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# Morphisms between toric lattices compatible with fans

This module is a part of the framework for toric varieties (`variety`, `fano_variety`). Its main purpose is to provide support for working with lattice morphisms compatible with fans via `FanMorphism` class.

AUTHORS:

- Andrey Novoseltsev (2010-10-17): initial version.

- Andrey Novoseltsev (2011-04-11): added tests for injectivity/surjectivity,

    fibration, bundle, as well as some related methods.

EXAMPLES:

Let’s consider the face and normal fans of the “diamond” and the projection to the \(x\)-axis:

    sage: diamond = lattice_polytope.cross_polytope(2)
    sage: face = FaceFan(diamond, lattice=ToricLattice(2))
    sage: normal = NormalFan(diamond)
    sage: N = face.lattice()
    sage: H = End(N)
    sage: phi = H([N.0, 0])
    sage: phi
    Free module morphism defined by the matrix
    [1 0]
    [0 0]
    Domain: 2-d lattice N
    Codomain: 2-d lattice N
    sage: FanMorphism(phi, normal, face)
    Traceback (most recent call last):
    ...
    ValueError: the image of generating cone #1 of the domain fan
    is not contained in a single cone of the codomain fan!

Some of the cones of the normal fan fail to be mapped to a single cone of the face fan. We can rectify the situation in the following way:

    sage: fm = FanMorphism(phi, normal, face, subdivide=True)
    sage: fm
    Fan morphism defined by the matrix
    [1 0]
    [0 0]
    Domain fan: Rational polyhedral fan in 2-d lattice N
    Codomain fan: Rational polyhedral fan in 2-d lattice N
    sage: fm.domain_fan().rays()
    N(-1,  1),
    N( 1,  1),
    N(-1, -1),
    N( 1, -1),
    N( 0, -1),
    N( 0,  1)
    in 2-d lattice N
    sage: normal.rays()
    N(-1,  1),
    N( 1,  1),
    N(-1, -1),
    N( 1, -1)
    in 2-d lattice N

As you see, it was necessary to insert two new rays (to prevent “upper” and “lower” cones of the normal fan from being mapped to the whole \(x\)-axis).

class `sage.geometry.fan_morphism.``FanMorphism`(morphism, domain_fan, codomain=None, subdivide=False, check=True, verbose=False)

  Bases: `sage.modules.free_module_morphism.FreeModuleMorphism`

  Create a fan morphism.

  Let \(\Sigma_1\) and \(\Sigma_2\) be two fans in lattices \(N_1\) and \(N_2\) respectively. Let \(\phi\) be a morphism (i.e. a linear map) from \(N_1\) to \(N_2\). We say that \(\phi\) is compatible with \(\Sigma_1\) and \(\Sigma_2\) if every cone \(\sigma_1\in\Sigma_1\) is mapped by \(\phi\) into a single cone \(\sigma_2\in\Sigma_2\), i.e. \(\phi(\sigma_1)\subset\sigma_2\) (\(\sigma_2\) may be different for different \(\sigma_1\)).

  By a fan morphism we understand a morphism between two lattices compatible with specified fans in these lattices. Such morphisms behave in exactly the same way as “regular” morphisms between lattices, but:

  - fan morphisms have a special constructor allowing some automatic adjustments to the initial fans (see below);
  - fan morphisms are aware of the associated fans and they can be accessed via `codomain_fan()` and `domain_fan()`;
  - fan morphisms can efficiently compute `image_cone()` of a given cone of the domain fan and `preimage_cones()` of a given cone of the codomain fan.

  INPUT:

  - `morphism` – either a morphism between domain and codomain, or an integral matrix defining such a morphism;
  - `domain_fan` – a `fan` in the domain;
  - `codomain` – (default: `None`) either a codomain lattice or a fan in the codomain. If the codomain fan is not given, the image fan (fan generated by images of generating cones) of `domain_fan` will be used, if possible;
  - `subdivide` – (default: `False`) if `True` and `domain_fan` is not compatible with the codomain fan because it is too coarse, it will be automatically refined to become compatible (the minimal refinement is canonical, so there are no choices involved);
  - `check` – (default: `True`) if `False`, given fans and morphism will be assumed to be compatible. Be careful when using this option, since wrong assumptions can lead to wrong and hard-to-detect errors. On the other hand, this option may save you some time;
  - `verbose` – (default: `False`) if `True`, some information may be printed during construction of the fan morphism.

  OUTPUT:

  - a fan morphism.

