Morphisms of toric varieties#
There are three “obvious” ways to map toric varieties to toric varieties:
Polynomial maps in local coordinates, the usual morphisms in algebraic geometry.
Polynomial maps in the (global) homogeneous coordinates.
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 [CLS2011]). 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 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 SchemeMorphism...
classes. Instead, use the
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
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
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 [BB2013]:
which requires square roots.
Fibrations#
If a toric morphism 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
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()[0]]))
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
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: fm = single_chart.fan_morphism()
sage: exceptional_cones = fm.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
[1 : z1*z2 : z0^2*z1*z2^3]
sage: phi_d.codomain().fan().rays()
N(-1, -1, 0),
N( 0, 1, 0),
N( 1, 0, 0)
in Sublattice <N(1, 0, 0), N(0, 1, 0)>
sage: for c in phi_d.codomain().fan():
....: c.ambient_ray_indices()
(0, 1)
(0, 2)
(1, 2)
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 github issue #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 (2,)
(1,) |-> 2-d affine toric variety (2 rays, multiplicity 1) over (1, 2)
(2,) |-> 2-d affine toric variety (2 rays, multiplicity 1) over (1, 2)
(0, 1) |-> 1-d affine toric variety (1 rays, multiplicity 1) over (1, 2)
(1, 2) |-> 1-d affine toric variety (1 rays, multiplicity 1) over (1, 2)
(0, 2) |-> 1-d affine toric variety (1 rays, multiplicity 1) over (1, 2)
(0, 1, 2) |-> 0-d affine toric variety (0 rays, multiplicity 1) over (1, 2)
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.
None connected components over (0,), each with 0 irreducible components.
None connected components over (1,), each with 0 irreducible components.
1 connected components over (2,), each with 1 irreducible components.
None connected components over (0, 1), each with 0 irreducible components.
None connected components over (0, 2), each with 0 irreducible components.
1 connected components over (1, 2), each with 2 irreducible components.
- class sage.schemes.toric.morphism.SchemeMorphism_fan_fiber_component_toric_variety(toric_morphism, defining_cone)#
Bases:
SchemeMorphism
The embedding of a fiber component of a toric morphism.
Note that the embedding map of a fiber component of a toric morphism is itself not a toric morphism!
INPUT:
toric_morphism
– a toric morphism. The toric morphism whose fiber component we are describing.defining_cone
– a cone of the fan of the domain oftoric_morphism
. Seefiber_component()
for details.
EXAMPLES:
sage: polytope = Polyhedron( ....: [(-3,0,-1,-1),(-1,2,-1,-1),(0,-1,0,0),(0,0,0,1),(0,0,1,0), ....: (0,1,0,0),(0,2,-1,-1),(1,0,0,0),(2,0,-1,-1)]) sage: coarse_fan = FaceFan(polytope, lattice=ToricLattice(4)) sage: P2 = toric_varieties.P2() sage: proj24 = matrix([[0,0],[1,0],[0,0],[0,1]]) sage: fm = FanMorphism(proj24, coarse_fan, P2.fan(), subdivide=True) sage: fibration = ToricVariety(fm.domain_fan()).hom(fm, P2) sage: ffm = fibration.fan_morphism() sage: primitive_cones = ffm.primitive_preimage_cones(P2.fan(1)[0]) sage: primitive_cone = primitive_cones[0] sage: fiber_component = fibration.fiber_component(primitive_cone) sage: fiber_component 2-d toric variety covered by 4 affine patches sage: fiber_component.embedding_morphism() Scheme morphism: From: 2-d toric variety covered by 4 affine patches To: 4-d toric variety covered by 23 affine patches Defn: Defined by embedding a fiber component corresponding to 1-d cone of Rational polyhedral fan in 4-d lattice N. sage: fiber_component.embedding_morphism().as_polynomial_map() Scheme morphism: From: 2-d toric variety covered by 4 affine patches To: 4-d toric variety covered by 23 affine patches Defn: Defined on coordinates by sending [z0 : z1 : z2 : z3] to [1 : 1 : 1 : 1 : z2 : 0 : 1 : z3 : 1 : 1 : 1 : z1 : z0 : 1 : 1] sage: type(fiber_component.embedding_morphism()) <class 'sage.schemes.toric.morphism.SchemeMorphism_fan_fiber_component_toric_variety'>
- as_polynomial_map()#
Express the embedding morphism via homogeneous polynomials.
