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# Toric varieties

This module provides support for (normal) toric varieties, corresponding to `rational polyhedral fans`. See also `fano_variety` for a more restrictive class of (weak) Fano toric varieties.

An excellent reference on toric varieties is the book “Toric Varieties” by David A. Cox, John B. Little, and Hal Schenck [CLS].

The interface to this module is provided through functions `AffineToricVariety()` and `ToricVariety()`, although you may also be interested in `normalize_names()`.

Note

We do NOT build “general toric varieties” from affine toric varieties. Instead, we are using the quotient representation of toric varieties with the homogeneous coordinate ring (a.k.a. Cox’s ring or the total coordinate ring). This description works best for simplicial fans of the full dimension.

REFERENCES:

[CLS] | (1, 2) David A. Cox, John B. Little, Hal Schenck, “Toric Varieties”, Graduate Studies in Mathematics, Amer. Math. Soc., Providence, RI, 2011

AUTHORS:

- Andrey Novoseltsev (2010-05-17): initial version.
- Volker Braun (2010-07-24): Cohomology and characteristic classes added.

EXAMPLES:

We start with constructing the affine plane as an affine toric variety. First, we need to have a corresponding cone:

    sage: quadrant = Cone([(1,0), (0,1)])

If you don’t care about variable names and the base field, that’s all we need for now:

    sage: A2 = AffineToricVariety(quadrant)
    sage: A2
    2-d affine toric variety
    sage: origin = A2(0,0)
    sage: origin
    [0 : 0]

Only affine toric varieties have points whose (homogeneous) coordinates are all zero.

    sage: parent(origin)
    Set of rational points of 2-d affine toric variety

As you can see, by default toric varieties live over the field of rational numbers:

    sage: A2.base_ring()
    Rational Field

While usually toric varieties are considered over the field of complex numbers, for computational purposes it is more convenient to work with fields that have exact representation on computers. You can also always do

    sage: C2 = AffineToricVariety(quadrant, base_field=CC)
    sage: C2.base_ring()
    Complex Field with 53 bits of precision
    sage: C2(1,2+i)
    [1.00000000000000 : 2.00000000000000 + 1.00000000000000*I]

or even

    sage: F = CC["a, b"].fraction_field()
    sage: F.inject_variables()
    Defining a, b
    sage: A2 = AffineToricVariety(quadrant, base_field=F)
    sage: A2(a,b)
    [a : b]

OK, if you need to work only with affine spaces, `AffineSpace()` may be a better way to construct them. Our next example is the product of two projective lines realized as the toric variety associated to the `face fan` of the “diamond”:

    sage: diamond = lattice_polytope.cross_polytope(2)
    sage: diamond.vertices_pc()
    M( 1,  0),
    M( 0,  1),
    M(-1,  0),
    M( 0, -1)
    in 2-d lattice M
    sage: fan = FaceFan(diamond)
    sage: P1xP1 = ToricVariety(fan)
    sage: P1xP1
    2-d toric variety covered by 4 affine patches
    sage: P1xP1.fan().rays()
    M( 1,  0),
    M( 0,  1),
    M(-1,  0),
    M( 0, -1)
    in 2-d lattice M
    sage: P1xP1.gens()
    (z0, z1, z2, z3)

We got four coordinates - two for each of the projective lines, but their names are perhaps not very well chosen. Let’s make \((x,y)\) to be coordinates on the first line and \((s,t)\) on the second one:

    sage: P1xP1 = ToricVariety(fan, coordinate_names="x s y t")
    sage: P1xP1.gens()
    (x, s, y, t)

Now, if we want to define subschemes of this variety, the defining polynomials must be homogeneous in each of these pairs:

    sage: P1xP1.inject_variables()
    Defining x, s, y, t
    sage: P1xP1.subscheme(x)
    Closed subscheme of 2-d toric variety
    covered by 4 affine patches defined by:
      x
    sage: P1xP1.subscheme(x^2 + y^2)
    Closed subscheme of 2-d toric variety
    covered by 4 affine patches defined by:
      x^2 + y^2
    sage: P1xP1.subscheme(x^2 + s^2)
    Traceback (most recent call last):
    ...
    ValueError: x^2 + s^2 is not homogeneous
    on 2-d toric variety covered by 4 affine patches!
    sage: P1xP1.subscheme([x^2*s^2 + x*y*t^2 +y^2*t^2, s^3 + t^3])
    Closed subscheme of 2-d toric variety
    covered by 4 affine patches defined by:
      x^2*s^2 + x*y*t^2 + y^2*t^2,
      s^3 + t^3

While we don’t build toric varieties from affine toric varieties, we still can access the “building pieces”:

    sage: patch = P1xP1.affine_patch(2)
    sage: patch
    2-d affine toric variety
    sage: patch.fan().rays()
    M(1, 0),
    M(0, 1)
    in 2-d lattice M
    sage: patch.embedding_morphism()
    Scheme morphism:
      From: 2-d affine toric variety
      To:   2-d toric variety covered by 4 affine patches
      Defn: Defined on coordinates by sending [x : s] to
            [x : s : 1 : 1]

The patch above was specifically chosen to coincide with our representation of the affine plane before, but you can get the other three patches as well. (While any cone of a fan will correspond to an affine toric variety, the main interest is usually in the generating fans as “the biggest” affine subvarieties, and these are precisely the patches that you can get from `affine_patch()`.)

All two-dimensional toric varieties are “quite nice” because any two-dimensional cone is generated by exactly two rays. From the point of view of the corresponding toric varieties, this means that they have at worst quotient singularities:

    sage: P1xP1.is_orbifold()
    True
    sage: P1xP1.is_smooth()
    True
    sage: TV = ToricVariety(NormalFan(diamond))
    sage: TV.fan().rays()
    N(-1,  1),
    N( 1,  1),
    N(-1, -1),
    N( 1, -1)
    in 2-d lattice N
    sage: TV.is_orbifold()
    True
    sage: TV.is_smooth()
    False

In higher dimensions worse things can happen:

    sage: TV3 = ToricVariety(NormalFan(lattice_polytope.cross_polytope(3)))
    sage: TV3.fan().rays()
    N(-1, -1,  1),
    N( 1, -1,  1),
    N(-1,  1,  1),
    N( 1,  1,  1),
    N(-1, -1, -1),
    N( 1, -1, -1),
    N(-1,  1, -1),
    N( 1,  1, -1)
    in 3-d lattice N
    sage: TV3.is_orbifold()
    False

Fortunately, we can perform a (partial) resolution:

    sage: TV3_res = TV3.resolve_to_orbifold()
    sage: TV3_res.is_orbifold()
    True
    sage: TV3_res.fan().ngenerating_cones()
    12
    sage: TV3.fan().ngenerating_cones()
    6

In this example we had to double the number of affine patches. The result is still singular:

    sage: TV3_res.is_smooth()
    False

You can resolve it further using `resolve()` method, but (at least for now) you will have to specify which rays should be inserted into the fan. See also `CPRFanoToricVariety()`, which can construct some other “nice partial resolutions.”

The intersection theory on toric varieties is very well understood, and there are explicit algorithms to compute many quantities of interest. The most important tools are the `cohomology ring` and the `Chow group`. For \(d\)-dimensional compact toric varieties with at most orbifold singularities, the rational cohomology ring \(H^*(X,\QQ)\) and the rational Chow ring \(A^*(X,\QQ) = A_{d-*}(X)\otimes \QQ\) are isomorphic except for a doubling in degree. More precisely, the Chow group has the same rank

> \[A_{d-k}(X) \otimes \QQ \simeq H^{2k}(X,\QQ)\]

and the intersection in of Chow cycles matches the cup product in cohomology.

In this case, you should work with the cohomology ring description because it is much faster. For example, here is a weighted projective space with a curve of \(\ZZ_3\)-orbifold singularities:

    sage: P4_11133 = toric_varieties.P4_11133()
    sage: P4_11133.is_smooth(), P4_11133.is_orbifold()
    (False, True)
    sage: cone = P4_11133.fan(3)[8]
    sage: cone.is_smooth(), cone.is_simplicial()
    (False, True)
    sage: HH = P4_11133.cohomology_ring();  HH
    Rational cohomology ring of a 4-d CPR-Fano toric variety covered by 5 affine patches
    sage: P4_11133.cohomology_basis()
    (([1],), ([z4],), ([z4^2],), ([z4^3],), ([z4^4],))

Every cone defines a torus orbit closure, and hence a (co)homology class:

    sage: HH.gens()
    ([3*z4], [3*z4], [z4], [z4], [z4])
    sage: map(HH, P4_11133.fan(1))
    [[3*z4], [3*z4], [z4], [z4], [z4]]
    sage: map(HH, P4_11133.fan(4) )
    [[9*z4^4], [9*z4^4], [9*z4^4], [9*z4^4], [9*z4^4]]
    sage: HH(cone)
    [3*z4^3]

We can compute intersection numbers by integrating top-dimensional cohomology classes:

    sage: D = P4_11133.divisor(0)
    sage: HH(D)
    [3*z4]
    sage: P4_11133.integrate( HH(D)^4 )
    9
    sage: P4_11133.integrate( HH(D) * HH(cone) )
    1