  EXAMPLES:

  Here we consider the face and normal fans of the “diamond” and the projection to the \(x\)-axis:

      sage: diamond = lattice_polytope.cross_polytope(2)
      sage: face = FaceFan(diamond, lattice=ToricLattice(2))
      sage: normal = NormalFan(diamond)
      sage: N = face.lattice()
      sage: H = End(N)
      sage: phi = H([N.0, 0])
      sage: phi
      Free module morphism defined by the matrix
      [1 0]
      [0 0]
      Domain: 2-d lattice N
      Codomain: 2-d lattice N
      sage: fm = FanMorphism(phi, face, normal)
      sage: fm.domain_fan() is face
      True

  Note, that since `phi` is compatible with these fans, the returned fan is exactly the same object as the initial `domain_fan`.

      sage: FanMorphism(phi, normal, face)
      Traceback (most recent call last):
      ...
      ValueError: the image of generating cone #1 of the domain fan
      is not contained in a single cone of the codomain fan!
      sage: fm = FanMorphism(phi, normal, face, subdivide=True)
      sage: fm.domain_fan() is normal
      False
      sage: fm.domain_fan().ngenerating_cones()
      6

  We had to subdivide two of the four cones of the normal fan, since they were mapped by `phi` into non-strictly convex cones.

  It is possible to omit the codomain fan, in which case the image fan will be used instead of it:

      sage: fm = FanMorphism(phi, face)
      sage: fm.codomain_fan()
      Rational polyhedral fan in 2-d lattice N
      sage: fm.codomain_fan().rays()
      N( 1, 0),
      N(-1, 0)
      in 2-d lattice N

  Now we demonstrate a more subtle example. We take the first quadrant as our domain fan. Then we divide the first quadrant into three cones, throw away the middle one and take the other two as our codomain fan. These fans are incompatible with the identity lattice morphism since the image of the domain fan is out of the support of the codomain fan:

      sage: N = ToricLattice(2)
      sage: phi = End(N).identity()
      sage: F1 = Fan(cones=[(0,1)], rays=[(1,0), (0,1)])
      sage: F2 = Fan(cones=[(0,1), (2,3)],
      ...            rays=[(1,0), (2,1), (1,2), (0,1)])
      sage: FanMorphism(phi, F1, F2)
      Traceback (most recent call last):
      ...
      ValueError: the image of generating cone #0 of the domain fan
      is not contained in a single cone of the codomain fan!
      sage: FanMorphism(phi, F1, F2, subdivide=True)
      Traceback (most recent call last):
      ...
      ValueError: morphism defined by
      [1 0]
      [0 1]
      does not map
      Rational polyhedral fan in 2-d lattice N
      into the support of
      Rational polyhedral fan in 2-d lattice N!

  The problem was detected and handled correctly (i.e. an exception was raised). However, the used algorithm requires extra checks for this situation after constructing a potential subdivision and this can take significant time. You can save about half the time using `check=False` option, if you know in advance that it is possible to make fans compatible with the morphism by subdividing the domain fan. Of course, if your assumption was incorrect, the result will be wrong and you will get a fan which does map into the support of the codomain fan, but is not a subdivision of the domain fan. You can test it on the example above:

      sage: fm = FanMorphism(phi, F1, F2, subdivide=True,
      ...                    check=False, verbose=True)
      Placing ray images (... ms)
      Computing chambers (... ms)
      Number of domain cones: 1.
      Number of chambers: 2.
      Cone 0 sits in chambers 0 1 (... ms)
      sage: fm.domain_fan().is_equivalent(F2)
      True

  `codomain_fan`(dim=None, codim=None)

    Return the codomain fan of `self`.

    INPUT:

    - `dim` – dimension of the requested cones;
    - `codim` – codimension of the requested cones.

    OUTPUT:

    - `rational polyhedral fan` if no parameters were given, `tuple` of `cones` otherwise.

    EXAMPLES:

        sage: quadrant = Cone([(1,0), (0,1)])
        sage: quadrant = Fan([quadrant])
        sage: quadrant_bl = quadrant.subdivide([(1,1)])
        sage: fm = FanMorphism(identity_matrix(2), quadrant_bl, quadrant)
        sage: fm.codomain_fan()
        Rational polyhedral fan in 2-d lattice N
        sage: fm.codomain_fan() is quadrant
        True

  `domain_fan`(dim=None, codim=None)

    Return the codomain fan of `self`.

    INPUT:

    - `dim` – dimension of the requested cones;
    - `codim` – codimension of the requested cones.

    OUTPUT:

    - `rational polyhedral fan` if no parameters were given, `tuple` of `cones` otherwise.

    EXAMPLES:

        sage: quadrant = Cone([(1,0), (0,1)])
        sage: quadrant = Fan([quadrant])
        sage: quadrant_bl = quadrant.subdivide([(1,1)])
        sage: fm = FanMorphism(identity_matrix(2), quadrant_bl, quadrant)
        sage: fm.domain_fan()
        Rational polyhedral fan in 2-d lattice N
        sage: fm.domain_fan() is quadrant_bl
        True

  `factor`()

    Factor `self` into injective * birational * surjective morphisms.

