sage.all import * >>> FanMorphism(phi, normal, face) Traceback (most recent call last): ... ValueError: the image of generating cone #1 of the domain fan is not contained in a single cone of the codomain fan! >>> fm = FanMorphism(phi, normal, face, subdivide=True) >>> fm.domain_fan() is normal False >>> fm.domain_fan().ngenerating_cones() 6
We had to subdivide two of the four cones of the normal fan, since they were mapped by
phi
into non-strictly convex cones.It is possible to omit the codomain fan, in which case the image fan will be used instead of it:
sage: fm = FanMorphism(phi, face) sage: fm.codomain_fan() Rational polyhedral fan in 2-d lattice N sage: fm.codomain_fan().rays() N( 1, 0), N(-1, 0) in 2-d lattice N
>>> from sage.all import * >>> fm = FanMorphism(phi, face) >>> fm.codomain_fan() Rational polyhedral fan in 2-d lattice N >>> fm.codomain_fan().rays() N( 1, 0), N(-1, 0) in 2-d lattice N
Now we demonstrate a more subtle example. We take the first quadrant as our domain fan. Then we divide the first quadrant into three cones, throw away the middle one and take the other two as our codomain fan. These fans are incompatible with the identity lattice morphism since the image of the domain fan is out of the support of the codomain fan:
sage: N = ToricLattice(2) sage: phi = End(N).identity() sage: F1 = Fan(cones=[(0,1)], rays=[(1,0), (0,1)]) sage: F2 = Fan(cones=[(0,1), (2,3)], ....: rays=[(1,0), (2,1), (1,2), (0,1)]) sage: FanMorphism(phi, F1, F2) Traceback (most recent call last): ... ValueError: the image of generating cone #0 of the domain fan is not contained in a single cone of the codomain fan! sage: FanMorphism(phi, F1, F2, subdivide=True) Traceback (most recent call last): ... ValueError: morphism defined by [1 0] [0 1] does not map Rational polyhedral fan in 2-d lattice N into the support of Rational polyhedral fan in 2-d lattice N!
>>> from sage.all import * >>> N = ToricLattice(Integer(2)) >>> phi = End(N).identity() >>> F1 = Fan(cones=[(Integer(0),Integer(1))], rays=[(Integer(1),Integer(0)), (Integer(0),Integer(1))]) >>> F2 = Fan(cones=[(Integer(0),Integer(1)), (Integer(2),Integer(3))], ... rays=[(Integer(1),Integer(0)), (Integer(2),Integer(1)), (Integer(1),Integer(2)), (Integer(0),Integer(1))]) >>> FanMorphism(phi, F1, F2) Traceback (most recent call last): ... ValueError: the image of generating cone #0 of the domain fan is not contained in a single cone of the codomain fan! >>> FanMorphism(phi, F1, F2, subdivide=True) Traceback (most recent call last): ... ValueError: morphism defined by [1 0] [0 1] does not map Rational polyhedral fan in 2-d lattice N into the support of Rational polyhedral fan in 2-d lattice N!
The problem was detected and handled correctly (i.e. an exception was raised). However, the used algorithm requires extra checks for this situation after constructing a potential subdivision and this can take significant time. You can save about half the time using
check=False
option, if you know in advance that it is possible to make fans compatible with the morphism by subdividing the domain fan. Of course, if your assumption was incorrect, the result will be wrong and you will get a fan which does map into the support of the codomain fan, but is not a subdivision of the domain fan. You can test it on the example above:sage: fm = FanMorphism(phi, F1, F2, subdivide=True, ....: check=False, verbose=True) Placing ray images (... ms) Computing chambers (... ms) Number of domain cones: 1. Number of chambers: 2. Cone 0 sits in chambers 0 1 (... ms) sage: fm.domain_fan().is_equivalent(F2) True
>>> from sage.all import * >>> fm = FanMorphism(phi, F1, F2, subdivide=True, ... check=False, verbose=True) Placing ray images (... ms) Computing chambers (... ms) Number of domain cones: 1. Number of chambers: 2. Cone 0 sits in chambers 0 1 (... ms) >>> fm.domain_fan().is_equivalent(F2) True
- codomain_fan(dim=None, codim=None)[source]#
Return the codomain fan of
self
.INPUT:
dim
– dimension of the requested cones;codim
– codimension of the requested cones.
OUTPUT:
rational polyhedral fan
if no parameters were given,tuple
ofcones
otherwise.
EXAMPLES:
sage: quadrant = Cone([(1,0), (0,1)]) sage: quadrant = Fan([quadrant]) sage: quadrant_bl = quadrant.subdivide([(1,1)]) sage: fm = FanMorphism(identity_matrix(2), quadrant_bl, quadrant) sage: fm.codomain_fan() Rational polyhedral fan in 2-d lattice N sage: fm.codomain_fan() is quadrant True
>>> from sage.all import * >>> quadrant = Cone([(Integer(1),Integer(0)), (Integer(0),Integer(1))]) >>> quadrant = Fan([quadrant]) >>> quadrant_bl = quadrant.subdivide([(Integer(1),Integer(1))]) >>> fm = FanMorphism(identity_matrix(Integer(2)), quadrant_bl, quadrant) >>> fm.codomain_fan() Rational polyhedral fan in 2-d lattice N >>> fm.codomain_fan() is quadrant True
- domain_fan(dim=None, codim=None)[source]#
Return the codomain fan of
self
.INPUT:
dim
– dimension of the requested cones;codim
– codimension of the requested cones.