Although computationally less efficient, we can do the same computations with the rational Chow group:

    sage: AA = P4_11133.Chow_group(QQ)
    sage: map(AA, P4_11133.fan(1))  # long time (5s on sage.math, 2012)
    [( 0 | 0 | 0 | 3 | 0 ), ( 0 | 0 | 0 | 3 | 0 ), ( 0 | 0 | 0 | 1 | 0 ), ( 0 | 0 | 0 | 1 | 0 ), ( 0 | 0 | 0 | 1 | 0 )]
    sage: map(AA, P4_11133.fan(4))  # long time (5s on sage.math, 2012)
    [( 1 | 0 | 0 | 0 | 0 ), ( 1 | 0 | 0 | 0 | 0 ), ( 1 | 0 | 0 | 0 | 0 ), ( 1 | 0 | 0 | 0 | 0 ), ( 1 | 0 | 0 | 0 | 0 )]
    sage: AA(cone).intersection_with_divisor(D)  # long time (4s on sage.math, 2013)
    ( 1 | 0 | 0 | 0 | 0 )
    sage: AA(cone).intersection_with_divisor(D).count_points()  # long time
    1

The real advantage of the Chow group is that

> - it works just as well over \(\ZZ\), so torsion information is also easily available, and
> - its combinatorial description also works over worse-than-orbifold singularities. By contrast, the cohomology groups can become very complicated to compute in this case, and one usually only has a spectral sequence but no toric algorithm.

Below you will find detailed descriptions of available functions. If you are familiar with toric geometry, you will likely see that many important objects and operations are unavailable. However, this module is under active development and hopefully will improve in future releases of Sage. If there are some particular features that you would like to see implemented ASAP, please consider reporting them to the Sage Development Team or even implementing them on your own as a patch for inclusion!

`sage.schemes.toric.variety.``AffineToricVariety`(cone, *args, **kwds)

  Construct an affine toric variety.

  INPUT:

  - `cone` – `strictly convex rational polyhedral cone`.

  This cone will be used to construct a `rational polyhedral fan`, which will be passed to `ToricVariety()` with the rest of positional and keyword arguments.

  OUTPUT:

  - `toric variety`.

  Note

  The generating rays of the fan of this variety are guaranteed to be listed in the same order as the rays of the original cone.

  EXAMPLES:

  We will create the affine plane as an affine toric variety:

      sage: quadrant = Cone([(1,0), (0,1)])
      sage: A2 = AffineToricVariety(quadrant)
      sage: origin = A2(0,0)
      sage: origin
      [0 : 0]
      sage: parent(origin)
      Set of rational points of 2-d affine toric variety

  Only affine toric varieties have points whose (homogeneous) coordinates are all zero.

class `sage.schemes.toric.variety.``CohomologyClass`(cohomology_ring, representative)

  Bases: `sage.rings.quotient_ring_element.QuotientRingElement`

  An element of the `CohomologyRing`.

  Warning

  You should not create instances of this class manually. The generators of the cohomology ring as well as the cohomology classes associated to cones of the fan can be obtained from `ToricVariety_field.cohomology_ring()`.

  EXAMPLES:

      sage: P2 = toric_varieties.P2()
      sage: P2.cohomology_ring().gen(0)
      [z]
      sage: HH = P2.cohomology_ring()
      sage: HH.gen(0)
      [z]
      sage: cone = P2.fan(1)[0];  HH(cone)
      [z]

  `deg`()

    The degree of the cohomology class.

    OUTPUT:

    An integer \(d\) such that the cohomology class is in degree \(2d\). If the cohomology class is of mixed degree, the highest degree is returned.

    EXAMPLES:

        sage: P2 = toric_varieties.P2()
        sage: P2.cohomology_ring().gen(0).deg()
        1
        sage: P2.cohomology_ring().zero().deg()
        -1

  `exp`()

    Exponentiate `self`.

    Note

    The exponential \(\exp(x)\) of a rational number \(x\) is usually not rational. Therefore, the cohomology class must not have a constant (degree zero) part. The coefficients in the Taylor series of \(\exp\) are rational, so any cohomology class without constant term can be exponentiated.

    OUTPUT

    The cohomology class \(\exp(\) `self` \()\) if the constant part vanishes, otherwise a `ValueError` is raised.

    EXAMPLES:

        sage: P2 = toric_varieties.P2()
        sage: H_class = P2.cohomology_ring().gen(0)
        sage: H_class
        [z]
        sage: H_class.exp()
        [1/2*z^2 + z + 1]

  `part_of_degree`(d)

    Project the (mixed-degree) cohomology class to the given degree.

    \[\mathop{pr}\nolimits_d:~ H^\bullet(X_\Delta,\QQ) \to H^{2d}(X_\Delta,\QQ)\]

    INPUT:

    - An integer `d`

    OUTPUT:

    - The degree-`2d` part of the cohomology class.

    EXAMPLES:

        sage: P1xP1 = toric_varieties.P1xP1()
        sage: t = P1xP1.cohomology_ring().gen(0)
        sage: y = P1xP1.cohomology_ring().gen(2)
        sage: 3*t+4*t^2*y+y+t*y+t+1
        [t*y + 4*t + y + 1]
        sage: (3*t+4*t^2*y+y+t*y+t+1).part_of_degree(1)
        [4*t + y]

class `sage.schemes.toric.variety.``CohomologyRing`(variety)

  Bases: `sage.rings.quotient_ring.QuotientRing_generic`, `sage.structure.unique_representation.UniqueRepresentation`

  The (even) cohomology ring of a toric variety.

  Irregardles of the variety’s base ring, we always work with the variety over \(\CC\) and its topology.

  The cohomology is always the singular cohomology with \(\QQ\)-coefficients. Note, however, that the cohomology of smooth toric varieties is torsion-free, so there is no loss of information in that case.

  Currently, the toric variety must not be “too singular”. See `ToricVariety_field.cohomology_ring()` for a detailed description of which toric varieties are admissible. For such varieties the odd-dimensional cohomology groups vanish.

  Warning

  You should not create instances of this class manually. Use `ToricVariety_field.cohomology_ring()` to generate the cohomology ring.

  INPUT:

  - `variety` – a toric variety. Currently, the toric variety must be at least an orbifold. See `ToricVariety_field.cohomology_ring()` for a detailed description of which toric varieties are admissible.

  EXAMPLES:

      sage: P2 = toric_varieties.P2()
      sage: P2.cohomology_ring()
      Rational cohomology ring of a 2-d CPR-Fano toric variety covered by 3 affine patches

  This is equivalent to:

      sage: from sage.schemes.toric.variety import CohomologyRing
      sage: CohomologyRing(P2)
      Rational cohomology ring of a 2-d CPR-Fano toric variety covered by 3 affine patches

  `gen`(i)

    Return the generators of the cohomology ring.

    INPUT:

    - `i` – integer.

    OUTPUT:

    The `i`-th generator of the cohomology ring. If we denote the toric variety by `X`, then this generator is associated to the ray `X.fan().ray(i)`, which spans the one-cone `X.fan(1)[i]`

    EXAMPLES:

        sage: P2 = toric_varieties.P2()
        sage: P2.cohomology_ring().gen(2)
        [z]

  `gens`()

    Return the generators of the cohomology ring.

    OUTPUT:

    A tuple of generators, one for each toric divisor of the toric variety `X`. The order is the same as the ordering of the rays of the fan `X.fan().rays()`, which is also the same as the ordering of the one-cones in `X.fan(1)`

    EXAMPLES:

        sage: P2 = toric_varieties.P2()
        sage: P2.cohomology_ring().gens()
        ([z], [z], [z])

`sage.schemes.toric.variety.``ToricVariety`(fan, coordinate_names=None, names=None, coordinate_indices=None, base_ring=Rational Field, base_field=None)

  Construct a toric variety.

  INPUT:

  - `fan` – `rational polyhedral fan`;
  - `coordinate_names` – names of variables for the coordinate ring, see `normalize_names()` for acceptable formats. If not given, indexed variable names will be created automatically;
  - `names` – an alias of `coordinate_names` for internal use. You may specify either `names` or `coordinate_names`, but not both;
  - `coordinate_indices` – list of integers, indices for indexed variables. If not given, the index of each variable will coincide with the index of the corresponding ray of the fan;
  - `base_ring` – base ring of the toric variety (default: \(\QQ\)). Must be a field.
  - `base_field` – alias for `base_ring`. Takes precedence if both are specified.

  OUTPUT:

  - `toric variety`.

  EXAMPLES:

  We will create the product of two projective lines:

      sage: fan = FaceFan(lattice_polytope.cross_polytope(2))
      sage: fan.rays()
      M( 1,  0),
      M( 0,  1),
      M(-1,  0),
      M( 0, -1)
      in 2-d lattice M
      sage: P1xP1 = ToricVariety(fan)
      sage: P1xP1.gens()
      (z0, z1, z2, z3)

  Let’s create some points:

      sage: P1xP1(1,1,1,1)
      [1 : 1 : 1 : 1]
      sage: P1xP1(0,1,1,1)
      [0 : 1 : 1 : 1]
      sage: P1xP1(0,1,0,1)
      Traceback (most recent call last):
      ...
      TypeError: coordinates (0, 1, 0, 1)
      are in the exceptional set!