    OUTPUT:

    - a triple of `FanMorphism` \((\phi_i, \phi_b, \phi_s)\), such that \(\phi_s\) is surjective, \(\phi_b\) is birational, \(\phi_i\) is injective, and `self` is equal to \(\phi_i \circ \phi_b \circ \phi_s\).

    Intermediate fans live in the saturation of the image of `self` as a map between lattices and are the image of the `domain_fan()` and the restriction of the `codomain_fan()`, i.e. if `self` maps \(\Sigma \to \Sigma'\), then we have factorization into

    \[\Sigma \twoheadrightarrow \Sigma_s \to \Sigma_i \hookrightarrow \Sigma.\]

    Note

    - \(\Sigma_s\) is the finest fan with the smallest support that is compatible with `self`: any fan morphism from \(\Sigma\) given by the same map of lattices as `self` factors through \(\Sigma_s\).
    - \(\Sigma_i\) is the coarsest fan of the largest support that is compatible with `self`: any fan morphism into \(\Sigma'\) given by the same map of lattices as `self` factors though \(\Sigma_i\).

    EXAMPLES:

    We map an affine plane into a projective 3-space in such a way, that it becomes “a double cover of a chart of the blow up of one of the coordinate planes”:

        sage: A2 = toric_varieties.A2()
        sage: P3 = toric_varieties.P(3)
        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]

    Now we will work with the underlying fan morphism:

        sage: phi = phi.fan_morphism()
        sage: phi
        Fan morphism defined by the matrix
        [2 0 0]
        [1 1 0]
        Domain fan: Rational polyhedral fan in 2-d lattice N
        Codomain fan: Rational polyhedral fan in 3-d lattice N
        sage: phi.is_surjective(), phi.is_birational(), phi.is_injective()
        (False, False, False)
        sage: phi_i, phi_b, phi_s = phi.factor()
        sage: phi_s.is_surjective(), phi_b.is_birational(), phi_i.is_injective()
        (True, True, True)
        sage: prod(phi.factor()) == phi
        True

    Double cover (surjective):

        sage: A2.fan().rays()
        N(1, 0),
        N(0, 1)
        in 2-d lattice N
        sage: phi_s
        Fan morphism defined by the matrix
        [2 0]
        [1 1]
        Domain fan: Rational polyhedral fan in 2-d lattice N
        Codomain fan: Rational polyhedral fan in Sublattice <N(1, 0, 0), N(0, 1, 0)>
        sage: phi_s.codomain_fan().rays()
        N(1, 0, 0),
        N(1, 1, 0)
        in Sublattice <N(1, 0, 0), N(0, 1, 0)>

    Blowup chart (birational):

        sage: phi_b
        Fan morphism defined by the matrix
        [1 0]
        [0 1]
        Domain fan: Rational polyhedral fan in Sublattice <N(1, 0, 0), N(0, 1, 0)>
        Codomain fan: Rational polyhedral fan in Sublattice <N(1, 0, 0), N(0, 1, 0)>
        sage: phi_b.codomain_fan().rays()
        N( 1,  0, 0),
        N( 0,  1, 0),
        N(-1, -1, 0)
        in Sublattice <N(1, 0, 0), N(0, 1, 0)>

    Coordinate plane inclusion (injective):

        sage: phi_i
        Fan morphism defined by the matrix
        [1 0 0]
        [0 1 0]
        Domain fan: Rational polyhedral fan in Sublattice <N(1, 0, 0), N(0, 1, 0)>
        Codomain fan: Rational polyhedral fan in 3-d lattice N
        sage: phi.codomain_fan().rays()
        N( 1,  0,  0),
        N( 0,  1,  0),
        N( 0,  0,  1),
        N(-1, -1, -1)
        in 3-d lattice N

    TESTS:

        sage: phi_s.matrix() * phi_b.matrix() * phi_i.matrix() == m
        True

        sage: phi.domain_fan() is phi_s.domain_fan()
        True
        sage: phi_s.codomain_fan() is phi_b.domain_fan()
        True
        sage: phi_b.codomain_fan() is phi_i.domain_fan()
        True
        sage: phi_i.codomain_fan() is phi.codomain_fan()
        True

        sage: trivialfan2 = Fan([],[],lattice=ToricLattice(2))
        sage: trivialfan3 = Fan([],[],lattice=ToricLattice(3))
        sage: f = FanMorphism(zero_matrix(2,3), trivialfan2, trivialfan3)
        sage: [phi.matrix().dimensions() for phi in f.factor()]
        [(0, 3), (0, 0), (2, 0)]

  `image_cone`(cone)

    Return the cone of the codomain fan containing the image of `cone`.