OUTPUT:
rational polyhedral fan
if no parameters were given,tuple
ofcones
otherwise.
EXAMPLES:
sage: quadrant = Cone([(1,0), (0,1)]) sage: quadrant = Fan([quadrant]) sage: quadrant_bl = quadrant.subdivide([(1,1)]) sage: fm = FanMorphism(identity_matrix(2), quadrant_bl, quadrant) sage: fm.domain_fan() Rational polyhedral fan in 2-d lattice N sage: fm.domain_fan() is quadrant_bl True
>>> from sage.all import * >>> quadrant = Cone([(Integer(1),Integer(0)), (Integer(0),Integer(1))]) >>> quadrant = Fan([quadrant]) >>> quadrant_bl = quadrant.subdivide([(Integer(1),Integer(1))]) >>> fm = FanMorphism(identity_matrix(Integer(2)), quadrant_bl, quadrant) >>> fm.domain_fan() Rational polyhedral fan in 2-d lattice N >>> fm.domain_fan() is quadrant_bl True
- factor()[source]#
Factor
self
into injective * birational * surjective morphisms.OUTPUT:
a triple of
FanMorphism
\((\phi_i, \phi_b, \phi_s)\), such that \(\phi_s\) is surjective, \(\phi_b\) is birational, \(\phi_i\) is injective, andself
is equal to \(\phi_i \circ \phi_b \circ \phi_s\).
Intermediate fans live in the saturation of the image of
self
as a map between lattices and are the image of thedomain_fan()
and the restriction of thecodomain_fan()
, i.e. ifself
maps \(\Sigma \to \Sigma'\), then we have factorization into\[\Sigma \twoheadrightarrow \Sigma_s \to \Sigma_i \hookrightarrow \Sigma.\]Note
\(\Sigma_s\) is the finest fan with the smallest support that is compatible with
self
: any fan morphism from \(\Sigma\) given by the same map of lattices asself
factors through \(\Sigma_s\).\(\Sigma_i\) is the coarsest fan of the largest support that is compatible with
self
: any fan morphism into \(\Sigma'\) given by the same map of lattices asself
factors though \(\Sigma_i\).
EXAMPLES:
We map an affine plane into a projective 3-space in such a way, that it becomes “a double cover of a chart of the blow up of one of the coordinate planes”:
sage: A2 = toric_varieties.A2() sage: P3 = toric_varieties.P(3) # needs palp sage: m = matrix([(2,0,0), (1,1,0)]) sage: phi = A2.hom(m, P3) # needs palp sage: phi.as_polynomial_map() # needs palp Scheme morphism: From: 2-d affine toric variety To: 3-d CPR-Fano toric variety covered by 4 affine patches Defn: Defined on coordinates by sending [x : y] to [x^2*y : y : 1 : 1]
>>> from sage.all import * >>> A2 = toric_varieties.A2() >>> P3 = toric_varieties.P(Integer(3)) # needs palp >>> m = matrix([(Integer(2),Integer(0),Integer(0)), (Integer(1),Integer(1),Integer(0))]) >>> phi = A2.hom(m, P3) # needs palp >>> phi.as_polynomial_map() # needs palp Scheme morphism: From: 2-d affine toric variety To: 3-d CPR-Fano toric variety covered by 4 affine patches Defn: Defined on coordinates by sending [x : y] to [x^2*y : y : 1 : 1]
Now we will work with the underlying fan morphism:
sage: # needs palp sage: phi = phi.fan_morphism(); phi Fan morphism defined by the matrix [2 0 0] [1 1 0] Domain fan: Rational polyhedral fan in 2-d lattice N Codomain fan: Rational polyhedral fan in 3-d lattice N sage: phi.is_surjective(), phi.is_birational(), phi.is_injective() (False, False, False) sage: phi_i, phi_b, phi_s = phi.factor() sage: phi_s.is_surjective(), phi_b.is_birational(), phi_i.is_injective() (True, True, True) sage: prod(phi.factor()) == phi True
>>> from sage.all import * >>> # needs palp >>> phi = phi.fan_morphism(); phi Fan morphism defined by the matrix [2 0 0] [1 1 0] Domain fan: Rational polyhedral fan in 2-d lattice N Codomain fan: Rational polyhedral fan in 3-d lattice N >>> phi.is_surjective(), phi.is_birational(), phi.is_injective() (False, False, False) >>> phi_i, phi_b, phi_s = phi.factor() >>> phi_s.is_surjective(), phi_b.is_birational(), phi_i.is_injective() (True, True, True) >>> prod(phi.factor()) == phi True