OUTPUT:
A
SchemeMorphism_polynomial_toric_variety
. Raises aValueError
if the morphism cannot be written in terms of homogeneous polynomials.EXAMPLES:
sage: polytope = Polyhedron( ....: [(-3,0,-1,-1),(-1,2,-1,-1),(0,-1,0,0),(0,0,0,1),(0,0,1,0), ....: (0,1,0,0),(0,2,-1,-1),(1,0,0,0),(2,0,-1,-1)]) sage: coarse_fan = FaceFan(polytope, lattice=ToricLattice(4)) sage: P2 = toric_varieties.P2() sage: proj24 = matrix([[0,0], [1,0], [0,0], [0,1]]) sage: fm = FanMorphism(proj24, coarse_fan, P2.fan(), subdivide=True) sage: fibration = ToricVariety(fm.domain_fan()).hom(fm, P2) sage: primitive_cone = Cone([(0, 1, 0, 0)]) sage: f = fibration.fiber_component(primitive_cone).embedding_morphism() sage: f.as_polynomial_map() Scheme morphism: From: 2-d toric variety covered by 4 affine patches To: 4-d toric variety covered by 23 affine patches Defn: Defined on coordinates by sending [z0 : z1 : z2 : z3] to [1 : 1 : 1 : 1 : z2 : 0 : 1 : z3 : 1 : 1 : 1 : z1 : z0 : 1 : 1] sage: primitive_cone = Cone([(-1, 2, -1, 0)]) sage: f = fibration.fiber_component(primitive_cone).embedding_morphism() sage: f.as_polynomial_map() Traceback (most recent call last): ... ValueError: the morphism cannot be written using homogeneous polynomials
- base_cone()#
Return the base cone \(\sigma\).
The fiber is constant over the base orbit closure \(V(\sigma)\).
OUTPUT:
A cone of the base of the toric fibration.
EXAMPLES:
sage: P1xP1 = toric_varieties.P1xP1() sage: P1 = toric_varieties.P1() sage: fc = P1xP1.hom(matrix([[1],[0]]), P1).fiber_component(Cone([(1,0)])) sage: f = fc.embedding_morphism() sage: f.defining_cone().rays() N(1, 0) in 2-d lattice N sage: f.base_cone().rays() N(1) in 1-d lattice N
- defining_cone()#
Return the cone corresponding to the fiber torus orbit.
OUTPUT:
A cone of the fan of the total space of the toric fibration.
EXAMPLES:
sage: P1xP1 = toric_varieties.P1xP1() sage: P1 = toric_varieties.P1() sage: fc = P1xP1.hom(matrix([[1],[0]]), P1).fiber_component(Cone([(1,0)])) sage: f = fc.embedding_morphism() sage: f.defining_cone().rays() N(1, 0) in 2-d lattice N sage: f.base_cone().rays() N(1) in 1-d lattice N
- pullback_divisor(divisor)#
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 (irreducible component of a fiber of a toric morphism) that is isomorphic to the pull-back divisor \(f^*(D)\) but with possibly different linearization.
EXAMPLES:
sage: A1 = toric_varieties.A1() sage: fan = Fan([(0,1,2)], [(1,1,0),(1,0,1),(1,-1,-1)]).subdivide(new_rays=[(1,0,0)]) sage: f = ToricVariety(fan).hom(matrix([[1],[0],[0]]), A1) sage: D = f.domain().divisor([1,1,3,4]); D V(z0) + V(z1) + 3*V(z2) + 4*V(z3) sage: fc = f.fiber_component(Cone([(1,1,0)])) sage: fc.embedding_morphism().pullback_divisor(D) 4*V(z0) + V(z1) + 4*V(z2) sage: fc = f.fiber_component(Cone([(1,0,0)])) sage: fc.embedding_morphism().pullback_divisor(D) -V(z0) - 3*V(z1) - 3*V(z2)
- class sage.schemes.toric.morphism.SchemeMorphism_fan_toric_variety(parent, fan_morphism, check=True)#
Bases:
SchemeMorphism
,Morphism
Construct a morphism determined by a fan morphism
Warning
You should not create objects of this class directly. Use the
hom()
method oftoric varieties
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 theparent
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
factoring the morphism
. SeeSchemeMorphism_fan_toric_variety_dominant
for additional functionality for fibrations.OUTPUT: A
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.