  We cannot set to zero both coordinates of the same projective line!

  Let’s change the names of the variables. We have to re-create our toric variety:

      sage: P1xP1 = ToricVariety(fan, "x s y t")
      sage: P1xP1.gens()
      (x, s, y, t)

  Now \((x, y)\) correspond to one line and \((s, t)\) to the other one.

      sage: P1xP1.inject_variables()
      Defining x, s, y, t
      sage: P1xP1.subscheme(x*s-y*t)
      Closed subscheme of 2-d toric variety
      covered by 4 affine patches defined by:
        x*s - y*t

  Here is a shorthand for defining the toric variety and homogeneous coordinates in one go:

      sage: P1xP1.<a,b,c,d> = ToricVariety(fan)
      sage: (a^2+b^2) * (c+d)
      a^2*c + b^2*c + a^2*d + b^2*d

class `sage.schemes.toric.variety.``ToricVariety_field`(fan, coordinate_names, coordinate_indices, base_field)

  Bases: `sage.misc.cachefunc.ClearCacheOnPickle`, `sage.schemes.generic.ambient_space.AmbientSpace`

  Construct a toric variety associated to a rational polyhedral fan.

  Warning

  This class does not perform any checks of correctness of input. Use `ToricVariety()` and `AffineToricVariety()` to construct toric varieties.

  INPUT:

  - `fan` – `rational polyhedral fan`;
  - `coordinate_names` – names of variables, see `normalize_names()` for acceptable formats. If `None`, indexed variable names will be created automatically;
  - `coordinate_indices` – list of integers, indices for indexed variables. If `None`, the index of each variable will coincide with the index of the corresponding ray of the fan;
  - `base_field` – base field of the toric variety.

  OUTPUT:

  - `toric variety`.

  TESTS:

      sage: fan = FaceFan(lattice_polytope.cross_polytope(2))
      sage: P1xP1 = ToricVariety(fan)

  `Aut_dimension`()

    Return the dimension of the automorphism group

    There are three kinds of symmetries of toric varieties:

    > - Toric automorphisms (rescaling of homogeneous coordinates)
    > - Demazure roots. These are translations \(x_i \to x_i + \epsilon x^m\) of a homogeneous coordinate \(x_i\) by a monomial \(x^m\) of the same homogeneous degree.
    > - Symmetries of the fan. These yield discrete subgroups.

    OUTPUT:

    An integer. The dimension of the automorphism group. Equals the dimension of the \(M\)-lattice plus the number of Demazure roots.

    EXAMPLES:

        sage: P2 = toric_varieties.P2()
        sage: P2.Aut_dimension()
        8

    TESTS:

        sage: toric_varieties.A1().Aut_dimension()
        Traceback (most recent call last):
        ...
        NotImplementedError: Aut_dimension() is only implemented for complete toric varieties.

  `Chern_character`(deg=None)

    Return the Chern character (of the tangent bundle) of the toric variety.

    INPUT:

    - `deg` – integer (optional). The degree of the Chern character.

    OUTPUT:

    - If the degree is specified, the degree-`deg` part of the Chern character.
    - If no degree is specified, the total Chern character.

    REFERENCES:

    > http://en.wikipedia.org/wiki/Chern_character#The_Chern_character

    EXAMPLES:

        sage: dP6 = toric_varieties.dP6()
        sage: dP6.Chern_character()
        [3*w^2 + y + 2*v + 2*z + w + 2]
        sage: dP6.ch()
        [3*w^2 + y + 2*v + 2*z + w + 2]
        sage: dP6.ch(1) == dP6.c(1)
        True

  `Chern_class`(deg=None)

    Return Chern classes of the (tangent bundle of the) toric variety.

    INPUT:

    - `deg` – integer (optional). The degree of the Chern class.

    OUTPUT:

    - If the degree is specified, the `deg`-th Chern class.
    - If no degree is specified, the total Chern class.

    REFERENCES:

    > http://en.wikipedia.org/wiki/Chern_class

    EXAMPLES:

        sage: X = toric_varieties.dP6()
        sage: X.Chern_class()
        [-6*w^2 + y + 2*v + 2*z + w + 1]
        sage: X.c()
        [-6*w^2 + y + 2*v + 2*z + w + 1]
        sage: X.c(1)
        [y + 2*v + 2*z + w]
        sage: X.c(2)
        [-6*w^2]
        sage: X.integrate( X.c(2) )
        6
        sage: X.integrate( X.c(2) ) == X.Euler_number()
        True

  `Chow_group`(base_ring=Integer Ring)

    Return the toric Chow group.

    INPUT:

    - `base_ring` – either `ZZ` (default) or `QQ`. The coefficient ring of the Chow group.

    OUTPUT:

    A `sage.schemes.toric.chow_group.ChowGroup_class`

    EXAMPLES:

        sage: A = toric_varieties.P2().Chow_group(); A
        Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches
        sage: A.gens()
        (( 1 | 0 | 0 ), ( 0 | 1 | 0 ), ( 0 | 0 | 1 ))

  `Demazure_roots`()

    Return the Demazure roots.

    OUTPUT:

    The roots as points of the \(M\)-lattice.

    REFERENCES:

    [Demazure] | M. Demazure Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. Ecole Norm. Sup. 1970, 3, 507–588.

    [Bazhov] | Ivan Bazhov: On orbits of the automorphism group on a complete toric variety. Arxiv 1110.4275, doi:10.1007/s13366-011-0084-0.

    EXAMPLE:

        sage: P2 = toric_varieties.P2()
        sage: P2.Demazure_roots()
        (M(-1, 0), M(-1, 1), M(0, -1), M(0, 1), M(1, -1), M(1, 0))

    Here are the remaining three examples listed in [Bazhov], Example 2.1 and 2.3:

        sage: s = 3
        sage: cones = [(0,1),(1,2),(2,3),(3,0)]
        sage: Hs = ToricVariety(Fan(rays=[(1,0),(0,-1),(-1,s),(0,1)], cones=cones))
        sage: Hs.Demazure_roots()
        (M(-1, 0), M(1, 0), M(0, 1), M(1, 1), M(2, 1), M(3, 1))

        sage: P11s = ToricVariety(Fan(rays=[(1,0),(0,-1),(-1,s)], cones=[(0,1),(1,2),(2,0)]))
        sage: P11s.Demazure_roots()
        (M(-1, 0), M(1, 0), M(0, 1), M(1, 1), M(2, 1), M(3, 1))
        sage: P11s.Demazure_roots() == Hs.Demazure_roots()
        True

        sage: Bs = ToricVariety(Fan(rays=[(s,1),(s,-1),(-s,-1),(-s,1)], cones=cones))
        sage: Bs.Demazure_roots()
        ()

    TESTS:

        sage: toric_varieties.A1().Demazure_roots()
        Traceback (most recent call last):
        ...
        NotImplementedError: Demazure_roots() is only implemented for complete toric varieties.

  `Euler_number`()

    Return the topological Euler number of the toric variety.

    Sometimes, this is also called the Euler characteristic. `chi()` is a synonym for `Euler_number()`.

    REFERENCES:

    > http://en.wikipedia.org/wiki/Euler_characteristic

    EXAMPLES:

        sage: P1xP1 = toric_varieties.P1xP1()
        sage: P1xP1.Euler_number()
        4
        sage: P1xP1.chi()
        4

  `K`()

    Returns the canonical divisor of the toric variety.

    EXAMPLES:

    Lets test that the del Pezzo surface \(dP_6\) has degree 6, as its name implies:

        sage: dP6 = toric_varieties.dP6()
        sage: HH = dP6.cohomology_ring()
        sage: dP6.K()
        -V(x) - V(u) - V(y) - V(v) - V(z) - V(w)
        sage: dP6.integrate( HH(dP6.K())^2 )
        6

  `Kaehler_cone`()

    Return the closure of the Kähler cone of `self`.

    OUTPUT:

    - `cone`.

    Note

    This cone sits in the rational divisor class group of `self` and the choice of coordinates agrees with `rational_class_group()`.

    EXAMPLES:

        sage: P1xP1 = toric_varieties.P1xP1()
        sage: Kc = P1xP1.Kaehler_cone()
        sage: Kc
        2-d cone in 2-d lattice
        sage: Kc.rays()
        Divisor class [0, 1],
        Divisor class [1, 0]
        in Basis lattice of The toric rational divisor class group
        of a 2-d CPR-Fano toric variety covered by 4 affine patches
        sage: [ divisor_class.lift() for divisor_class in Kc.rays() ]
        [V(x), V(s)]
        sage: Kc.lattice()
        Basis lattice of The toric rational divisor class group of a
        2-d CPR-Fano toric variety covered by 4 affine patches

  `Mori_cone`()

    Returns the Mori cone of `self`.

    OUTPUT:

    - `cone`.

