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:

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:

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:

OUTPUT:

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:

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.

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.

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:

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:

OUTPUT:

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:

OUTPUT:

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.

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:

OUTPUT:

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.

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:

OUTPUT:

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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