- as_polynomial_map()#
Express the morphism via homogeneous polynomials.
OUTPUT: A
SchemeMorphism_polynomial_toric_variety
.Raises a
TypeError
if the morphism cannot be written in terms of homogeneous polynomials.EXAMPLES:
sage: A1 = toric_varieties.A1() sage: square = A1.hom(matrix([[2]]), A1) sage: square.as_polynomial_map() Scheme endomorphism of 1-d affine toric variety Defn: Defined on coordinates by sending [z] to [z^2] sage: P1 = toric_varieties.P1() sage: patch = A1.hom(matrix([[1]]), P1) sage: patch.as_polynomial_map() Scheme morphism: From: 1-d affine toric variety To: 1-d CPR-Fano toric variety covered by 2 affine patches Defn: Defined on coordinates by sending [z] to [z : 1]
- factor()#
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,\[X \twoheadrightarrow X_s \to X_i \hookrightarrow X'.\]Then any toric morphism from \(X\) coinciding with
self
on the maximal torus factors through \(X_s\) and any toric morphism into \(X'\) coinciding withself
on the maximal torus factors through \(X_i\). In particular, \(X_i\) is the closure of the image ofself
in \(X'\).See
factor()
for a description of the toric algorithm.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) sage: m = matrix([(2,0,0), (1,1,0)]) sage: phi = A2.hom(m, P3) sage: phi.as_polynomial_map() 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] 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
Double cover (surjective):
sage: phi_s.as_polynomial_map() Scheme morphism: From: 2-d affine toric variety To: 2-d affine toric variety Defn: Defined on coordinates by sending [x : y] to [x^2 : y]
Blowup chart (birational):
sage: phi_b.as_polynomial_map() Scheme morphism: From: 2-d affine toric variety To: 2-d toric variety covered by 3 affine patches Defn: Defined on coordinates by sending [z0 : z1] to [1 : z1 : z0*z1]
Coordinate plane inclusion (injective):
sage: phi_i.as_polynomial_map() Scheme morphism: From: 2-d toric variety covered by 3 affine patches To: 3-d CPR-Fano toric variety covered by 4 affine patches Defn: Defined on coordinates by sending [z0 : z1 : z2] to [z2 : z1 : z0 : z0]
- fan_morphism()#
Return the defining fan morphism.
OUTPUT: A
FanMorphism
.EXAMPLES:
sage: P1xP1 = toric_varieties.P1xP1() sage: P1 = toric_varieties.P1() sage: f = P1xP1.hom(matrix([[1],[0]]), P1) sage: f.fan_morphism() Fan morphism defined by the matrix [1] [0] Domain fan: Rational polyhedral fan in 2-d lattice N Codomain fan: Rational polyhedral fan in 1-d lattice N
- is_birational()#
Check if
self
is birational.See
is_birational()
for fan morphisms for a description of the toric algorithm.OUTPUT:
Boolean. Whether
self
is birational.EXAMPLES:
sage: dP8 = toric_varieties.dP8() sage: P2 = toric_varieties.P2() sage: dP8.hom(identity_matrix(2), P2).is_birational() True sage: X = toric_varieties.A(2) sage: Y = ToricVariety(Fan([Cone([(1,0), (1,1)])])) sage: m = identity_matrix(2) sage: f = Y.hom(m, X) sage: f.is_birational() True
- is_bundle()#
Check if
self
is a bundle.See
is_bundle()
for fan morphisms for details.OUTPUT:
True
ifself
is a bundle,False
otherwise.
EXAMPLES:
sage: P1xP1 = toric_varieties.P1xP1() sage: P1 = toric_varieties.P1() sage: P1xP1.hom(matrix([[1],[0]]), P1).is_bundle() True
- is_dominant()#
Return whether
self
is dominant.See
is_dominant()
for fan morphisms for a description of the toric algorithm.OUTPUT:
Boolean. Whether
self
is a dominant scheme morphism.EXAMPLES:
sage: P1 = toric_varieties.P1() sage: A1 = toric_varieties.A1() sage: phi = A1.hom(identity_matrix(1), P1); phi Scheme morphism: From: 1-d affine toric variety To: 1-d CPR-Fano toric variety covered by 2 affine patches Defn: Defined by sending Rational polyhedral fan in 1-d lattice N to Rational polyhedral fan in 1-d lattice N. sage: phi.is_dominant() True sage: phi.is_surjective() False
- is_fibration()#
Check if
self
is a fibration.See
is_fibration()
for fan morphisms for details.OUTPUT:
True
ifself
is a fibration,False
otherwise.