    Note

    - The Mori cone is dual to the Kähler cone.
    - We think of the Mori cone as living inside the row span of the Gale transform matrix (computed by `self.fan().Gale_transform()`).
    - The points in the Mori cone are the effective curves in the variety.
    - The `i`-th entry in each Mori vector is the intersection number of the curve corresponding to the generator of the `i`-th ray of the fan with the corresponding divisor class. The very last entry is associated to the orgin of the fan lattice.
    - The Mori vectors are also known as the gauged linear sigma model charge vectors.

    EXAMPLES:

        sage: P4_11169 = toric_varieties.P4_11169_resolved()
        sage: P4_11169.Mori_cone()
        2-d cone in 7-d lattice
        sage: P4_11169.Mori_cone().rays()
        (3, 2, 0, 0, 0,  1, -6),
        (0, 0, 1, 1, 1, -3,  0)
        in Ambient free module of rank 7
        over the principal ideal domain Integer Ring

  `Spec`(cone=None, names=None)

    Return the spectrum associated to the dual cone.

    Let \(\sigma \in N_\RR\) be a cone and \(\sigma^\vee \cap M\) the associated semigroup of lattice points in the dual cone. Then

    \[S = \CC[\sigma^\vee \cap M]\]

    is a \(\CC\)-algebra. It is spanned over \(\CC\) by the points of \(\sigma \cap N\), addition is formal linear combination of lattice points, and multiplication of lattice points is the semigroup law (that is, addition of lattice points). The \(\CC\)-algebra \(S\) then defines a scheme \(\mathop{Spec}(S)\).

    For example, if \(\sigma=\{(x,y)|x\geq 0,y\geq 0\}\) is the first quadrant then \(S\) is the polynomial ring in two variables. The associated scheme is \(\mathop{Spec}(S) = \CC^2\).

    The same construction works over any base field, this introduction only used \(\CC\) for simplicity.

    INPUT:

    - `cone` – a `Cone`. Can be omitted for an affine toric variety, in which case the (unique) generating cone is used.
    - `names` – (optional). Names of variables for the semigroup ring, see `normalize_names()` for acceptable formats. If not given, indexed variable names will be created automatically.

    Output:

    The spectrum of the semigroup ring \(\CC[\sigma^\vee \cap M]\).

    EXAMPLES:

        sage: quadrant = Cone([(1,0),(0,1)])
        sage: AffineToricVariety(quadrant).Spec()
        Spectrum of Multivariate Polynomial Ring in z0, z1 over Rational Field

    A more interesting example:

        sage: A2Z2 = Cone([(0,1),(2,1)])
        sage: AffineToricVariety(A2Z2).Spec(names='u,v,t')
        Spectrum of Quotient of Multivariate Polynomial Ring
        in u, v, t over Rational Field by the ideal (-u*v + t^2)

  `Stanley_Reisner_ideal`()

    Return the Stanley-Reisner ideal.

    OUTPUT:

    - The Stanley-Reisner ideal in the polynomial ring over \(\QQ\) generated by the homogeneous coordinates.

    EXAMPLES:

        sage: fan = Fan([[0,1,3],[3,4],[2,0],[1,2,4]], [(-3, -2, 1), (0, 0, 1), (3, -2, 1), (-1, -1, 1), (1, -1, 1)])
        sage: X = ToricVariety(fan, coordinate_names='A B C D E', base_field=GF(5))
        sage: SR = X.Stanley_Reisner_ideal(); SR
        Ideal (A*E, C*D, A*B*C, B*D*E) of Multivariate Polynomial Ring in A, B, C, D, E over Rational Field

  `Td`(deg=None)

    Return the Todd class (of the tangent bundle) of the toric variety.

    INPUT:

    - `deg` – integer (optional). The desired degree part.

    OUTPUT:

    - If the degree is specified, the degree-`deg` part of the Todd class.
    - If no degree is specified, the total Todd class.

    REFERENCES:

    > http://en.wikipedia.org/wiki/Todd_class

    EXAMPLES:

        sage: dP6 = toric_varieties.dP6()
        sage: dP6.Todd_class()
        [-w^2 + 1/2*y + v + z + 1/2*w + 1]
        sage: dP6.Td()
        [-w^2 + 1/2*y + v + z + 1/2*w + 1]
        sage: dP6.integrate( dP6.Td() )
        1

  `Todd_class`(deg=None)

    Return the Todd class (of the tangent bundle) of the toric variety.

    INPUT:

    - `deg` – integer (optional). The desired degree part.

    OUTPUT:

    - If the degree is specified, the degree-`deg` part of the Todd class.
    - If no degree is specified, the total Todd class.

    REFERENCES:

    > http://en.wikipedia.org/wiki/Todd_class

    EXAMPLES:

        sage: dP6 = toric_varieties.dP6()
        sage: dP6.Todd_class()
        [-w^2 + 1/2*y + v + z + 1/2*w + 1]
        sage: dP6.Td()
        [-w^2 + 1/2*y + v + z + 1/2*w + 1]
        sage: dP6.integrate( dP6.Td() )
        1

  `affine_algebraic_patch`(cone=None, names=None)

    Return the patch corresponding to `cone` as an affine algebraic subvariety.

    INPUT:

    - `cone` – a `Cone` \(\sigma\) of the fan. It can be omitted for an affine toric variety, in which case the single generating cone is used.

    OUTPUT:

    A `affine algebraic subscheme` corresponding to the patch \(\mathop{Spec}(\sigma^\vee \cap M)\) associated to the cone \(\sigma\).

    See also `affine_patch()`, which expresses the patches as subvarieties of affine toric varieties instead.

    EXAMPLES:

        sage: cone = Cone([(0,1),(2,1)])
        sage: A2Z2 = AffineToricVariety(cone)
        sage: A2Z2.affine_algebraic_patch()
        Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
          -z0*z1 + z2^2
        sage: A2Z2.affine_algebraic_patch(Cone([(0,1)]), names='x, y, t')
        Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
          1

  `affine_patch`(i)

    Return the `i`-th affine patch of `self`.

    INPUT:

    - `i` – integer, index of a generating cone of the fan of `self`.

    OUTPUT:

    - affine `toric variety` corresponding to the `i`-th generating cone of the fan of `self`.

    The result is cached, so the `i`-th patch is always the same object in memory.

    See also `affine_algebraic_patch()`, which expresses the patches as subvarieties of affine space instead.

    EXAMPLES:

        sage: fan = FaceFan(lattice_polytope.cross_polytope(2))
        sage: P1xP1 = ToricVariety(fan, "x s y t")
        sage: patch0 = P1xP1.affine_patch(0)
        sage: patch0
        2-d affine toric variety
        sage: patch0.embedding_morphism()
        Scheme morphism:
          From: 2-d affine toric variety
          To:   2-d toric variety covered by 4 affine patches
          Defn: Defined on coordinates by sending [x : t] to
                [x : 1 : 1 : t]
        sage: patch1 = P1xP1.affine_patch(1)
        sage: patch1.embedding_morphism()
        Scheme morphism:
          From: 2-d affine toric variety
          To:   2-d toric variety covered by 4 affine patches
          Defn: Defined on coordinates by sending [y : t] to
                [1 : 1 : y : t]
        sage: patch1 is P1xP1.affine_patch(1)
        True

  `c`(deg=None)

    Return Chern classes of the (tangent bundle of the) toric variety.

    INPUT:

    - `deg` – integer (optional). The degree of the Chern class.

    OUTPUT:

    - If the degree is specified, the `deg`-th Chern class.
    - If no degree is specified, the total Chern class.

    REFERENCES:

    > http://en.wikipedia.org/wiki/Chern_class

    EXAMPLES:

        sage: X = toric_varieties.dP6()
        sage: X.Chern_class()
        [-6*w^2 + y + 2*v + 2*z + w + 1]
        sage: X.c()
        [-6*w^2 + y + 2*v + 2*z + w + 1]
        sage: X.c(1)
        [y + 2*v + 2*z + w]
        sage: X.c(2)
        [-6*w^2]
        sage: X.integrate( X.c(2) )
        6
        sage: X.integrate( X.c(2) ) == X.Euler_number()
        True

  `cartesian_product`(other, coordinate_names=None, coordinate_indices=None)

    Return the Cartesian product of `self` with `other`.

    INPUT:

    - `other` – a `toric variety`;
    - `coordinate_names` – names of variables for the coordinate ring, see `normalize_names()` for acceptable formats. If not given, indexed variable names will be created automatically;
    - `coordinate_indices` – list of integers, indices for indexed variables. If not given, the index of each variable will coincide with the index of the corresponding ray of the fan.

    OUTPUT:

    – a `toric variety`.

    EXAMPLES:

        sage: P1 = ToricVariety(Fan([Cone([(1,)]), Cone([(-1,)])]))
        sage: P1xP1 = P1.cartesian_product(P1); P1xP1
        2-d toric variety covered by 4 affine patches
        sage: P1xP1.fan().rays()
        N+N(-1,  0),
        N+N( 1,  0),
        N+N( 0, -1),
        N+N( 0,  1)
        in 2-d lattice N+N

  `ch`(deg=None)

    Return the Chern character (of the tangent bundle) of the toric variety.

