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.

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_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.

OUPUT:

A SchemeMorphism_fan_toric_variety.

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'>

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 \(X_s\), \(X_i\) seat in between, i.e.

\[X \twoheadrightarrow X_s \to X_i \hookrightarrow X'.\]

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 a description of the toric algorithm.

OUTPUT:

Boolean. Whether self is birational.

EXAMPLES:

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_injective()

Check if self is injective.

See is_injective() for a description of the toric algorithm.

OUTPUT:

Boolean. Whether self is injective.

EXAMPLES:

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 a description of the toric algorithm.

OUTPUT:

Boolean. Whether self is surjective.

EXAMPLES:

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
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: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
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: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1.inject_variables()
Defining z0, z1, z2, z3
sage: P1 = P1xP1.subscheme(z0-z2)
sage: H = P1xP1.Hom(P1)
sage: import sage.schemes.toric.morphism as MOR
sage: MOR.SchemeMorphism_polynomial_toric_variety(H, [z0,z1,z0,z3])
Scheme morphism:
  From: 2-d toric variety covered by 4 affine patches
  To:   Closed subscheme of 2-d toric variety
        covered by 4 affine patches defined by:
  z0 - z2
  Defn: Defined on coordinates by sending
        [z0 : z1 : z2 : z3] to [z0 : z1 : z0 : z3]
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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