    INPUT:

    - `cone` – a `cone` equivalent to a cone of the `domain_fan()` of `self`.

    OUTPUT:

    - a `cone` of the `codomain_fan()` of `self`.

    EXAMPLES:

        sage: quadrant = Cone([(1,0), (0,1)])
        sage: quadrant = Fan([quadrant])
        sage: quadrant_bl = quadrant.subdivide([(1,1)])
        sage: fm = FanMorphism(identity_matrix(2), quadrant_bl, quadrant)
        sage: fm.image_cone(Cone([(1,0)]))
        1-d cone of Rational polyhedral fan in 2-d lattice N
        sage: fm.image_cone(Cone([(1,1)]))
        2-d cone of Rational polyhedral fan in 2-d lattice N

    TESTS:

    We check that complete codomain fans are handled correctly, since a different algorithm is used in this case:

        sage: diamond = lattice_polytope.cross_polytope(2)
        sage: face = FaceFan(diamond, lattice=ToricLattice(2))
        sage: normal = NormalFan(diamond)
        sage: N = face.lattice()
        sage: fm = FanMorphism(identity_matrix(2),
        ...           normal, face, subdivide=True)
        sage: fm.image_cone(Cone([(1,0)]))
        1-d cone of Rational polyhedral fan in 2-d lattice N
        sage: fm.image_cone(Cone([(1,1)]))
        2-d cone of Rational polyhedral fan in 2-d lattice N

  `index`(cone=None)

    Return the index of `self` as a map between lattices.

    INPUT:

    - `cone` – (default: `None`) a `cone` of the `codomain_fan()` of `self`.

    OUTPUT:

    - an integer, infinity, or `None`.

    If no cone was specified, this function computes the index of the image of `self` in the codomain. If a cone \(\sigma\) was given, the index of `self` over \(\sigma\) is computed in the sense of Definition 2.1.7 of [HLY]: if \(\sigma'\) is any cone of the `domain_fan()` of `self` whose relative interior is mapped to the relative interior of \(\sigma\), it is the index of the image of \(N'(\sigma')\) in \(N(\sigma)\), where \(N'\) and \(N\) are domain and codomain lattices respectively. While that definition was formulated for the case of the finite index only, we extend it to the infinite one as well and return `None` if there is no \(\sigma'\) at all. See examples below for situations when such things happen. Note also that the index of `self` is the same as index over the trivial cone.

    EXAMPLES:

        sage: Sigma = toric_varieties.dP8().fan()
        sage: Sigma_p = toric_varieties.P1().fan()
        sage: phi = FanMorphism(matrix([[1], [-1]]), Sigma, Sigma_p)
        sage: phi.index()
        1
        sage: psi = FanMorphism(matrix([[2], [-2]]), Sigma, Sigma_p)
        sage: psi.index()
        2
        sage: xi = FanMorphism(matrix([[1, 0]]), Sigma_p, Sigma)
        sage: xi.index()
        +Infinity

    Infinite index in the last example indicates that the image has positive codimension in the codomain. Let’s look at the rays of our fans:

        sage: Sigma_p.rays()
        N( 1),
        N(-1)
        in 1-d lattice N
        sage: Sigma.rays()
        N( 1,  1),
        N( 0,  1),
        N(-1, -1),
        N( 1,  0)
        in 2-d lattice N
        sage: xi.factor()[0].domain_fan().rays()
        N( 1, 0),
        N(-1, 0)
        in Sublattice <N(1, 0)>

    We see that one of the rays of the fan of `P1` is mapped to a ray, while the other one to the interior of some 2-d cone. Both rays correspond to single points on `P1`, yet one is mapped to the distinguished point of a torus invariant curve of `dP8` (with the rest of this curve being uncovered) and the other to a fixed point of `dP8` (thus completely covering this torus orbit in `dP8`).

    We should therefore expect the following behaviour: all indices over 1-d cones are `None`, except for one which is infinite, and all indices over 2-d cones are `None`, except for one which is 1:

        sage: [xi.index(cone) for cone in Sigma(1)]
        [None, None, None, +Infinity]
        sage: [xi.index(cone) for cone in Sigma(2)]
        [None, 1, None, None]

    TESTS:

        sage: Sigma = toric_varieties.dP8().fan()
        sage: Sigma_p = toric_varieties.Cube_nonpolyhedral().fan()
        sage: m = matrix([[2,6,10], [7,11,13]])
        sage: zeta = FanMorphism(m, Sigma, Sigma_p, subdivide=True)
        sage: [zeta.index(cone) for cone in flatten(Sigma_p.cones())]
        [+Infinity, None, None, None, None, None, None, None, None, None,
         4, 4, None, 4, None, None, 2, None, 4, None, 4, 1, 1, 1, 1, 1, 1]
        sage: zeta = prod(zeta.factor()[1:])
        sage: Sigma_p = zeta.codomain_fan()
        sage: [zeta.index(cone) for cone in flatten(Sigma_p.cones())]
        [4, 4, 1, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1]
        sage: zeta.index() == zeta.index(Sigma_p(0)[0])
        True

  `is_birational`()

    Check if `self` is birational.