Double cover (surjective):
sage: A2.fan().rays() N(1, 0), N(0, 1) in 2-d lattice N sage: phi_s # needs palp Fan morphism defined by the matrix [2 0] [1 1] Domain fan: Rational polyhedral fan in 2-d lattice N Codomain fan: Rational polyhedral fan in Sublattice <N(1, 0, 0), N(0, 1, 0)> sage: phi_s.codomain_fan().rays() # needs palp N(1, 0, 0), N(1, 1, 0) in Sublattice <N(1, 0, 0), N(0, 1, 0)>
>>> from sage.all import * >>> A2.fan().rays() N(1, 0), N(0, 1) in 2-d lattice N >>> phi_s # needs palp Fan morphism defined by the matrix [2 0] [1 1] Domain fan: Rational polyhedral fan in 2-d lattice N Codomain fan: Rational polyhedral fan in Sublattice <N(1, 0, 0), N(0, 1, 0)> >>> phi_s.codomain_fan().rays() # needs palp N(1, 0, 0), N(1, 1, 0) in Sublattice <N(1, 0, 0), N(0, 1, 0)>
Blowup chart (birational):
sage: phi_b # needs palp Fan morphism defined by the matrix [1 0] [0 1] Domain fan: Rational polyhedral fan in Sublattice <N(1, 0, 0), N(0, 1, 0)> Codomain fan: Rational polyhedral fan in Sublattice <N(1, 0, 0), N(0, 1, 0)> sage: phi_b.codomain_fan().rays() # needs palp N(-1, -1, 0), N( 0, 1, 0), N( 1, 0, 0) in Sublattice <N(1, 0, 0), N(0, 1, 0)>
>>> from sage.all import * >>> phi_b # needs palp Fan morphism defined by the matrix [1 0] [0 1] Domain fan: Rational polyhedral fan in Sublattice <N(1, 0, 0), N(0, 1, 0)> Codomain fan: Rational polyhedral fan in Sublattice <N(1, 0, 0), N(0, 1, 0)> >>> phi_b.codomain_fan().rays() # needs palp N(-1, -1, 0), N( 0, 1, 0), N( 1, 0, 0) in Sublattice <N(1, 0, 0), N(0, 1, 0)>
Coordinate plane inclusion (injective):
sage: phi_i # needs palp Fan morphism defined by the matrix [1 0 0] [0 1 0] Domain fan: Rational polyhedral fan in Sublattice <N(1, 0, 0), N(0, 1, 0)> Codomain fan: Rational polyhedral fan in 3-d lattice N sage: phi.codomain_fan().rays() # needs palp N( 1, 0, 0), N( 0, 1, 0), N( 0, 0, 1), N(-1, -1, -1) in 3-d lattice N
>>> from sage.all import * >>> phi_i # needs palp Fan morphism defined by the matrix [1 0 0] [0 1 0] Domain fan: Rational polyhedral fan in Sublattice <N(1, 0, 0), N(0, 1, 0)> Codomain fan: Rational polyhedral fan in 3-d lattice N >>> phi.codomain_fan().rays() # needs palp N( 1, 0, 0), N( 0, 1, 0), N( 0, 0, 1), N(-1, -1, -1) in 3-d lattice N
- image_cone(cone)[source]#
Return the cone of the codomain fan containing the image of
cone
.INPUT:
cone
– acone
equivalent to a cone of thedomain_fan()
ofself
.
OUTPUT:
a
cone
of thecodomain_fan()
ofself
.
EXAMPLES:
sage: quadrant = Cone([(1,0), (0,1)]) sage: quadrant = Fan([quadrant]) sage: quadrant_bl = quadrant.subdivide([(1,1)]) sage: fm = FanMorphism(identity_matrix(2), quadrant_bl, quadrant) sage: fm.image_cone(Cone([(1,0)])) 1-d cone of Rational polyhedral fan in 2-d lattice N sage: fm.image_cone(Cone([(1,1)])) 2-d cone of Rational polyhedral fan in 2-d lattice N
>>> from sage.all import * >>> quadrant = Cone([(Integer(1),Integer(0)), (Integer(0),Integer(1))]) >>> quadrant = Fan([quadrant]) >>> quadrant_bl = quadrant.subdivide([(Integer(1),Integer(1))]) >>> fm = FanMorphism(identity_matrix(Integer(2)), quadrant_bl, quadrant) >>> fm.image_cone(Cone([(Integer(1),Integer(0))])) 1-d cone of Rational polyhedral fan in 2-d lattice N >>> fm.image_cone(Cone([(Integer(1),Integer(1))])) 2-d cone of Rational polyhedral fan in 2-d lattice N
- index(cone=None)[source]#
Return the index of
self
as a map between lattices.INPUT:
cone
– (default:None
) acone
of thecodomain_fan()
ofself
.
OUTPUT:
an integer, infinity, or
None
.
