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# Library of toric varieties

This module provides a simple way to construct often-used toric varieties. Please see the help for the individual methods of `toric_varieties` for a more detailed description of which varieties can be constructed.

AUTHORS:

- Volker Braun (2010-07-02): initial version

EXAMPLES:

    sage: toric_varieties.dP6()
    2-d CPR-Fano toric variety covered by 6 affine patches

You can assign the homogeneous coordinates to Sage variables either with `inject_variables()` or immediately during assignment like this:

    sage: P2.<x,y,z> = toric_varieties.P2()
    sage: x^2 + y^2 + z^2
    x^2 + y^2 + z^2
    sage: P2.coordinate_ring()
    Multivariate Polynomial Ring in x, y, z over Rational Field

class `sage.schemes.toric.library.``ToricVarietyFactory`

  Bases: `sage.structure.sage_object.SageObject`

  The methods of this class construct toric varieties.

  Warning

  You need not create instances of this class. Use the already-provided object `toric_varieties` instead.

  `A`(n, names='z+', base_ring=Rational Field)

    Construct the `n`-dimensional affine space.

    INPUT:

    - `n` – positive integer. The dimension of the affine space.
    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `toric variety`.

    EXAMPLES:

        sage: A3 = toric_varieties.A(3)
        sage: A3
        3-d affine toric variety
        sage: A3.fan().rays()
        N(1, 0, 0),
        N(0, 1, 0),
        N(0, 0, 1)
        in 3-d lattice N
        sage: A3.gens()
        (z0, z1, z2)

  `A1`(names='z', base_ring=Rational Field)

    Construct the affine line \(\mathbb{A}^1\) as a toric variety.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `toric variety`.

    EXAMPLES:

        sage: A1 = toric_varieties.A1()
        sage: A1
        1-d affine toric variety
        sage: A1.fan().rays()
        N(1)
        in 1-d lattice N
        sage: A1.gens()
        (z,)

  `A2`(names='x y', base_ring=Rational Field)

    Construct the affine plane \(\mathbb{A}^2\) as a toric variety.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `toric variety`.

    EXAMPLES:

        sage: A2 = toric_varieties.A2()
        sage: A2
        2-d affine toric variety
        sage: A2.fan().rays()
        N(1, 0),
        N(0, 1)
        in 2-d lattice N
        sage: A2.gens()
        (x, y)

  `A2_Z2`(names='x y', base_ring=Rational Field)

    Construct the orbifold \(\mathbb{A}^2 / \ZZ_2\) as a toric variety.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `toric variety`.

    EXAMPLES:

        sage: A2_Z2 = toric_varieties.A2_Z2()
        sage: A2_Z2
        2-d affine toric variety
        sage: A2_Z2.fan().rays()
        N(1, 0),
        N(1, 2)
        in 2-d lattice N
        sage: A2_Z2.gens()
        (x, y)

  `BCdlOG`(names='v1 v2 c1 c2 v4 v5 b e1 e2 e3 f g v6', base_ring=Rational Field)

    Construct the 5-dimensional toric variety studied in [BCdlOG], [HLY]

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: X = toric_varieties.BCdlOG()
        sage: X
        5-d CPR-Fano toric variety covered by 54 affine patches
        sage: X.fan().rays()
        N(-1,  0,  0,  2,  3),
        N( 0, -1,  0,  2,  3),
        N( 0,  0, -1,  2,  3),
        N( 0,  0, -1,  1,  2),
        N( 0,  0,  0, -1,  0),
        N( 0,  0,  0,  0, -1),
        N( 0,  0,  0,  2,  3),
        N( 0,  0,  1,  2,  3),
        N( 0,  0,  2,  2,  3),
        N( 0,  0,  1,  1,  1),
        N( 0,  1,  2,  2,  3),
        N( 0,  1,  3,  2,  3),
        N( 1,  0,  4,  2,  3)
        in 5-d lattice N
        sage: X.gens()
        (v1, v2, c1, c2, v4, v5, b, e1, e2, e3, f, g, v6)

    REFERENCES:

    [BCdlOG] | Volker Braun, Philip Candelas, Xendia de la Ossa, Antonella Grassi, “Toric Calabi-Yau Fourfolds, Duality Between N=1 Theories and Divisors that Contribute to the Superpotential”, http://arxiv.org/abs/hep-th/0001208

    [HLY] | Yi Hu, Chien-Hao Liu, Shing-Tung Yau, “Toric morphisms and fibrations of toric Calabi-Yau hypersurfaces”, http://arxiv.org/abs/math/0010082

  `BCdlOG_base`(names='d4 d3 r2 r1 d2 u d1', base_ring=Rational Field)

    Construct the base of the \(\mathbb{P}^2(1,2,3)\) fibration `BCdlOG()`.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `toric variety`.