    OUTPUT:

    - `True` if `self` is birational, `False` otherwise.

    For fan morphisms this check is equivalent to `self.index() == 1` and means that the corresponding map between toric varieties is birational.

    EXAMPLES:

        sage: Sigma = toric_varieties.dP8().fan()
        sage: Sigma_p = toric_varieties.P1().fan()
        sage: phi = FanMorphism(matrix([[1], [-1]]), Sigma, Sigma_p)
        sage: psi = FanMorphism(matrix([[2], [-2]]), Sigma, Sigma_p)
        sage: xi = FanMorphism(matrix([[1, 0]]), Sigma_p, Sigma)
        sage: phi.index(), psi.index(), xi.index()
        (1, 2, +Infinity)
        sage: phi.is_birational(), psi.is_birational(), xi.is_birational()
        (True, False, False)

  `is_bundle`()

    Check if `self` is a bundle.

    OUTPUT:

    - `True` if `self` is a bundle, `False` otherwise.

    Let \(\phi: \Sigma \to \Sigma'\) be a fan morphism such that the underlying lattice morphism \(\phi: N \to N'\) is surjective. Let \(\Sigma_0\) be the kernel fan of \(\phi\). Then \(\phi\) is a bundle (or splitting) if there is a subfan \(\widehat{\Sigma}\) of \(\Sigma\) such that the following two conditions are satisfied:

    > 1. Cones of \(\Sigma\) are precisely the cones of the form \(\sigma_0 + \widehat{\sigma}\), where \(\sigma_0 \in \Sigma_0\) and \(\widehat{\sigma} \in \widehat{\Sigma}\).
    > 2. Cones of \(\widehat{\Sigma}\) are in bijection with cones of \(\Sigma'\) induced by \(\phi\) and \(\phi\) maps lattice points in every cone \(\widehat{\sigma} \in \widehat{\Sigma}\) bijectively onto lattice points in \(\phi(\widehat{\sigma})\).

    If a fan morphism \(\phi: \Sigma \to \Sigma'\) is a bundle, then \(X_\Sigma\) is a fiber bundle over \(X_{\Sigma'}\) with fibers \(X_{\Sigma_0, N_0}\), where \(N_0\) is the kernel lattice of \(\phi\). See [CLS11] for more details.

    See also

    `is_fibration()`, `kernel_fan()`.

    REFERENCES:

    [CLS11] | David A. Cox, John Little, and Hal Schenck. Toric Varieties. Volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011.

    EXAMPLES:

    We consider several maps between fans of a del Pezzo surface and the projective line:

        sage: Sigma = toric_varieties.dP8().fan()
        sage: Sigma_p = toric_varieties.P1().fan()
        sage: phi = FanMorphism(matrix([[1], [-1]]), Sigma, Sigma_p)
        sage: psi = FanMorphism(matrix([[2], [-2]]), Sigma, Sigma_p)
        sage: xi = FanMorphism(matrix([[1, 0]]), Sigma_p, Sigma)
        sage: phi.is_bundle()
        True
        sage: phi.is_fibration()
        True
        sage: phi.index()
        1
        sage: psi.is_bundle()
        False
        sage: psi.is_fibration()
        True
        sage: psi.index()
        2
        sage: xi.is_fibration()
        False
        sage: xi.index()
        +Infinity

    The first of these maps induces not only a fibration, but a fiber bundle structure. The second map is very similar, yet it fails to be a bundle, as its index is 2. The last map is not even a fibration.

  `is_dominant`()

    Return whether the fan morphism is dominant.

    A fan morphism \(\phi\) is dominant if it is surjective as a map of vector spaces. That is, \(\phi_\RR: N_\RR \to N'_\RR\) is surjective.

    If the domain fan is `complete`, then this implies that the fan morphism is `surjective`.

    If the fan morphism is dominant, then the associated morphism of toric varieties is dominant in the algebraic-geometric sense (that is, surjective onto a dense subset).

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: P1 = toric_varieties.P1()
        sage: A1 = toric_varieties.A1()
        sage: phi = FanMorphism(matrix([[1]]), A1.fan(), P1.fan())
        sage: phi.is_dominant()
        True
        sage: phi.is_surjective()
        False

  `is_fibration`()

    Check if `self` is a fibration.