EXAMPLES:
sage: P1xP1 = toric_varieties.P1xP1() sage: P1 = toric_varieties.P1() sage: P1xP1.hom(matrix([[1],[0]]), P1).is_fibration() True
- is_injective()#
Check if
self
is injective.See
is_injective()
for fan morphisms for a description of the toric algorithm.OUTPUT:
Boolean. Whether
self
is injective.EXAMPLES:
sage: P1xP1 = toric_varieties.P1xP1() sage: P1 = toric_varieties.P1() sage: P1xP1.hom(matrix([[1],[0]]), P1).is_injective() False sage: X = toric_varieties.A(2) sage: m = identity_matrix(2) sage: f = X.hom(m, X) sage: f.is_injective() True sage: Y = ToricVariety(Fan([Cone([(1,0), (1,1)])])) sage: f = Y.hom(m, X) sage: f.is_injective() False
- is_surjective()#
Check if
self
is surjective.See
is_surjective()
for fan morphisms for a description of the toric algorithm.OUTPUT:
Boolean. Whether
self
is surjective.EXAMPLES:
sage: P1xP1 = toric_varieties.P1xP1() sage: P1 = toric_varieties.P1() sage: P1xP1.hom(matrix([[1],[0]]), P1).is_surjective() True sage: X = toric_varieties.A(2) sage: m = identity_matrix(2) sage: f = X.hom(m, X) sage: f.is_surjective() True sage: Y = ToricVariety(Fan([Cone([(1,0), (1,1)])])) sage: f = Y.hom(m, X) sage: f.is_surjective() False
- pullback_divisor(divisor)#
Pull back a toric divisor.
INPUT:
divisor
– a torus-invariant \(\QQ\)-Cartier divisor on the codomain ofself
.
OUTPUT:
The pull-back divisor \(f^*(D)\).
EXAMPLES:
sage: A2_Z2 = toric_varieties.A2_Z2() sage: A2 = toric_varieties.A2() sage: f = A2.hom(matrix([[1,0], [1,2]]), A2_Z2) sage: f.pullback_divisor(A2_Z2.divisor(0)) V(x) sage: A1 = toric_varieties.A1() sage: square = A1.hom(matrix([[2]]), A1) sage: D = A1.divisor(0); D V(z) sage: square.pullback_divisor(D) 2*V(z)
- class sage.schemes.toric.morphism.SchemeMorphism_fan_toric_variety_dominant(parent, fan_morphism, check=True)#
Bases:
SchemeMorphism_fan_toric_variety
Construct a morphism determined by a dominant fan morphism.
A dominant morphism is one that is surjective onto a dense subset. In the context of toric morphisms, this means that it is onto the big torus orbit.
Warning
You should not create objects of this class directly. Use the
hom()
method oftoric varieties
instead.INPUT:
See
SchemeMorphism_fan_toric_variety
. The given fan morphismmust be dominant
.OUTPUT: A
SchemeMorphism_fan_toric_variety_dominant
.EXAMPLES:
sage: P2 = toric_varieties.P2() sage: dP8 = toric_varieties.dP8() sage: f = dP8.hom(identity_matrix(2), P2); f 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. sage: type(f) <class 'sage.schemes.toric.morphism.SchemeMorphism_fan_toric_variety_dominant'>
- fiber_component(domain_cone, multiplicity=False)#
Return a fiber component corresponding to
domain_cone
.INPUT:
domain_cone
– a cone of the domain fan ofself
.multiplicity
(default:False
) – whether to return the number of fiber components corresponding todomain_cone
as well.
OUTPUT:
either \(X\) or a tuple \((X, n)\), where \(X\) is a
toric variety
with the embedding morphism into domain ofself
and \(n\) is an integer.