If no cone was specified, this function computes the index of the image of
self
in the codomain. If a cone \(\sigma\) was given, the index ofself
over \(\sigma\) is computed in the sense of Definition 2.1.7 of [HLY2002]: if \(\sigma'\) is any cone of thedomain_fan()
ofself
whose relative interior is mapped to the relative interior of \(\sigma\), it is the index of the image of \(N'(\sigma')\) in \(N(\sigma)\), where \(N'\) and \(N\) are domain and codomain lattices respectively. While that definition was formulated for the case of the finite index only, we extend it to the infinite one as well and returnNone
if there is no \(\sigma'\) at all. See examples below for situations when such things happen. Note also that the index ofself
is the same as index over the trivial cone.EXAMPLES:
sage: # needs palp sage: Sigma = toric_varieties.dP8().fan() sage: Sigma_p = toric_varieties.P1().fan() sage: phi = FanMorphism(matrix([[1], [-1]]), Sigma, Sigma_p) sage: phi.index() 1 sage: psi = FanMorphism(matrix([[2], [-2]]), Sigma, Sigma_p) sage: psi.index() 2 sage: xi = FanMorphism(matrix([[1, 0]]), Sigma_p, Sigma) sage: xi.index() +Infinity
>>> from sage.all import * >>> # needs palp >>> Sigma = toric_varieties.dP8().fan() >>> Sigma_p = toric_varieties.P1().fan() >>> phi = FanMorphism(matrix([[Integer(1)], [-Integer(1)]]), Sigma, Sigma_p) >>> phi.index() 1 >>> psi = FanMorphism(matrix([[Integer(2)], [-Integer(2)]]), Sigma, Sigma_p) >>> psi.index() 2 >>> xi = FanMorphism(matrix([[Integer(1), Integer(0)]]), Sigma_p, Sigma) >>> xi.index() +Infinity
Infinite index in the last example indicates that the image has positive codimension in the codomain. Let’s look at the rays of our fans:
sage: Sigma_p.rays() # needs palp N( 1), N(-1) in 1-d lattice N sage: Sigma.rays() # needs palp N( 1, 1), N( 0, 1), N(-1, -1), N( 1, 0) in 2-d lattice N sage: xi.factor()[0].domain_fan().rays() # needs palp N(-1, 0), N( 1, 0) in Sublattice <N(1, 0)>
>>> from sage.all import * >>> Sigma_p.rays() # needs palp N( 1), N(-1) in 1-d lattice N >>> Sigma.rays() # needs palp N( 1, 1), N( 0, 1), N(-1, -1), N( 1, 0) in 2-d lattice N >>> xi.factor()[Integer(0)].domain_fan().rays() # needs palp N(-1, 0), N( 1, 0) in Sublattice <N(1, 0)>
We see that one of the rays of the fan of
P1
is mapped to a ray, while the other one to the interior of some 2-d cone. Both rays correspond to single points onP1
, yet one is mapped to the distinguished point of a torus invariant curve ofdP8
(with the rest of this curve being uncovered) and the other to a fixed point ofdP8
(thus completely covering this torus orbit indP8
).We should therefore expect the following behaviour: all indices over 1-d cones are
None
, except for one which is infinite, and all indices over 2-d cones areNone
, except for one which is 1:sage: [xi.index(cone) for cone in Sigma(1)] # needs palp [None, None, None, +Infinity] sage: [xi.index(cone) for cone in Sigma(2)] # needs palp [None, 1, None, None]
>>> from sage.all import * >>> [xi.index(cone) for cone in Sigma(Integer(1))] # needs palp [None, None, None, +Infinity] >>> [xi.index(cone) for cone in Sigma(Integer(2))] # needs palp [None, 1, None, None]
- is_birational()[source]#
Check if
self
is birational.OUTPUT:
True
ifself
is birational,False
otherwise.
For fan morphisms this check is equivalent to
self.index() == 1
and means that the corresponding map between toric varieties is birational.EXAMPLES:
sage: # needs palp sage: Sigma = toric_varieties.dP8().fan() sage: Sigma_p = toric_varieties.P1().fan() sage: phi = FanMorphism(matrix([[1], [-1]]), Sigma, Sigma_p) sage: psi = FanMorphism(matrix([[2], [-2]]), Sigma, Sigma_p) sage: xi = FanMorphism(matrix([[1, 0]]), Sigma_p, Sigma) sage: phi.index(), psi.index(), xi.index() (1, 2, +Infinity) sage: phi.is_birational(), psi.is_birational(), xi.is_birational() (True, False, False)
>>> from sage.all import * >>> # needs palp >>> Sigma = toric_varieties.dP8().fan() >>> Sigma_p = toric_varieties.P1().fan() >>> phi = FanMorphism(matrix([[Integer(1)], [-Integer(1)]]), Sigma, Sigma_p) >>> psi = FanMorphism(matrix([[Integer(2)], [-Integer(2)]]), Sigma, Sigma_p) >>> xi = FanMorphism(matrix([[Integer(1), Integer(0)]]), Sigma_p, Sigma) >>> phi.index(), psi.index(), xi.index() (1, 2, +Infinity) >>> phi.is_birational(), psi.is_birational(), xi.is_birational() (True, False, False)
- is_bundle()[source]#
Check if
self
is a bundle.OUTPUT:
True
ifself
is a bundle,False
otherwise.
Let \(\phi: \Sigma \to \Sigma'\) be a fan morphism such that the underlying lattice morphism \(\phi: N \to N'\) is surjective. Let \(\Sigma_0\) be the kernel fan of \(\phi\). Then \(\phi\) is a bundle (or splitting) if there is a subfan \(\widehat{\Sigma}\) of \(\Sigma\) such that the following two conditions are satisfied:
Cones of \(\Sigma\) are precisely the cones of the form \(\sigma_0 + \widehat{\sigma}\), where \(\sigma_0 \in \Sigma_0\) and \(\widehat{\sigma} \in \widehat{\Sigma}\).
Cones of \(\widehat{\Sigma}\) are in bijection with cones of \(\Sigma'\) induced by \(\phi\) and \(\phi\) maps lattice points in every cone \(\widehat{\sigma} \in \widehat{\Sigma}\) bijectively onto lattice points in \(\phi(\widehat{\sigma})\).