    OUTPUT:

    - `True` if `self` is a fibration, `False` otherwise.

    A fan morphism \(\phi: \Sigma \to \Sigma'\) is a fibration if for any cone \(\sigma' \in \Sigma'\) and any primitive preimage cone \(\sigma \in \Sigma\) corresponding to \(\sigma'\) the linear map of vector spaces \(\phi_\RR\) induces a bijection between \(\sigma\) and \(\sigma'\), and, in addition, \(phi\) is `dominant` (that is, \(\phi_\RR: N_\RR \to N'_\RR\) is surjective).

    If a fan morphism \(\phi: \Sigma \to \Sigma'\) is a fibration, then the associated morphism between toric varieties \(\tilde{\phi}: X_\Sigma \to X_{\Sigma'}\) is a fibration in the sense that it is surjective and all of its fibers have the same dimension, namely \(\dim X_\Sigma - \dim X_{\Sigma'}\). These fibers do not have to be isomorphic, i.e. a fibration is not necessarily a fiber bundle. See [HLY02] for more details.

    See also

    `is_bundle()`, `primitive_preimage_cones()`.

    REFERENCES:

    [HLY02] | Yi Hu, Chien-Hao Liu, and Shing-Tung Yau. Toric morphisms and fibrations of toric Calabi-Yau hypersurfaces. Adv. Theor. Math. Phys., 6(3):457-506, 2002. arXiv:math/0010082v2 [math.AG].

    EXAMPLES:

    We consider several maps between fans of a del Pezzo surface and the projective line:

        sage: Sigma = toric_varieties.dP8().fan()
        sage: Sigma_p = toric_varieties.P1().fan()
        sage: phi = FanMorphism(matrix([[1], [-1]]), Sigma, Sigma_p)
        sage: psi = FanMorphism(matrix([[2], [-2]]), Sigma, Sigma_p)
        sage: xi = FanMorphism(matrix([[1, 0]]), Sigma_p, Sigma)
        sage: phi.is_bundle()
        True
        sage: phi.is_fibration()
        True
        sage: phi.index()
        1
        sage: psi.is_bundle()
        False
        sage: psi.is_fibration()
        True
        sage: psi.index()
        2
        sage: xi.is_fibration()
        False
        sage: xi.index()
        +Infinity

    The first of these maps induces not only a fibration, but a fiber bundle structure. The second map is very similar, yet it fails to be a bundle, as its index is 2. The last map is not even a fibration.

    TESTS:

    We check that reviewer’s example on Trac 11200 works as expected:

        sage: P1 = toric_varieties.P1()
        sage: A1 = toric_varieties.A1()
        sage: phi = FanMorphism(matrix([[1]]), A1.fan(), P1.fan())
        sage: phi.is_fibration()
        False

  `is_injective`()

    Check if `self` is injective.

    OUTPUT:

    - `True` if `self` is injective, `False` otherwise.

    Let \(\phi: \Sigma \to \Sigma'\) be a fan morphism such that the underlying lattice morphism \(\phi: N \to N'\) bijectively maps \(N\) to a saturated sublattice of \(N'\). Let \(\psi: \Sigma \to \Sigma'_0\) be the restriction of \(\phi\) to the image. Then \(\phi\) is injective if the map between cones corresponding to \(\psi\) (injectively) maps each cone of \(\Sigma\) to a cone of the same dimension.

    If a fan morphism \(\phi: \Sigma \to \Sigma'\) is injective, then the associated morphism between toric varieties \(\tilde{\phi}: X_\Sigma \to X_{\Sigma'}\) is injective.

    See also

    `factor()`.

    EXAMPLES:

    Consider the fan of the affine plane:

        sage: A2 = toric_varieties.A(2).fan()

    We will map several fans consisting of a single ray into the interior of the 2-cone:

        sage: Sigma = Fan([Cone([(1,1)])])
        sage: m = identity_matrix(2)
        sage: FanMorphism(m, Sigma, A2).is_injective()
        False

    This morphism was not injective since (in the toric varieties interpretation) the 1-dimensional orbit corresponding to the ray was mapped to the 0-dimensional orbit corresponding to the 2-cone.

        sage: Sigma = Fan([Cone([(1,)])])
        sage: m = matrix(1, 2, [1,1])
        sage: FanMorphism(m, Sigma, A2).is_injective()
        True

    While the fans in this example are close to the previous one, here the ray corresponds to a 0-dimensional orbit.

        sage: Sigma = Fan([Cone([(1,)])])
        sage: m = matrix(1, 2, [2,2])
        sage: FanMorphism(m, Sigma, A2).is_injective()
        False

    Here the problem is that `m` maps the domain lattice to a non-saturated sublattice of the codomain. The corresponding map of the toric varieties is a two-sheeted cover of its image.

