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 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 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

\[\CC^\times \times \mathbb{P}^1, (\mu,[u:v]) \mapsto [u:\mu v] \quad\text{and}\quad \left(\CC^\times\right)^2 \times \mathbb{P}^2, ((\alpha,\beta),[x:y:z]) \mapsto [x:\alpha y:\beta z] ,\]

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 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]:

\[f: O_{\mathbb{P}^1}(2) \to \CC^2/\ZZ_2, \quad (x,t,y) \mapsto \left( x\sqrt{t}, y\sqrt{t} \right)\]

which requires square roots.

Fibrations

If a toric morphism is 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()[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 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 ticket #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
class sage.schemes.toric.morphism.SchemeMorphism_fan_fiber_component_toric_variety(toric_morphism, defining_cone)

Bases: sage.schemes.generic.morphism.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 of toric_morphism. See fiber_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: primitive_cones = fibration.fan_morphism().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 : z1 : 0 : 1 : z0 : 1 : 1 : 1 : z2 : z3 : 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 a ValueError 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 : z1 : 0 : 1 : z0 : 1 : 1 : 1 : z2 : z3 : 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)
3*V(z0) + 2*V(z2)
sage: fc = f.fiber_component(Cone([(1,0,0)]))
sage: fc.embedding_morphism().pullback_divisor(D)
-3*V(z0) - 3*V(z1) - V(z2)
class sage.schemes.toric.morphism.SchemeMorphism_fan_toric_variety(parent, fan_morphism, check=True)

Bases: sage.schemes.generic.morphism.SchemeMorphism, sage.categories.morphism.Morphism

Construct a morphism determined by a fan morphism

Warning

You should not create objects of this class directly. Use the hom() method of toric 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 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 factoring the morphism. See SchemeMorphism_fan_toric_variety_dominant for additional functionality for fibrations.

OUPUT:

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 with self on the maximal torus factors through \(X_i\). In particular, \(X_i\) is the closure of the image of self 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
        [z0*z1 : z1 : 1]

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
        [z0 : z1 : z2 : z2]
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 if self 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 if self 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 of self.

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: sage.schemes.toric.morphism.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 of toric varieties instead.

INPUT:

See SchemeMorphism_fan_toric_variety. The given fan morphism must be dominant.

OUPUT:

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 of self.
  • multiplicity (default: False) – whether to return the number of fiber components corresponding to domain_cone as well.

OUTPUT:

  • either \(X\) or a tuple \((X, n)\), where \(X\) is a toric variety with the embedding morphism into domain of self and \(n\) is an integer.

Let \(\phi: \Sigma \to \Sigma'\) be the fan morphism corresponding to self. Let \(\sigma \in \Sigma\) and \(\sigma' \in \Sigma'\) be the image_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 to preimage_cones() of \(\sigma'\) and \(X\) is one of the \(n\) components corresponding to \(\sigma\). The irreducible components correspond to primitive_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: primitive_cones = fibration.fan_morphism().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 of self 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 the kernel_fan() of the defining fan morphism. By contrast, embeddings of fiber components over lower-dimensional torus orbits of the image are not toric morphisms. Use fiber_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

fiber_component().

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: sage.schemes.generic.morphism.SchemeMorphism, sage.categories.morphism.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 of toric 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.

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
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.

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: sage.schemes.generic.morphism.SchemeMorphism_point, sage.categories.morphism.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 of toric varieties 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 SchemeMorphism_point_toric_field.

TESTS:

sage: P1xP1 = toric_varieties.P1xP1()
sage: P1xP1(1,2,3,4)
[1 : 2 : 3 : 4]
class sage.schemes.toric.morphism.SchemeMorphism_polynomial_toric_variety(parent, polynomials, check=True)

Bases: sage.schemes.generic.morphism.SchemeMorphism_polynomial, sage.categories.morphism.Morphism

A morphism determined by homogeneous polynomials.

Warning

You should not create objects of this class directly. Use the hom() method of toric varieties instead.

INPUT:

Same as for SchemeMorphism_polynomial.

OUPUT:

A SchemeMorphism_polynomial_toric_variety.

TESTS:

sage: P1xP1 = toric_varieties.P1xP1()
sage: P1xP1.inject_variables()
Defining s, t, x, y
sage: P1 = P1xP1.subscheme(s-t)
sage: H = P1xP1.Hom(P1)
sage: import sage.schemes.toric.morphism as MOR
sage: MOR.SchemeMorphism_polynomial_toric_variety(H, [s, s, x, y])
Scheme morphism:
  From: 2-d CPR-Fano toric variety covered by 4 affine patches
  To:   Closed subscheme of 2-d CPR-Fano toric variety
        covered by 4 affine patches defined by:
  s - t
  Defn: Defined on coordinates by sending [s : t : x : y] to
        [s : s : x : y]
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!

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