If a fan morphism \(\phi: \Sigma \to \Sigma'\) is a bundle, then \(X_\Sigma\) is a fiber bundle over \(X_{\Sigma'}\) with fibers \(X_{\Sigma_0, N_0}\), where \(N_0\) is the kernel lattice of \(\phi\). See [CLS2011] for more details.
See also
EXAMPLES:
We consider several maps between fans of a del Pezzo surface and the projective line:
sage: # needs palp sage: Sigma = toric_varieties.dP8().fan() sage: Sigma_p = toric_varieties.P1().fan() sage: phi = FanMorphism(matrix([[1], [-1]]), Sigma, Sigma_p) sage: psi = FanMorphism(matrix([[2], [-2]]), Sigma, Sigma_p) sage: xi = FanMorphism(matrix([[1, 0]]), Sigma_p, Sigma) sage: phi.is_bundle() True sage: phi.is_fibration() True sage: phi.index() 1 sage: psi.is_bundle() False sage: psi.is_fibration() True sage: psi.index() 2 sage: xi.is_fibration() False sage: xi.index() +Infinity
>>> from sage.all import * >>> # needs palp >>> Sigma = toric_varieties.dP8().fan() >>> Sigma_p = toric_varieties.P1().fan() >>> phi = FanMorphism(matrix([[Integer(1)], [-Integer(1)]]), Sigma, Sigma_p) >>> psi = FanMorphism(matrix([[Integer(2)], [-Integer(2)]]), Sigma, Sigma_p) >>> xi = FanMorphism(matrix([[Integer(1), Integer(0)]]), Sigma_p, Sigma) >>> phi.is_bundle() True >>> phi.is_fibration() True >>> phi.index() 1 >>> psi.is_bundle() False >>> psi.is_fibration() True >>> psi.index() 2 >>> xi.is_fibration() False >>> xi.index() +Infinity
The first of these maps induces not only a fibration, but a fiber bundle structure. The second map is very similar, yet it fails to be a bundle, as its index is 2. The last map is not even a fibration.
- is_dominant()[source]#
Return whether the fan morphism is dominant.
A fan morphism \(\phi\) is dominant if it is surjective as a map of vector spaces. That is, \(\phi_\RR: N_\RR \to N'_\RR\) is surjective.
If the domain fan is
complete
, then this implies that the fan morphism issurjective
.If the fan morphism is dominant, then the associated morphism of toric varieties is dominant in the algebraic-geometric sense (that is, surjective onto a dense subset).
OUTPUT:
Boolean.
EXAMPLES:
sage: P1 = toric_varieties.P1() sage: A1 = toric_varieties.A1() sage: phi = FanMorphism(matrix([[1]]), A1.fan(), P1.fan()) sage: phi.is_dominant() True sage: phi.is_surjective() False
>>> from sage.all import * >>> P1 = toric_varieties.P1() >>> A1 = toric_varieties.A1() >>> phi = FanMorphism(matrix([[Integer(1)]]), A1.fan(), P1.fan()) >>> phi.is_dominant() True >>> phi.is_surjective() False
- is_fibration()[source]#
Check if
self
is a fibration.OUTPUT:
True
ifself
is a fibration,False
otherwise.
A fan morphism \(\phi: \Sigma \to \Sigma'\) is a fibration if for any cone \(\sigma' \in \Sigma'\) and any primitive preimage cone \(\sigma \in \Sigma\) corresponding to \(\sigma'\) the linear map of vector spaces \(\phi_\RR\) induces a bijection between \(\sigma\) and \(\sigma'\), and, in addition, \(\phi\) is
dominant
(that is, \(\phi_\RR: N_\RR \to N'_\RR\) is surjective).If a fan morphism \(\phi: \Sigma \to \Sigma'\) is a fibration, then the associated morphism between toric varieties \(\tilde{\phi}: X_\Sigma \to X_{\Sigma'}\) is a fibration in the sense that it is surjective and all of its fibers have the same dimension, namely \(\dim X_\Sigma - \dim X_{\Sigma'}\). These fibers do not have to be isomorphic, i.e. a fibration is not necessarily a fiber bundle. See [HLY2002] for more details.
See also
EXAMPLES:
We consider several maps between fans of a del Pezzo surface and the projective line:
sage: # needs palp sage: Sigma = toric_varieties.dP8().fan() sage: Sigma_p = toric_varieties.P1().fan() sage: phi = FanMorphism(matrix([[1], [-1]]), Sigma, Sigma_p) sage: psi = FanMorphism(matrix([[2], [-2]]), Sigma, Sigma_p) sage: xi = FanMorphism(matrix([[1, 0]]), Sigma_p, Sigma) sage: phi.is_bundle() True sage: phi.is_fibration() True sage: phi.index() 1 sage: psi.is_bundle() False sage: psi.is_fibration() True sage: psi.index() 2 sage: xi.is_fibration() False sage: xi.index() +Infinity
>>> from sage.all import * >>> # needs palp >>> Sigma = toric_varieties.dP8().fan() >>> Sigma_p = toric_varieties.P1().fan() >>> phi = FanMorphism(matrix([[Integer(1)], [-Integer(1)]]), Sigma, Sigma_p) >>> psi = FanMorphism(matrix([[Integer(2)], [-Integer(2)]]), Sigma, Sigma_p) >>> xi = FanMorphism(matrix([[Integer(1), Integer(0)]]), Sigma_p, Sigma) >>> phi.is_bundle() True >>> phi.is_fibration() True >>> phi.index() 1 >>> psi.is_bundle() False >>> psi.is_fibration() True >>> psi.index() 2 >>> xi.is_fibration() False >>> xi.index() +Infinity
The first of these maps induces not only a fibration, but a fiber bundle structure. The second map is very similar, yet it fails to be a bundle, as its index is 2. The last map is not even a fibration.