    INPUT:

    - `deg` – integer (optional). The degree of the Chern character.

    OUTPUT:

    - If the degree is specified, the degree-`deg` part of the Chern character.
    - If no degree is specified, the total Chern character.

    REFERENCES:

    > http://en.wikipedia.org/wiki/Chern_character#The_Chern_character

    EXAMPLES:

        sage: dP6 = toric_varieties.dP6()
        sage: dP6.Chern_character()
        [3*w^2 + y + 2*v + 2*z + w + 2]
        sage: dP6.ch()
        [3*w^2 + y + 2*v + 2*z + w + 2]
        sage: dP6.ch(1) == dP6.c(1)
        True

  `change_ring`(F)

    Return a toric variety over `F` and otherwise the same as `self`.

    INPUT:

    - `F` – field.

    OUTPUT:

    - `toric variety` over `F`.

    Note

    There is no need to have any relation between `F` and the base field of `self`. If you do want to have such a relation, use `base_extend()` instead.

    EXAMPLES:

        sage: P1xA1 = toric_varieties.P1xA1()
        sage: P1xA1.base_ring()
        Rational Field
        sage: P1xA1_RR = P1xA1.change_ring(RR)
        sage: P1xA1_RR.base_ring()
        Real Field with 53 bits of precision
        sage: P1xA1_QQ = P1xA1_RR.change_ring(QQ)
        sage: P1xA1_QQ.base_ring()
        Rational Field
        sage: P1xA1_RR.base_extend(QQ)
        Traceback (most recent call last):
        ...
        ValueError: no natural map from the base ring
        (=Real Field with 53 bits of precision)
        to R (=Rational Field)!
        sage: R = PolynomialRing(QQ, 2, 'a')
        sage: P1xA1.change_ring(R)
        Traceback (most recent call last):
        ...
        TypeError: need a field to construct a toric variety!
         Got Multivariate Polynomial Ring in a0, a1 over Rational Field

  `chi`()

    Return the topological Euler number of the toric variety.

    Sometimes, this is also called the Euler characteristic. `chi()` is a synonym for `Euler_number()`.

    REFERENCES:

    > http://en.wikipedia.org/wiki/Euler_characteristic

    EXAMPLES:

        sage: P1xP1 = toric_varieties.P1xP1()
        sage: P1xP1.Euler_number()
        4
        sage: P1xP1.chi()
        4

  `cohomology_basis`(d=None)

    Return a basis for the cohomology of the toric variety.

    INPUT:

    - `d` (optional) – integer.

    OUTPUT:

    - Without the optional argument, a list whose d-th entry is a basis for \(H^{2d}(X,\QQ)\)
    - If the argument is an integer `d`, returns basis for \(H^{2d}(X,\QQ)\)

    EXAMPLES:

        sage: X = toric_varieties.dP8()
        sage: X.cohomology_basis()
        (([1],), ([z], [y]), ([y*z],))
        sage: X.cohomology_basis(1)
        ([z], [y])
        sage: X.cohomology_basis(dimension(X))[0] == X.volume_class()
        True

  `cohomology_ring`()

    Return the cohomology ring of the toric variety.

    OUTPUT:

    - If the toric variety is is over \(\CC\) and has at most finite orbifold singularities: \(H^\bullet(X,\QQ)\) as a polynomial quotient ring.
    - Other cases are not handled yet.

    Note

    - Toric varieties over any field of characteristic 0 are treated as if they were varieties over \(\CC\).
    - The integral cohomology of smooth toric varieties is torsion-free, so in this case there is no loss of information when going to rational coefficients.
    - `self.cohomology_ring().gen(i)` is the divisor class corresponding to the `i`-th ray of the fan.

    EXAMPLES:

        sage: X = toric_varieties.dP6()
        sage: X.cohomology_ring()
        Rational cohomology ring of a 2-d CPR-Fano toric variety covered by 6 affine patches
        sage: X.cohomology_ring().defining_ideal()
        Ideal (-u - y + z + w, x - y - v + w, x*y, x*v, x*z, u*v, u*z, u*w, y*z, y*w, v*w) of Multivariate Polynomial Ring in x, u, y, v, z, w over Rational Field
        sage: X.cohomology_ring().defining_ideal().ring()
        Multivariate Polynomial Ring in x, u, y, v, z, w over Rational Field
        sage: X.variable_names()
        ('x', 'u', 'y', 'v', 'z', 'w')
        sage: X.cohomology_ring().gens()
        ([y + v - w], [-y + z + w], [y], [v], [z], [w])

    TESTS:

    The cohomology ring is a circular reference that is potentially troublesome on unpickling, see trac ticket #15050 and trac ticket #15149

        sage: variety = toric_varieties.P(1)
        sage: a = [variety.cohomology_ring(), variety.cohomology_basis(), variety.volume_class()]
        sage: b = [variety.Todd_class(), variety.Chern_class(), variety.Chern_character(), variety.Kaehler_cone(), variety.Mori_cone()]
        sage: loads(dumps(variety)) == variety
        True

  `coordinate_ring`()

    Return the coordinate ring of `self`.

    For toric varieties this is the homogeneous coordinate ring (a.k.a. Cox’s ring and total ring).

    OUTPUT:

    - polynomial ring.

    EXAMPLES:

        sage: P1xP1 = toric_varieties.P1xP1()
        sage: P1xP1.coordinate_ring()
        Multivariate Polynomial Ring in s, t, x, y
        over Rational Field

    TESTS:

        sage: R = toric_varieties.A1().coordinate_ring();  R
        Multivariate Polynomial Ring in z over Rational Field
        sage: type(R)
        <type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular'>

  `count_points`()

    Return the number of points of `self`.

    This is an alias for `point_set().cardinality()`, see `cardinality()` for details.

    EXAMPLES:

        sage: o = lattice_polytope.cross_polytope(3)
        sage: V = ToricVariety(FaceFan(o))
        sage: V2 = V.change_ring(GF(2))
        sage: V2.point_set().cardinality()
        27
        sage: V2.count_points()
        27

  `dimension_singularities`()

    Return the dimension of the singular set.

    OUTPUT:

    Integer. The dimension of the singular set of the toric variety. Often the singular set is a reducible subvariety, and this method will return the dimension of the largest-dimensional component.

    Returns -1 if the toric variety is smooth.

    EXAMPLES:

        sage: toric_varieties.P4_11169().dimension_singularities()
        1
        sage: toric_varieties.Conifold().dimension_singularities()
        0
        sage: toric_varieties.P2().dimension_singularities()
        -1

  `divisor`(arg, base_ring=None, check=True, reduce=True)

    Return a divisor.

    INPUT:

    The arguments are the same as in `sage.schemes.toric.divisor.ToricDivisor()`, with the exception of defining a divisor with a single integer: this method considers it to be the index of a ray of the `fan()` of `self`.

    OUTPUT:

    - A `sage.schemes.toric.divisor.ToricDivisor_generic`

    EXAMPLES:

        sage: dP6 = toric_varieties.dP6()
        sage: dP6.coordinate_ring()
        Multivariate Polynomial Ring in x, u, y, v, z, w
        over Rational Field
        sage: dP6.divisor(range(6))
        V(u) + 2*V(y) + 3*V(v) + 4*V(z) + 5*V(w)
        sage: dP6.inject_variables()
        Defining x, u, y, v, z, w
        sage: dP6.divisor(x*u^3)
        V(x) + 3*V(u)

    You can also construct divisors based on ray indices:

        sage: dP6.divisor(0)
        V(x)
        sage: for i in range(0,dP6.fan().nrays()):
        ...       print dP6.divisor(i),
        ...       print ': generated by ray',
        ...       dP6.fan().ray(i)
        V(x) : generated by ray N(0, 1)
        V(u) : generated by ray N(-1, 0)
        V(y) : generated by ray N(-1, -1)
        V(v) : generated by ray N(0, -1)
        V(z) : generated by ray N(1, 0)
        V(w) : generated by ray N(1, 1)

    TESTS:

    We check that the issue trac ticket #12812 is resolved:

        sage: sum(dP6.divisor(i) for i in range(3))
        V(x) + V(u) + V(y)

  `divisor_group`(base_ring=Integer Ring)

    Return the group of Weil divisors.

    INPUT:

    - `base_ring` – the coefficient ring, usually `ZZ` (default) or `QQ`.

    OUTPUT:

    The (free abelian) group of Cartier divisors, that is, formal linear combinations of polynomial equations over the coefficient ring `base_ring`.

    These need not be toric (=defined by monomials), but allow general polynomials. The output will be an instance of `sage.schemes.generic.divisor_group.DivisorGroup_generic`.

    Warning

    You almost certainly want the group of toric divisors, see `toric_divisor_group()`. The toric divisor group is generated by the rays of the fan. The general divisor group has no toric functionality implemented.

    EXAMPLES:

        sage: dP6 = toric_varieties.dP6()
        sage: Div = dP6.divisor_group(); Div
        Group of ZZ-Divisors on 2-d CPR-Fano toric variety
        covered by 6 affine patches
        sage: Div(x)
        V(x)

  `embedding_morphism`()

    Return the default embedding morphism of `self`.