Let \(\phi: \Sigma \to \Sigma'\) be the
fan morphism
corresponding toself
. Let \(\sigma \in \Sigma\) and \(\sigma' \in \Sigma'\) be theimage_cone()
of \(\sigma\). The fiber over any point of the torus orbit corresponding to \(\sigma'\) consists of \(n\) isomorphic connected components with each component being a union of toric varieties intersecting along their torus invariant subvarieties. The latter correspond topreimage_cones()
of \(\sigma'\) and \(X\) is one of the \(n\) components corresponding to \(\sigma\). The irreducible components correspond toprimitive_preimage_cones()
.EXAMPLES:
sage: polytope = LatticePolytope( ....: [(-3,0,-1,-1),(-1,2,-1,-1),(0,-1,0,0),(0,0,0,1),(0,0,1,0), ....: (0,1,0,0),(0,2,-1,-1),(1,0,0,0),(2,0,-1,-1)]) sage: coarse_fan = FaceFan(polytope) sage: P2 = toric_varieties.P2() sage: proj24 = matrix([[0,0], [1,0], [0,0], [0,1]]) sage: fm = FanMorphism(proj24, coarse_fan, P2.fan(), subdivide=True) sage: fibration = ToricVariety(fm.domain_fan()).hom(fm, P2) sage: ffm = fibration.fan_morphism() sage: primitive_cones = ffm.primitive_preimage_cones(P2.fan(1)[0]) sage: primitive_cone = primitive_cones[0] sage: fibration.fiber_component(primitive_cone) 2-d toric variety covered by 4 affine patches sage: fibration.fiber_component(primitive_cone, True) (2-d toric variety covered by 4 affine patches, 1) sage: for primitive_cone in primitive_cones: ....: print(fibration.fiber_component(primitive_cone)) 2-d toric variety covered by 4 affine patches 2-d toric variety covered by 3 affine patches 2-d toric variety covered by 3 affine patches
- fiber_dimension(codomain_cone)#
Return the dimension of the fiber over a particular torus orbit in the base.
INPUT:
codomain_cone
– a cone \(\sigma\) of the codomain, specifying a torus orbit \(O(\sigma)\).
OUTPUT:
An integer. The dimension of the fiber over the torus orbit corresponding to
codomain_cone
. If the fiber is the empty set,-1
is returned. Note that all fibers over this torus orbit are isomorphic, and therefore have the same dimension.EXAMPLES:
sage: P1xP1 = toric_varieties.P1xP1() sage: P1 = toric_varieties.P1() sage: f = P1xP1.hom(matrix([[1],[0]]), P1) sage: f.fiber_dimension(P1.fan(0)[0]) 1 sage: f.fiber_dimension(P1.fan(1)[0]) 1 sage: f.fiber_dimension(P1.fan(1)[1]) 1
Here is a more complicated example that is not a flat fibration:
sage: A2_Z2 = toric_varieties.A2_Z2() sage: O2_P1 = A2_Z2.resolve(new_rays=[(1,1)]) sage: blowup = O2_P1.hom(identity_matrix(2), A2_Z2) sage: blowup.fiber_dimension(A2_Z2.fan(0)[0]) 0 sage: blowup.fiber_dimension(A2_Z2.fan(1)[0]) 0 sage: blowup.fiber_dimension(A2_Z2.fan(2)[0]) 1
This corresponds to the three different fibers:
sage: blowup.fiber_generic() (0-d affine toric variety, 1) sage: blowup.fiber_component(Cone([(1,0)])) 0-d affine toric variety sage: blowup.fiber_component(Cone([(1,1)])) 1-d toric variety covered by 2 affine patches
- fiber_generic()#
Return the generic fiber.
OUTPUT:
a tuple \((X, n)\), where \(X\) is a
toric variety
with the embedding morphism into domain ofself
and \(n\) is an integer.
The fiber over the base point with homogeneous coordinates \([1:1:\cdots:1]\) consists of \(n\) disjoint toric varieties isomorphic to \(X\). Note that fibers of a dominant toric morphism are isomorphic over all points of a fixed torus orbit of its codomain, in particular over all points of the maximal torus, so it makes sense to talk about “the generic” fiber.