- is_injective()[source]#
Check if
self
is injective.OUTPUT:
True
ifself
is injective,False
otherwise.
Let \(\phi: \Sigma \to \Sigma'\) be a fan morphism such that the underlying lattice morphism \(\phi: N \to N'\) bijectively maps \(N\) to a saturated sublattice of \(N'\). Let \(\psi: \Sigma \to \Sigma'_0\) be the restriction of \(\phi\) to the image. Then \(\phi\) is injective if the map between cones corresponding to \(\psi\) (injectively) maps each cone of \(\Sigma\) to a cone of the same dimension.
If a fan morphism \(\phi: \Sigma \to \Sigma'\) is injective, then the associated morphism between toric varieties \(\tilde{\phi}: X_\Sigma \to X_{\Sigma'}\) is injective.
See also
EXAMPLES:
Consider the fan of the affine plane:
sage: A2 = toric_varieties.A(2).fan()
>>> from sage.all import * >>> A2 = toric_varieties.A(Integer(2)).fan()
We will map several fans consisting of a single ray into the interior of the 2-cone:
sage: Sigma = Fan([Cone([(1,1)])]) sage: m = identity_matrix(2) sage: FanMorphism(m, Sigma, A2).is_injective() False
>>> from sage.all import * >>> Sigma = Fan([Cone([(Integer(1),Integer(1))])]) >>> m = identity_matrix(Integer(2)) >>> FanMorphism(m, Sigma, A2).is_injective() False
This morphism was not injective since (in the toric varieties interpretation) the 1-dimensional orbit corresponding to the ray was mapped to the 0-dimensional orbit corresponding to the 2-cone.
sage: Sigma = Fan([Cone([(1,)])]) sage: m = matrix(1, 2, [1,1]) sage: FanMorphism(m, Sigma, A2).is_injective() True
>>> from sage.all import * >>> Sigma = Fan([Cone([(Integer(1),)])]) >>> m = matrix(Integer(1), Integer(2), [Integer(1),Integer(1)]) >>> FanMorphism(m, Sigma, A2).is_injective() True
While the fans in this example are close to the previous one, here the ray corresponds to a 0-dimensional orbit.
sage: Sigma = Fan([Cone([(1,)])]) sage: m = matrix(1, 2, [2,2]) sage: FanMorphism(m, Sigma, A2).is_injective() False
>>> from sage.all import * >>> Sigma = Fan([Cone([(Integer(1),)])]) >>> m = matrix(Integer(1), Integer(2), [Integer(2),Integer(2)]) >>> FanMorphism(m, Sigma, A2).is_injective() False
Here the problem is that
m
maps the domain lattice to a non-saturated sublattice of the codomain. The corresponding map of the toric varieties is a two-sheeted cover of its image.We also embed the affine plane into the projective one:
sage: P2 = toric_varieties.P(2).fan() # needs palp sage: m = identity_matrix(2) sage: FanMorphism(m, A2, P2).is_injective() # needs palp True
>>> from sage.all import * >>> P2 = toric_varieties.P(Integer(2)).fan() # needs palp >>> m = identity_matrix(Integer(2)) >>> FanMorphism(m, A2, P2).is_injective() # needs palp True
- is_surjective()[source]#
Check if
self
is surjective.OUTPUT:
True
ifself
is surjective,False
otherwise.
A fan morphism \(\phi: \Sigma \to \Sigma'\) is surjective if the corresponding map between cones is surjective, i.e. for each cone \(\sigma' \in \Sigma'\) there is at least one preimage cone \(\sigma \in \Sigma\) such that the relative interior of \(\sigma\) is mapped to the relative interior of \(\sigma'\) and, in addition, \(\phi_\RR: N_\RR \to N'_\RR\) is surjective.
If a fan morphism \(\phi: \Sigma \to \Sigma'\) is surjective, then the associated morphism between toric varieties \(\tilde{\phi}: X_\Sigma \to X_{\Sigma'}\) is surjective.
See also
is_bundle()
,is_fibration()
,preimage_cones()
,is_complete()
.EXAMPLES:
We check that the blow up of the affine plane at the origin is surjective:
sage: A2 = toric_varieties.A(2).fan() sage: Bl = A2.subdivide([(1,1)]) sage: m = identity_matrix(2) sage: FanMorphism(m, Bl, A2).is_surjective() True
>>> from sage.all import * >>> A2 = toric_varieties.A(Integer(2)).fan() >>> Bl = A2.subdivide([(Integer(1),Integer(1))]) >>> m = identity_matrix(Integer(2)) >>> FanMorphism(m, Bl, A2).is_surjective() True
It remains surjective if we throw away “south and north poles” of the exceptional divisor:
sage: FanMorphism(m, Fan(Bl.cones(1)), A2).is_surjective() True
>>> from sage.all import * >>> FanMorphism(m, Fan(Bl.cones(Integer(1))), A2).is_surjective() True
But a single patch of the blow up does not cover the plane:
sage: F = Fan([Bl.generating_cone(0)]) sage: FanMorphism(m, F, A2).is_surjective() False
>>> from sage.all import * >>> F = Fan([Bl.generating_cone(Integer(0))]) >>> FanMorphism(m, F, A2).is_surjective() False
- kernel_fan()[source]#
Return the subfan of the domain fan mapped into the origin.