    We also embed the affine plane into the projective one:

        sage: P2 = toric_varieties.P(2).fan()
        sage: m = identity_matrix(2)
        sage: FanMorphism(m, A2, P2).is_injective()
        True

  `is_surjective`()

    Check if `self` is surjective.

    OUTPUT:

    - `True` if `self` is surjective, `False` otherwise.

    A fan morphism \(\phi: \Sigma \to \Sigma'\) is surjective if the corresponding map between cones is surjective, i.e. for each cone \(\sigma' \in \Sigma'\) there is at least one preimage cone \(\sigma \in \Sigma\) such that the relative interior of \(\sigma\) is mapped to the relative interior of \(\sigma'\) and, in addition, \(\phi_\RR: N_\RR \to N'_\RR\) is surjective.

    If a fan morphism \(\phi: \Sigma \to \Sigma'\) is surjective, then the associated morphism between toric varieties \(\tilde{\phi}: X_\Sigma \to X_{\Sigma'}\) is surjective.

    See also

    `is_bundle()`, `is_fibration()`, `preimage_cones()`, `is_complete()`.

    EXAMPLES:

    We check that the blow up of the affine plane at the origin is surjective:

        sage: A2 = toric_varieties.A(2).fan()
        sage: Bl = A2.subdivide([(1,1)])
        sage: m = identity_matrix(2)
        sage: FanMorphism(m, Bl, A2).is_surjective()
        True

    It remains surjective if we throw away “south and north poles” of the exceptional divisor:

        sage: FanMorphism(m, Fan(Bl.cones(1)), A2).is_surjective()
        True

    But a single patch of the blow up does not cover the plane:

        sage: F = Fan([Bl.generating_cone(0)])
        sage: FanMorphism(m, F, A2).is_surjective()
        False

    TESTS:

    We check that reviewer’s example on trac ticket #11200 works as expected:

        sage: P1 = toric_varieties.P1()
        sage: A1 = toric_varieties.A1()
        sage: phi = FanMorphism(matrix([[1]]), A1.fan(), P1.fan())
        sage: phi.is_surjective()
        False

  `kernel_fan`()

    Return the subfan of the domain fan mapped into the origin.

    OUTPUT:

    - a `fan`.

    Note

    The lattice of the kernel fan is the `kernel()` sublattice of `self`.

    See also

    `preimage_fan()`.

    EXAMPLES:

        sage: fan = Fan(rays=[(1,0), (1,1), (0,1)], cones=[(0,1), (1,2)])
        sage: fm = FanMorphism(matrix(2, 1, [1,-1]), fan, ToricLattice(1))
        sage: fm.kernel_fan()
        Rational polyhedral fan in Sublattice <N(1, 1)>
        sage: _.rays()
        N(1, 1)
        in Sublattice <N(1, 1)>
        sage: fm.kernel_fan().cones()
        ((0-d cone of Rational polyhedral fan in Sublattice <N(1, 1)>,),
         (1-d cone of Rational polyhedral fan in Sublattice <N(1, 1)>,))

  `preimage_cones`(cone)

    Return cones of the domain fan whose `image_cone()` is `cone`.

    INPUT:

    - `cone` – a `cone` equivalent to a cone of the `codomain_fan()` of `self`.

    OUTPUT:

    - a `tuple` of `cones` of the `domain_fan()` of `self`, sorted by dimension.

    See also

    `preimage_fan()`.

    EXAMPLES:

        sage: quadrant = Cone([(1,0), (0,1)])
        sage: quadrant = Fan([quadrant])
        sage: quadrant_bl = quadrant.subdivide([(1,1)])
        sage: fm = FanMorphism(identity_matrix(2), quadrant_bl, quadrant)
        sage: fm.preimage_cones(Cone([(1,0)]))
        (1-d cone of Rational polyhedral fan in 2-d lattice N,)
        sage: fm.preimage_cones(Cone([(1,0), (0,1)]))
        (1-d cone of Rational polyhedral fan in 2-d lattice N,
         2-d cone of Rational polyhedral fan in 2-d lattice N,
         2-d cone of Rational polyhedral fan in 2-d lattice N)

    TESTS:

    We check that reviewer’s example from Trac #9972 is handled correctly:

        sage: N1 = ToricLattice(1)
        sage: N2 = ToricLattice(2)
        sage: Hom21 = Hom(N2, N1)
        sage: pr = Hom21([N1.0,0])
        sage: P1xP1 = toric_varieties.P1xP1()
        sage: f = FanMorphism(pr, P1xP1.fan())
        sage: c = f.image_cone(Cone([(1,0), (0,1)]))
        sage: c
        1-d cone of Rational polyhedral fan in 1-d lattice N
        sage: f.preimage_cones(c)
        (1-d cone of Rational polyhedral fan in 2-d lattice N,
         2-d cone of Rational polyhedral fan in 2-d lattice N,
         2-d cone of Rational polyhedral fan in 2-d lattice N)

  `preimage_fan`(cone)

    Return the subfan of the domain fan mapped into `cone`.