    Such a morphism is always defined for an affine patch of a toric variety (which is also a toric varieties itself).

    OUTPUT:

    - `scheme morphism` if the default embedding morphism was defined for `self`, otherwise a `ValueError` exception is raised.

    EXAMPLES:

        sage: fan = FaceFan(lattice_polytope.cross_polytope(2))
        sage: P1xP1 = ToricVariety(fan, "x s y t")
        sage: P1xP1.embedding_morphism()
        Traceback (most recent call last):
        ...
        ValueError: no default embedding was
        defined for this toric variety!
        sage: patch = P1xP1.affine_patch(0)
        sage: patch
        2-d affine toric variety
        sage: patch.embedding_morphism()
        Scheme morphism:
          From: 2-d affine toric variety
          To:   2-d toric variety covered by 4 affine patches
          Defn: Defined on coordinates by sending [x : t] to
                [x : 1 : 1 : t]

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

    Return the underlying fan of `self` or its cones.

    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: fan = FaceFan(lattice_polytope.cross_polytope(2))
        sage: P1xP1 = ToricVariety(fan)
        sage: P1xP1.fan()
        Rational polyhedral fan in 2-d lattice M
        sage: P1xP1.fan() is fan
        True
        sage: P1xP1.fan(1)[0]
        1-d cone of Rational polyhedral fan in 2-d lattice M

  `inject_coefficients`(scope=None, verbose=True)

    Inject generators of the base field of `self` into `scope`.

    This function is useful if the base field is the field of rational functions.

    INPUT:

    - `scope` – namespace (default: global, not just the scope from which this function was called);
    - `verbose` – if `True` (default), names of injected generators will be printed.

    OUTPUT:

    - none.

    EXAMPLES:

        sage: fan = FaceFan(lattice_polytope.cross_polytope(2))
        sage: F = QQ["a, b"].fraction_field()
        sage: P1xP1 = ToricVariety(fan, base_field=F)
        sage: P1xP1.inject_coefficients()
        Defining a, b

    We check that we can use names `a` and `b`, Trac #10498 is fixed:

        sage: a + b
        a + b
        sage: a + b in P1xP1.coordinate_ring()
        True

  `integrate`(cohomology_class)

    Integrate a cohomology class over the toric variety.

    INPUT:

    - `cohomology_class` – A cohomology class given as a polynomial in `self.cohomology_ring()`

    OUTPUT:

    The integral of the cohomology class over the variety. The volume normalization is given by `volume_class()`, that is, `self.integrate(self.volume_class())` is always one (if the volume class exists).

    EXAMPLES:

        sage: dP6 = toric_varieties.dP6()
        sage: HH = dP6.cohomology_ring()
        sage: D = [ HH(c) for c in dP6.fan(dim=1) ]
        sage: matrix([ [ D[i]*D[j] for i in range(0,6) ] for j in range(0,6) ])
        [ [w^2] [-w^2]    [0]    [0]    [0] [-w^2]]
        [[-w^2]  [w^2] [-w^2]    [0]    [0]    [0]]
        [   [0] [-w^2]  [w^2] [-w^2]    [0]    [0]]
        [   [0]    [0] [-w^2]  [w^2] [-w^2]    [0]]
        [   [0]    [0]    [0] [-w^2]  [w^2] [-w^2]]
        [[-w^2]    [0]    [0]    [0] [-w^2]  [w^2]]
        sage: matrix([ [ dP6.integrate(D[i]*D[j]) for i in range(0,6) ] for j in range(0,6) ])
        [-1  1  0  0  0  1]
        [ 1 -1  1  0  0  0]
        [ 0  1 -1  1  0  0]
        [ 0  0  1 -1  1  0]
        [ 0  0  0  1 -1  1]
        [ 1  0  0  0  1 -1]

    If the toric variety is an orbifold, the intersection numbers are usually fractional:

        sage: P2_123 = toric_varieties.P2_123()
        sage: HH = P2_123.cohomology_ring()
        sage: D = [ HH(c) for c in P2_123.fan(dim=1) ]
        sage: matrix([ [ P2_123.integrate(D[i]*D[j]) for i in range(0,3) ] for j in range(0,3) ])
        [2/3   1 1/3]
        [  1 3/2 1/2]
        [1/3 1/2 1/6]
        sage: A = P2_123.Chow_group(QQ)
        sage: matrix([ [ A(P2_123.divisor(i))
        ...              .intersection_with_divisor(P2_123.divisor(j))
        ...              .count_points() for i in range(0,3) ] for j in range(0,3) ])
        [2/3   1 1/3]
        [  1 3/2 1/2]
        [1/3 1/2 1/6]

  `is_affine`()

    Check if `self` is an affine toric variety.

    An affine toric variety is a toric variety whose fan is the face lattice of a single cone. See also `AffineToricVariety()`.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: toric_varieties.A2().is_affine()
        True
        sage: toric_varieties.P1xA1().is_affine()
        False

  `is_complete`()

    Check if `self` is complete.

    OUTPUT:

    - `True` if `self` is complete and `False` otherwise.

    EXAMPLES:

        sage: P1xP1 = toric_varieties.P1xP1()
        sage: P1xP1.is_complete()
        True
        sage: P1xP1.affine_patch(0).is_complete()
        False

  `is_homogeneous`(polynomial)

    Check if `polynomial` is homogeneous.

    The coordinate ring of a toric variety is multigraded by relations between generating rays of the underlying fan.

    INPUT:

    - `polynomial` – polynomial in the coordinate ring of `self` or its quotient.

    OUTPUT:

    - `True` if `polynomial` is homogeneous and `False` otherwise.

    EXAMPLES:

    We will use the product of two projective lines with coordinates \((x, y)\) for one and \((s, t)\) for the other:

        sage: P1xP1.<x,y,s,t> = toric_varieties.P1xP1()
        sage: P1xP1.is_homogeneous(x - y)
        True
        sage: P1xP1.is_homogeneous(x*s + y*t)
        True
        sage: P1xP1.is_homogeneous(x - t)
        False
        sage: P1xP1.is_homogeneous(1)
        True

    Note that by homogeneous, we mean well-defined with respect to the homogeneous rescalings of `self`. So a polynomial that you would usually not call homogeneous can be homogeneous if there are no homogeneous rescalings, for example:

        sage: A1.<z> = toric_varieties.A1()
        sage: A1.is_homogeneous(z^3+z^7)
        True

    Finally, the degree group is really the Chow group \(A_{d-1}(X)\) and can contain torsion. For example, take \(\CC^2/\ZZ_2\). Here, the Chow group is \(A_{d-1}(\CC^2/\ZZ_2) = \ZZ_2\) and distinguishes even-degree homogeneous polynomials from odd-degree homogeneous polynomials:

        sage: A2_Z2.<x,y> = toric_varieties.A2_Z2()
        sage: A2_Z2.is_homogeneous(x+y+x^3+y^5+x^3*y^4)
        True
        sage: A2_Z2.is_homogeneous(x^2+x*y+y^4+(x*y)^5+x^4*y^4)
        True
        sage: A2_Z2.is_homogeneous(x+y^2)
        False

  `is_isomorphic`(another)

    Check if `self` is isomorphic to `another`.

    INPUT:

    - `another` - `toric variety`.

    OUTPUT:

    - `True` if `self` and `another` are isomorphic, `False` otherwise.

    EXAMPLES:

        sage: TV1 = toric_varieties.P1xA1()
        sage: TV2 = toric_varieties.P1xP1()

    Only the most trivial case is implemented so far:

        sage: TV1.is_isomorphic(TV1)
        True
        sage: TV1.is_isomorphic(TV2)
        Traceback (most recent call last):
        ...
        NotImplementedError:
        isomorphism check is not yet implemented!

  `is_orbifold`()

    Check if `self` has only quotient singularities.

    A toric variety with at most orbifold singularities (in this sense) is often called a simplicial toric variety. In this package, we generally try to avoid this term since it mixes up differential geometry and cone terminology.

    OUTPUT:

    - `True` if `self` has at most quotient singularities by finite groups, `False` otherwise.

    EXAMPLES:

        sage: fan1 = FaceFan(lattice_polytope.cross_polytope(2))
        sage: P1xP1 = ToricVariety(fan1)
        sage: P1xP1.is_orbifold()
        True
        sage: fan2 = NormalFan(lattice_polytope.cross_polytope(3))
        sage: TV = ToricVariety(fan2)
        sage: TV.is_orbifold()
        False

  `is_smooth`()

    Check if `self` is smooth.

    OUTPUT:

    - `True` if `self` is smooth and `False` otherwise.

    EXAMPLES:

        sage: fan1 = FaceFan(lattice_polytope.cross_polytope(2))
        sage: P1xP1 = ToricVariety(fan1)
        sage: P1xP1.is_smooth()
        True
        sage: fan2 = NormalFan(lattice_polytope.cross_polytope(2))
        sage: TV = ToricVariety(fan2)
        sage: TV.is_smooth()
        False

  `linear_equivalence_ideal`()

    Return the ideal generated by linear relations

    OUTPUT:

    - The ideal generated by the linear relations of the rays in the polynomial ring over \(\QQ\) generated by the homogeneous coordinates.