OUTPUT:
a
fan
.
Note
The lattice of the kernel fan is the
kernel()
sublattice ofself
.See also
EXAMPLES:
sage: fan = Fan(rays=[(1,0), (1,1), (0,1)], cones=[(0,1), (1,2)]) sage: fm = FanMorphism(matrix(2, 1, [1,-1]), fan, ToricLattice(1)) sage: fm.kernel_fan() Rational polyhedral fan in Sublattice <N(1, 1)> sage: _.rays() N(1, 1) in Sublattice <N(1, 1)> sage: fm.kernel_fan().cones() ((0-d cone of Rational polyhedral fan in Sublattice <N(1, 1)>,), (1-d cone of Rational polyhedral fan in Sublattice <N(1, 1)>,))
>>> from sage.all import * >>> fan = Fan(rays=[(Integer(1),Integer(0)), (Integer(1),Integer(1)), (Integer(0),Integer(1))], cones=[(Integer(0),Integer(1)), (Integer(1),Integer(2))]) >>> fm = FanMorphism(matrix(Integer(2), Integer(1), [Integer(1),-Integer(1)]), fan, ToricLattice(Integer(1))) >>> fm.kernel_fan() Rational polyhedral fan in Sublattice <N(1, 1)> >>> _.rays() N(1, 1) in Sublattice <N(1, 1)> >>> fm.kernel_fan().cones() ((0-d cone of Rational polyhedral fan in Sublattice <N(1, 1)>,), (1-d cone of Rational polyhedral fan in Sublattice <N(1, 1)>,))
- preimage_cones(cone)[source]#
Return cones of the domain fan whose
image_cone()
iscone
.INPUT:
cone
– acone
equivalent to a cone of thecodomain_fan()
ofself
.
OUTPUT:
a
tuple
ofcones
of thedomain_fan()
ofself
, sorted by dimension.
See also
EXAMPLES:
sage: quadrant = Cone([(1,0), (0,1)]) sage: quadrant = Fan([quadrant]) sage: quadrant_bl = quadrant.subdivide([(1,1)]) sage: fm = FanMorphism(identity_matrix(2), quadrant_bl, quadrant) sage: fm.preimage_cones(Cone([(1,0)])) (1-d cone of Rational polyhedral fan in 2-d lattice N,) sage: fm.preimage_cones(Cone([(1,0), (0,1)])) (1-d cone of Rational polyhedral fan in 2-d lattice N, 2-d cone of Rational polyhedral fan in 2-d lattice N, 2-d cone of Rational polyhedral fan in 2-d lattice N)
>>> from sage.all import * >>> quadrant = Cone([(Integer(1),Integer(0)), (Integer(0),Integer(1))]) >>> quadrant = Fan([quadrant]) >>> quadrant_bl = quadrant.subdivide([(Integer(1),Integer(1))]) >>> fm = FanMorphism(identity_matrix(Integer(2)), quadrant_bl, quadrant) >>> fm.preimage_cones(Cone([(Integer(1),Integer(0))])) (1-d cone of Rational polyhedral fan in 2-d lattice N,) >>> fm.preimage_cones(Cone([(Integer(1),Integer(0)), (Integer(0),Integer(1))])) (1-d cone of Rational polyhedral fan in 2-d lattice N, 2-d cone of Rational polyhedral fan in 2-d lattice N, 2-d cone of Rational polyhedral fan in 2-d lattice N)
- preimage_fan(cone)[source]#
Return the subfan of the domain fan mapped into
cone
.INPUT:
cone
– acone
equivalent to a cone of thecodomain_fan()
ofself
.
OUTPUT:
a
fan
.
Note
The preimage fan of
cone
consists of all cones of thedomain_fan()
which are mapped intocone
, including those that are mapped into its boundary. So this fan is not necessarily generated bypreimage_cones()
ofcone
.See also
EXAMPLES:
sage: quadrant_cone = Cone([(1,0), (0,1)]) sage: quadrant_fan = Fan([quadrant_cone]) sage: quadrant_bl = quadrant_fan.subdivide([(1,1)]) sage: fm = FanMorphism(identity_matrix(2), ....: quadrant_bl, quadrant_fan) sage: fm.preimage_fan(Cone([(1,0)])).cones() ((0-d cone of Rational polyhedral fan in 2-d lattice N,), (1-d cone of Rational polyhedral fan in 2-d lattice N,)) sage: fm.preimage_fan(quadrant_cone).ngenerating_cones() 2 sage: len(fm.preimage_cones(quadrant_cone)) 3
>>> from sage.all import * >>> quadrant_cone = Cone([(Integer(1),Integer(0)), (Integer(0),Integer(1))]) >>> quadrant_fan = Fan([quadrant_cone]) >>> quadrant_bl = quadrant_fan.subdivide([(Integer(1),Integer(1))]) >>> fm = FanMorphism(identity_matrix(Integer(2)), ... quadrant_bl, quadrant_fan) >>> fm.preimage_fan(Cone([(Integer(1),Integer(0))])).cones() ((0-d cone of Rational polyhedral fan in 2-d lattice N,), (1-d cone of Rational polyhedral fan in 2-d lattice N,)) >>> fm.preimage_fan(quadrant_cone).ngenerating_cones() 2 >>> len(fm.preimage_cones(quadrant_cone)) 3
- primitive_preimage_cones(cone)[source]#
Return the primitive cones of the domain fan corresponding to
cone
.INPUT:
cone
– acone
equivalent to a cone of thecodomain_fan()
ofself
.