    INPUT:

    - `cone` – a `cone` equivalent to a cone of the `codomain_fan()` of `self`.

    OUTPUT:

    - a `fan`.

    Note

    The preimage fan of `cone` consists of all cones of the `domain_fan()` which are mapped into `cone`, including those that are mapped into its boundary. So this fan is not necessarily generated by `preimage_cones()` of `cone`.

    See also

    `kernel_fan()`, `preimage_cones()`.

    EXAMPLES:

        sage: quadrant_cone = Cone([(1,0), (0,1)])
        sage: quadrant_fan = Fan([quadrant_cone])
        sage: quadrant_bl = quadrant_fan.subdivide([(1,1)])
        sage: fm = FanMorphism(identity_matrix(2),
        ...                    quadrant_bl, quadrant_fan)
        sage: fm.preimage_fan(Cone([(1,0)])).cones()
        ((0-d cone of Rational polyhedral fan in 2-d lattice N,),
         (1-d cone of Rational polyhedral fan in 2-d lattice N,))
        sage: fm.preimage_fan(quadrant_cone).ngenerating_cones()
        2
        sage: len(fm.preimage_cones(quadrant_cone))
        3

  `primitive_preimage_cones`(cone)

    Return the primitive cones of the domain fan corresponding to `cone`.

    INPUT:

    - `cone` – a `cone` equivalent to a cone of the `codomain_fan()` of `self`.

    OUTPUT:

    - a `cone`.

    Let \(\phi: \Sigma \to \Sigma'\) be a fan morphism, let \(\sigma \in \Sigma\), and let \(\sigma' = \phi(\sigma)\). Then \(\sigma\) is a primitive cone corresponding to \(\sigma'\) if there is no proper face \(\tau\) of \(\sigma\) such that \(\phi(\tau) = \sigma'\).

    Primitive cones play an important role for fibration morphisms.

    See also

    `is_fibration()`, `preimage_cones()`, `preimage_fan()`.

    EXAMPLES:

    Consider a projection of a del Pezzo surface onto the projective line:

        sage: Sigma = toric_varieties.dP6().fan()
        sage: Sigma.rays()
        N( 0,  1),
        N(-1,  0),
        N(-1, -1),
        N( 0, -1),
        N( 1,  0),
        N( 1,  1)
        in 2-d lattice N
        sage: Sigma_p = toric_varieties.P1().fan()
        sage: phi = FanMorphism(matrix([[1], [-1]]), Sigma, Sigma_p)

    Under this map, one pair of rays is mapped to the origin, one in the positive direction, and one in the negative one. Also three 2-dimensional cones are mapped in the positive direction and three in the negative one, so there are 5 preimage cones corresponding to either of the rays of the codomain fan `Sigma_p`:

        sage: len(phi.preimage_cones(Cone([(1,)])))
        5

    Yet only rays are primitive:

        sage: phi.primitive_preimage_cones(Cone([(1,)]))
        (1-d cone of Rational polyhedral fan in 2-d lattice N,
         1-d cone of Rational polyhedral fan in 2-d lattice N)

    Since all primitive cones are mapped onto their images bijectively, we get a fibration:

        sage: phi.is_fibration()
        True

    But since there are several primitive cones corresponding to the same cone of the codomain fan, this map is not a bundle, even though its index is 1:

        sage: phi.is_bundle()
        False
        sage: phi.index()
        1

  `relative_star_generators`(domain_cone)

    Return the relative star generators of `domain_cone`.

    INPUT:

    - `domain_cone` – a cone of the `domain_fan()` of `self`.

    OUTPUT:

    - `star_generators()` of `domain_cone` viewed as a cone of `preimage_fan()` of `image_cone()` of `domain_cone`.

    EXAMPLES:

        sage: A2 = toric_varieties.A(2).fan()
        sage: Bl = A2.subdivide([(1,1)])
        sage: f = FanMorphism(identity_matrix(2), Bl, A2)
        sage: for c1 in Bl(1):
        ...       print f.relative_star_generators(c1)
        (1-d cone of Rational polyhedral fan in 2-d lattice N,)
        (1-d cone of Rational polyhedral fan in 2-d lattice N,)
        (2-d cone of Rational polyhedral fan in 2-d lattice N,
         2-d cone of Rational polyhedral fan in 2-d lattice N)

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