    EXAMPLES:

        sage: fan = Fan([[0,1,3],[3,4],[2,0],[1,2,4]], [(-3, -2, 1), (0, 0, 1), (3, -2, 1), (-1, -1, 1), (1, -1, 1)])
        sage: X = ToricVariety(fan, coordinate_names='A B C D E', base_field=GF(5))
        sage: lin = X.linear_equivalence_ideal(); lin
        Ideal (-3*A + 3*C - D + E, -2*A - 2*C - D - E, A + B + C + D + E) of Multivariate Polynomial Ring in A, B, C, D, E over Rational Field

  `orbit_closure`(cone)

    Return the orbit closure of `cone`.

    The cones \(\sigma\) of a fan \(\Sigma\) are in one-to-one correspondence with the torus orbits \(O(\sigma)\) of the corresponding toric variety \(X_\Sigma\). Each orbit is isomorphic to a lower dimensional torus (of dimension equal to the codimension of \(\sigma\)). Just like the toric variety \(X_\Sigma\) itself, these orbits are (partially) compactified by lower-dimensional orbits. In particular, one can define the closure \(V(\sigma)\) of the torus orbit \(O(\sigma)\) in the ambient toric variety \(X_\Sigma\), which is again a toric variety.

    See Proposition 3.2.7 of [CLS] for more details.

    INPUT:

    - `cone` – a `cone` of the fan.

    OUTPUT:

    - a torus orbit closure associated to `cone` as a `toric variety`.

    EXAMPLES:

        sage: P1xP1 = toric_varieties.P1xP1()
        sage: H = P1xP1.fan(1)[0]
        sage: V = P1xP1.orbit_closure(H);  V
        1-d toric variety covered by 2 affine patches
        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.
        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 4 affine patches
          Defn: Defined on coordinates by sending [z0 : z1] to
                [0 : 1 : z1 : z0]

    TESTS:

        sage: A2 = toric_varieties.A2()
        sage: A2.orbit_closure(A2.fan(2)[0])
        0-d affine toric variety

  `plot`(**options)

    Plot `self`, i.e. the corresponding fan.

    INPUT:

    - any options for toric plots (see `toric_plotter.options`), none are mandatory.

    OUTPUT:

    - a plot.

    Note

    The difference between `X.plot()` and `X.fan().plot()` is that in the first case default ray labels correspond to variables of `X`.

    EXAMPLES:

        sage: X = toric_varieties.Cube_deformation(4)
        sage: X.plot()
        Graphics3d Object

  `rational_class_group`()

    Return the rational divisor class group of `self`.

    Let \(X\) be a toric variety.

    The Weil divisor class group \(\mathop{Cl}(X)\) is a finitely generated abelian group and can contain torsion. Its rank equals the number of rays in the fan of \(X\) minus the dimension of \(X\).

    The rational divisor class group is \(\mathop{Cl}(X) \otimes_\ZZ \QQ\) and never includes torsion. If \(X\) is smooth, this equals the Picard group of \(X\), whose elements are the isomorphism classes of line bundles on \(X\). The group law (which we write as addition) is the tensor product of the line bundles. The Picard group of a toric variety is always torsion-free.

    OUTPUT:

    - `rational divisor class group`.

    Note

    - Coordinates correspond to the rows of `self.fan().gale_transform()`.
    - `Kaehler_cone()` yields a cone in this group.

    EXAMPLES:

        sage: P1xA1 = toric_varieties.P1xA1()
        sage: P1xA1.rational_class_group()
        The toric rational divisor class group
        of a 2-d toric variety covered by 2 affine patches

  `resolve`(**kwds)

    Construct a toric variety whose fan subdivides the fan of `self`.

    The name of this function reflects the fact that usually such subdivisions are done for resolving singularities of the original variety.

    INPUT:

    This function accepts only keyword arguments, none of which are mandatory.

    - `coordinate_names` – names for coordinates of the new variety. If not given, will be constructed from the coordinate names of `self` and necessary indexed ones. See `normalize_names()` for the description of acceptable formats;
    - `coordinate_indices` – coordinate indices which should be used for indexed variables of the new variety;
    - all other arguments will be passed to `subdivide()` method of the underlying `rational polyhedral fan`, see its documentation for the available options.

    OUTPUT:

    - `toric variety`.

    EXAMPLES:

    First we will “manually” resolve a simple orbifold singularity:

        sage: cone = Cone([(1,1), (-1,1)])
        sage: fan = Fan([cone])
        sage: TV = ToricVariety(fan)
        sage: TV.is_smooth()
        False
        sage: TV_res = TV.resolve(new_rays=[(0,1)])
        sage: TV_res.is_smooth()
        True
        sage: TV_res.fan().rays()
        N( 1, 1),
        N(-1, 1),
        N( 0, 1)
        in 2-d lattice N
        sage: [cone.ambient_ray_indices() for cone in TV_res.fan()]
        [(0, 2), (1, 2)]

    Now let’s “automatically” partially resolve a more complicated fan:

        sage: fan = NormalFan(lattice_polytope.cross_polytope(3))
        sage: TV = ToricVariety(fan)
        sage: TV.is_smooth()
        False
        sage: TV.is_orbifold()
        False
        sage: TV.fan().nrays()
        8
        sage: TV.fan().ngenerating_cones()
        6
        sage: TV_res = TV.resolve(make_simplicial=True)
        sage: TV_res.is_smooth()
        False
        sage: TV_res.is_orbifold()
        True
        sage: TV_res.fan().nrays()
        8
        sage: TV_res.fan().ngenerating_cones()
        12
        sage: TV.gens()
        (z0, z1, z2, z3, z4, z5, z6, z7)
        sage: TV_res.gens()
        (z0, z1, z2, z3, z4, z5, z6, z7)
        sage: TV_res = TV.resolve(coordinate_names="x+",
        ...                       make_simplicial=True)
        sage: TV_res.gens()
        (x0, x1, x2, x3, x4, x5, x6, x7)

  `resolve_to_orbifold`(**kwds)

    Construct an orbifold whose fan subdivides the fan of `self`.

    It is a synonym for `resolve()` with `make_simplicial=True` option.

    INPUT:

    - this function accepts only keyword arguments. See `resolve()` for documentation.

    OUTPUT:

    - `toric variety`.

    EXAMPLES:

        sage: fan = NormalFan(lattice_polytope.cross_polytope(3))
        sage: TV = ToricVariety(fan)
        sage: TV.is_orbifold()
        False
        sage: TV.fan().nrays()
        8
        sage: TV.fan().ngenerating_cones()
        6
        sage: TV_res = TV.resolve_to_orbifold()
        sage: TV_res.is_orbifold()
        True
        sage: TV_res.fan().nrays()
        8
        sage: TV_res.fan().ngenerating_cones()
        12

  `subscheme`(polynomials)

    Return the subscheme of `self` defined by `polynomials`.

    INPUT:

    - `polynomials` – list of polynomials in the coordinate ring of `self`.

    OUTPUT:

    - `subscheme of a toric variety`.

    EXAMPLES:

    We will construct a subscheme of the product of two projective lines with coordinates \((x, y)\) for one and \((s, t)\) for the other:

        sage: P1xP1.<x,y,s,t> = toric_varieties.P1xP1()
        sage: X = P1xP1.subscheme([x*s + y*t, x^3+y^3])
        sage: X
        Closed subscheme of 2-d CPR-Fano toric variety
        covered by 4 affine patches defined by:
          x*s + y*t,
          x^3 + y^3
        sage: X.defining_polynomials()
        (x*s + y*t, x^3 + y^3)
        sage: X.defining_ideal()
        Ideal (x*s + y*t, x^3 + y^3)
        of Multivariate Polynomial Ring in x, y, s, t
        over Rational Field
        sage: X.base_ring()
        Rational Field
        sage: X.base_scheme()
        Spectrum of Rational Field
        sage: X.structure_morphism()
        Scheme morphism:
          From: Closed subscheme of 2-d CPR-Fano toric variety
                covered by 4 affine patches defined by:
          x*s + y*t,
          x^3 + y^3
          To:   Spectrum of Rational Field
          Defn: Structure map

  `toric_divisor_group`(base_ring=Integer Ring)

    Return the group of toric (T-Weil) divisors.

    INPUT:

    - `base_ring` – the coefficient ring, usually `ZZ` (default) or `QQ`.

    OUTPUT:

    The free Abelian agroup of toric Weil divisors, that is, formal `base_ring`-linear combinations of codimension-one toric subvarieties. The output will be an instance of `sage.schemes.toric.divisor.ToricDivisorGroup`.

    The \(i\)-th generator of the divisor group is the divisor where the \(i\)-th homogeneous coordinate vanishes, \(\{z_i=0\}\).

    EXAMPLES:

        sage: dP6 = toric_varieties.dP6()
        sage: TDiv = dP6.toric_divisor_group(); TDiv
        Group of toric ZZ-Weil divisors on 2-d CPR-Fano toric variety
        covered by 6 affine patches
        sage: TDiv == dP6.toric_divisor_group()
        True
        sage: TDiv.gens()
        (V(x), V(u), V(y), V(v), V(z), V(w))
        sage: dP6.coordinate_ring()
        Multivariate Polynomial Ring in x, u, y, v, z, w over Rational Field

  `volume_class`()

    Return the cohomology class of the volume form on the toric variety.