The embedding of \(X\) is a toric morphism with the
domain_fan()
being thekernel_fan()
of the defining fan morphism. By contrast, embeddings of fiber components over lower-dimensional torus orbits of the image are not toric morphisms. Usefiber_component()
for the latter (non-generic) fibers.EXAMPLES:
sage: P1xP1 = toric_varieties.P1xP1() sage: P1 = toric_varieties.P1() sage: fiber = P1xP1.hom(matrix([[1],[0]]), P1).fiber_generic() sage: fiber (1-d toric variety covered by 2 affine patches, 1) sage: f = fiber[0].embedding_morphism(); f 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 sending Rational polyhedral fan in Sublattice <N(0, 1)> to Rational polyhedral fan in 2-d lattice N. sage: f.as_polynomial_map() 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 on coordinates by sending [z0 : z1] to [1 : 1 : z0 : z1] sage: A1 = toric_varieties.A1() sage: fan = Fan([(0,1,2)], [(1,1,0), (1,0,1), (1,-1,-1)]) sage: fan = fan.subdivide(new_rays=[(1,0,0)]) sage: f = ToricVariety(fan).hom(matrix([[1],[0],[0]]), A1) sage: f.fiber_generic() (2-d affine toric variety, 1) sage: _[0].fan().generating_cones() (0-d cone of Rational polyhedral fan in Sublattice <N(0, 1, 0), N(0, 0, 1)>,)
- fiber_graph(codomain_cone)#
Return the fiber over a given torus orbit in the codomain.
INPUT:
codomain_cone
– a cone \(\sigma\) of the codomain, specifying a torus orbit \(O(\sigma)\).
OUTPUT:
A graph whose nodes are the irreducible components of a connected component of the fiber over a point of \(O(\sigma)\). If two irreducible components intersect, the corresponding nodes of the graph are joined by an edge. Note that irreducible components do not have to be of the same dimension.
See also
EXAMPLES:
sage: polytope = Polyhedron( ....: [(-3,0,-1,-1),(-1,2,-1,-1),(0,-1,0,0),(0,0,0,1),(0,0,1,0), ....: (0,1,0,0),(0,2,-1,-1),(1,0,0,0),(2,0,-1,-1)]) sage: coarse_fan = FaceFan(polytope, lattice=ToricLattice(4)) sage: P2 = toric_varieties.P2() sage: proj34 = block_matrix(2, 1, [zero_matrix(2,2), ....: identity_matrix(2)]) sage: fm = FanMorphism(proj34, coarse_fan, P2.fan(), subdivide=True) sage: fibration = ToricVariety(fm.domain_fan()).hom(fm, P2) sage: fibration.fiber_graph(P2.fan(0)[0]) Graph on 1 vertex sage: for c1 in P2.fan(1): ....: fibration.fiber_graph(c1) Graph on 1 vertex Graph on 1 vertex Graph on 4 vertices sage: fibration.fiber_graph(P2.fan(1)[2]).get_vertices() {0: 2-d toric variety covered by 4 affine patches, 1: 2-d toric variety covered by 3 affine patches, 2: 2-d toric variety covered by 3 affine patches, 3: 2-d toric variety covered by 4 affine patches} sage: fibration Scheme morphism: From: 4-d toric variety covered by 18 affine patches To: 2-d CPR-Fano toric variety covered by 3 affine patches Defn: Defined by sending Rational polyhedral fan in 4-d lattice N to Rational polyhedral fan in 2-d lattice N.
- class sage.schemes.toric.morphism.SchemeMorphism_orbit_closure_toric_variety(parent, defining_cone, ray_map)#
Bases:
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
orbit_closure()
method oftoric varieties
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.
- as_polynomial_map()#
Express the morphism via homogeneous polynomials.
OUTPUT:
A
SchemeMorphism_polynomial_toric_variety
. Raises aTypeError
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
- defining_cone()#
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
- pullback_divisor(divisor)#
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)
- class sage.schemes.toric.morphism.SchemeMorphism_point_toric_field(X, coordinates, check=True)#
Bases:
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
hom()
method oftoric varieties
instead.INPUT:
X
– toric variety or subscheme of a toric variety.coordinates
– list of coordinates in the base field ofX
.check
– ifTrue
(default), the input will be checked for correctness.
OUTPUT: A
SchemeMorphism_point_toric_field
.
- class sage.schemes.toric.morphism.SchemeMorphism_polynomial_toric_variety(parent, polynomials, check=True)#
Bases:
SchemeMorphism_polynomial
,Morphism
A morphism determined by homogeneous polynomials.
Warning
You should not create objects of this class directly. Use the
hom()
method oftoric varieties
instead.INPUT:
Same as for
SchemeMorphism_polynomial
.OUTPUT: A
SchemeMorphism_polynomial_toric_variety
.- as_fan_morphism()#
Express the morphism as a map defined by a fan morphism.
OUTPUT: A
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