OUTPUT:
a
cone
.
Let \(\phi: \Sigma \to \Sigma'\) be a fan morphism, let \(\sigma \in \Sigma\), and let \(\sigma' = \phi(\sigma)\). Then \(\sigma\) is a primitive cone corresponding to \(\sigma'\) if there is no proper face \(\tau\) of \(\sigma\) such that \(\phi(\tau) = \sigma'\).
Primitive cones play an important role for fibration morphisms.
See also
EXAMPLES:
Consider a projection of a del Pezzo surface onto the projective line:
sage: Sigma = toric_varieties.dP6().fan() # needs palp sage: Sigma.rays() # needs palp N( 0, 1), N(-1, 0), N(-1, -1), N( 0, -1), N( 1, 0), N( 1, 1) in 2-d lattice N sage: Sigma_p = toric_varieties.P1().fan() sage: phi = FanMorphism(matrix([[1], [-1]]), Sigma, Sigma_p) # needs palp
>>> from sage.all import * >>> Sigma = toric_varieties.dP6().fan() # needs palp >>> Sigma.rays() # needs palp N( 0, 1), N(-1, 0), N(-1, -1), N( 0, -1), N( 1, 0), N( 1, 1) in 2-d lattice N >>> Sigma_p = toric_varieties.P1().fan() >>> phi = FanMorphism(matrix([[Integer(1)], [-Integer(1)]]), Sigma, Sigma_p) # needs palp
Under this map, one pair of rays is mapped to the origin, one in the positive direction, and one in the negative one. Also three 2-dimensional cones are mapped in the positive direction and three in the negative one, so there are 5 preimage cones corresponding to either of the rays of the codomain fan
Sigma_p
:sage: len(phi.preimage_cones(Cone([(1,)]))) # needs palp 5
>>> from sage.all import * >>> len(phi.preimage_cones(Cone([(Integer(1),)]))) # needs palp 5
Yet only rays are primitive:
sage: phi.primitive_preimage_cones(Cone([(1,)])) # needs palp (1-d cone of Rational polyhedral fan in 2-d lattice N, 1-d cone of Rational polyhedral fan in 2-d lattice N)
>>> from sage.all import * >>> phi.primitive_preimage_cones(Cone([(Integer(1),)])) # needs palp (1-d cone of Rational polyhedral fan in 2-d lattice N, 1-d cone of Rational polyhedral fan in 2-d lattice N)
Since all primitive cones are mapped onto their images bijectively, we get a fibration:
sage: phi.is_fibration() # needs palp True
>>> from sage.all import * >>> phi.is_fibration() # needs palp True
But since there are several primitive cones corresponding to the same cone of the codomain fan, this map is not a bundle, even though its index is 1:
sage: phi.is_bundle() # needs palp False sage: phi.index() # needs palp 1
>>> from sage.all import * >>> phi.is_bundle() # needs palp False >>> phi.index() # needs palp 1
- relative_star_generators(domain_cone)[source]#
Return the relative star generators of
domain_cone
.INPUT:
domain_cone
– a cone of thedomain_fan()
ofself
.
OUTPUT:
star_generators()
ofdomain_cone
viewed as a cone ofpreimage_fan()
ofimage_cone()
ofdomain_cone
.
EXAMPLES:
sage: A2 = toric_varieties.A(2).fan() sage: Bl = A2.subdivide([(1,1)]) sage: f = FanMorphism(identity_matrix(2), Bl, A2) sage: for c1 in Bl(1): ....: print(f.relative_star_generators(c1)) (1-d cone of Rational polyhedral fan in 2-d lattice N,) (1-d cone of Rational polyhedral fan in 2-d lattice N,) (2-d cone of Rational polyhedral fan in 2-d lattice N, 2-d cone of Rational polyhedral fan in 2-d lattice N)
>>> from sage.all import * >>> A2 = toric_varieties.A(Integer(2)).fan() >>> Bl = A2.subdivide([(Integer(1),Integer(1))]) >>> f = FanMorphism(identity_matrix(Integer(2)), Bl, A2) >>> for c1 in Bl(Integer(1)): ... print(f.relative_star_generators(c1)) (1-d cone of Rational polyhedral fan in 2-d lattice N,) (1-d cone of Rational polyhedral fan in 2-d lattice N,) (2-d cone of Rational polyhedral fan in 2-d lattice N, 2-d cone of Rational polyhedral fan in 2-d lattice N)