    Note that we are using cohomology with compact supports. If the variety is non-compact this is dual to homology without any support condition. In particular, for non-compact varieties the volume form \(\mathrm{dVol}=\wedge_i(dx_i \wedge dy_i)\) does not define a (non-zero) cohomology class.

    OUTPUT:

    A `CohomologyClass`. If it exists, it is the class of the (properly normalized) volume form, that is, it is the Poincare dual of a single point. If it does not exist, a `ValueError` is raised.

    EXAMPLES:

        sage: P2 = toric_varieties.P2()
        sage: P2.volume_class()
        [z^2]

        sage: A2_Z2 = toric_varieties.A2_Z2()
        sage: A2_Z2.volume_class()
        Traceback (most recent call last):
        ...
        ValueError: Volume class does not exist.

    If none of the maximal cones is smooth things get more tricky. In this case no torus-fixed point is smooth. If we want to count an ordinary point as \(1\), then a \(G\)-orbifold point needs to count as \(\frac{1}{|G|}\). For example, take \(\mathbb{P}^1\times\mathbb{P}^1\) with inhomogeneous coordinates \((t,y)\). Take the quotient by the action \((t,y)\mapsto (-t,-y)\). The \(\ZZ_2\)-invariant Weil divisors \(\{t=0\}\) and \(\{y=0\}\) intersect in a \(\ZZ_2\)-fixed point, so they ought to have intersection number \(\frac{1}{2}\). This means that the cohomology class \([t] \cap [y]\) should be \(\frac{1}{2}\) times the volume class. Note that this is different from the volume normalization chosen in [Schubert]:

        sage: P1xP1_Z2 = toric_varieties.P1xP1_Z2()
        sage: Dt = P1xP1_Z2.divisor(1);  Dt
        V(t)
        sage: Dy = P1xP1_Z2.divisor(3);  Dy
        V(y)
        sage: P1xP1_Z2.volume_class()
        [2*t*y]

        sage: HH = P1xP1_Z2.cohomology_ring()
        sage: HH(Dt) * HH(Dy) == 1/2 * P1xP1_Z2.volume_class()
        True

    The fractional coefficients are also necessary to match the normalization in the rational Chow group for simplicial toric varieties:

        sage: A = P1xP1_Z2.Chow_group(QQ)
        sage: A(Dt).intersection_with_divisor(Dy).count_points()
        1/2

    REFERENCES:

    [Schubert] | Sheldon Katz and Stein Arild Stromme, A Maple package for intersection theory and enumerative geometry.

`sage.schemes.toric.variety.``certify_names`(names)

  Make sure that `names` are valid in Python.

  INPUT:

  - `names` – list of strings.

  OUTPUT:

  - none, but a `ValueError` exception is raised if `names` are invalid.

  Each name must satisfy the following requirements:

  - Be non-empty.
  - Contain only (Latin) letters, digits, and underscores (“_”).
  - Start with a letter.

  In addition, all names must be distinct.

  EXAMPLES:

      sage: from sage.schemes.toric.variety import certify_names
      sage: certify_names([])
      sage: certify_names(["a", "x0", "x_45"])
      sage: certify_names(["", "x0", "x_45"])
      Traceback (most recent call last):
      ...
      ValueError: name must be nonempty!
      sage: certify_names(["a", "0", "x_45"])
      Traceback (most recent call last):
      ...
      ValueError: name must start with a letter! Got 0
      sage: certify_names(["a", "x0", "@_45"])
      Traceback (most recent call last):
      ...
      ValueError: name must be alphanumeric! Got @_45
      sage: certify_names(["a", "x0", "x0"])
      Traceback (most recent call last):
      ...
      ValueError: names must be distinct! Got: ['a', 'x0', 'x0']

`sage.schemes.toric.variety.``is_CohomologyClass`(x)

  Check whether `x` is a cohomology class of a toric variety.

  INPUT:

  - `x` – anything.

  OUTPUT:

  `True` or `False` depending on whether `x` is an instance of `CohomologyClass`

  EXAMPLES:

      sage: P2 = toric_varieties.P2()
      sage: HH = P2.cohomology_ring()
      sage: from sage.schemes.toric.variety import is_CohomologyClass
      sage: is_CohomologyClass( HH.one() )
      True
      sage: is_CohomologyClass( HH(P2.fan(1)[0]) )
      True
      sage: is_CohomologyClass('z')
      False

`sage.schemes.toric.variety.``is_ToricVariety`(x)

  Check if `x` is a toric variety.

  INPUT:

  - `x` – anything.

  OUTPUT:

  - `True` if `x` is a `toric variety` and `False` otherwise.

  Note

  While projective spaces are toric varieties mathematically, they are not toric varieties in Sage due to efficiency considerations, so this function will return `False`.

  EXAMPLES:

      sage: from sage.schemes.toric.variety import is_ToricVariety
      sage: is_ToricVariety(1)
      False
      sage: fan = FaceFan(lattice_polytope.cross_polytope(2))
      sage: P = ToricVariety(fan)
      sage: P
      2-d toric variety covered by 4 affine patches
      sage: is_ToricVariety(P)
      True
      sage: is_ToricVariety(ProjectiveSpace(2))
      False

`sage.schemes.toric.variety.``normalize_names`(names=None, ngens=None, prefix=None, indices=None, return_prefix=False)

  Return a list of names in the standard form.

  INPUT:

  All input parameters are optional.

  - `names` – names given either as a single string (with individual names separated by commas or spaces) or a list of strings with each string specifying a name. If the last name ends with the plus sign, “+”, this name will be used as `prefix` (even if `prefix` was given explicitly);
  - `ngens` – number of names to be returned;
  - `prefix` – prefix for the indexed names given as a string;
  - `indices` – list of integers (default: `range(ngens)`) used as indices for names with `prefix`. If given, must be of length `ngens`;
  - `return_prefix` – if `True`, the last element of the returned list will contain the prefix determined from `names` or given as the parameter `prefix`. This is useful if you may need more names in the future.

  OUTPUT:

  - list of names given as strings.

  These names are constructed in the following way:

  1. If necessary, split `names` into separate names.
  2. If the last name ends with “+”, put it into `prefix`.
  3. If `ngens` was given, add to the names obtained so far as many indexed names as necessary to get this number. If the `k`-th name of the total list of names is indexed, it is `prefix + str(indices[k])`. If there were already more names than `ngens`, discard “extra” ones.
  4. Check if constructed names are valid. See `certify_names()` for details.
  5. If the option `return_prefix=True` was given, add `prefix` to the end of the list.

  EXAMPLES:

  As promised, all parameters are optional:

      sage: from sage.schemes.toric.variety import normalize_names
      sage: normalize_names()
      []

  One of the most common uses is probably this one:

      sage: normalize_names("x+", 4)
      ['x0', 'x1', 'x2', 'x3']

  Now suppose that you want to enumerate your variables starting with one instead of zero:

      sage: normalize_names("x+", 4, indices=range(1,5))
      ['x1', 'x2', 'x3', 'x4']

  You may actually have an arbitrary enumeration scheme:

      sage: normalize_names("x+", 4, indices=[1, 10, 100, 1000])
      ['x1', 'x10', 'x100', 'x1000']

  Now let’s add some “explicit” names:

      sage: normalize_names("x y z t+", 4)
      ['x', 'y', 'z', 't3']

  Note that the “automatic” name is `t3` instead of `t0`. This may seem weird, but the reason for this behaviour is that the fourth name in this list will be the same no matter how many explicit names were given:

      sage: normalize_names("x y t+", 4)
      ['x', 'y', 't2', 't3']

  This is especially useful if you get `names` from a user but want to specify all default names:

      sage: normalize_names("x, y", 4, prefix="t")
      ['x', 'y', 't2', 't3']

  In this format, the user can easily override your choice for automatic names:

      sage: normalize_names("x y s+", 4, prefix="t")
      ['x', 'y', 's2', 's3']

  Let’s now use all parameters at once:

      sage: normalize_names("x, y, s+", 4, prefix="t",
      ...       indices=range(1,5), return_prefix=True)
      ['x', 'y', 's3', 's4', 's']

  Note that you still need to give indices for all names, even if some of the first ones will be “wasted” because of the explicit names. The reason is the same as before - this ensures consistency of automatically generated names, no matter how many explicit names were given.

  The prefix is discarded if `ngens` was not given:

      sage: normalize_names("alpha, beta, gamma, zeta+")
      ['alpha', 'beta', 'gamma']

  Finally, let’s take a look at some possible mistakes:

      sage: normalize_names("123")
      Traceback (most recent call last):
      ...
      ValueError: name must start with a letter! Got 123

  A more subtle one:

      sage: normalize_names("x1", 4, prefix="x")
      Traceback (most recent call last):
      ...
      ValueError: names must be distinct! Got: ['x1', 'x1', 'x2', 'x3']

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