    EXAMPLES:

        sage: base = toric_varieties.BCdlOG_base()
        sage: base
        3-d toric variety covered by 10 affine patches
        sage: base.fan().rays()
        N(-1,  0,  0),
        N( 0, -1,  0),
        N( 0,  0, -1),
        N( 0,  0,  1),
        N( 0,  1,  2),
        N( 0,  1,  3),
        N( 1,  0,  4)
        in 3-d lattice N
        sage: base.gens()
        (d4, d3, r2, r1, d2, u, d1)

  `Conifold`(names='u x y v', base_ring=Rational Field)

    Construct the conifold as a toric variety.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `toric variety`.

    EXAMPLES:

        sage: Conifold = toric_varieties.Conifold()
        sage: Conifold
        3-d affine toric variety
        sage: Conifold.fan().rays()
        N(0, 0, 1),
        N(0, 1, 1),
        N(1, 0, 1),
        N(1, 1, 1)
        in 3-d lattice N
        sage: Conifold.gens()
        (u, x, y, v)

  `Cube_deformation`(k, names=None, base_ring=Rational Field)

    Construct, for each \(k\in\ZZ_{\geq 0}\), a toric variety with \(\ZZ_k\)-torsion in the Chow group.

    The fans of this sequence of toric varieties all equal the face fan of a unit cube topologically, but the `(1,1,1)`-vertex is moved to `(1,1,2k+1)`. This example was studied in [FS].

    INPUT:

    - `k` – integer. The case `k=0` is the same as `Cube_face_fan()`.
    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `toric variety` \(X_k\). Its Chow group is \(A_1(X_k)=\ZZ_k\).

    EXAMPLES:

        sage: X_2 = toric_varieties.Cube_deformation(2)
        sage: X_2
        3-d toric variety covered by 6 affine patches
        sage: X_2.fan().rays()
        N( 1,  1,  5),
        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: X_2.gens()
        (z0, z1, z2, z3, z4, z5, z6, z7)

    REFERENCES:

    [FS] | William Fulton, Bernd Sturmfels, “Intersection Theory on Toric Varieties”, http://arxiv.org/abs/alg-geom/9403002

  `Cube_face_fan`(names='z+', base_ring=Rational Field)

    Construct the toric variety given by the face fan of the 3-dimensional unit lattice cube.

    This variety has 6 conifold singularities but the fan is still polyhedral.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: Cube_face_fan = toric_varieties.Cube_face_fan()
        sage: Cube_face_fan
        3-d CPR-Fano toric variety covered by 6 affine patches
        sage: Cube_face_fan.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: Cube_face_fan.gens()
        (z0, z1, z2, z3, z4, z5, z6, z7)

  `Cube_nonpolyhedral`(names='z+', base_ring=Rational Field)

    Construct the toric variety defined by a fan that is not the face fan of a polyhedron.

    This toric variety is defined by a fan that is topologically like the face fan of a 3-dimensional cube, but with a different N-lattice structure.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `toric variety`.

    NOTES:

    - This is an example of an non-polyhedral fan.
    - Its Chow group has torsion: \(A_2(X)=\ZZ^5 \oplus \ZZ_2\)

    EXAMPLES:

        sage: Cube_nonpolyhedral = toric_varieties.Cube_nonpolyhedral()
        sage: Cube_nonpolyhedral
        3-d toric variety covered by 6 affine patches
        sage: Cube_nonpolyhedral.fan().rays()
        N( 1,  2,  3),
        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: Cube_nonpolyhedral.gens()
        (z0, z1, z2, z3, z4, z5, z6, z7)

  `Cube_sublattice`(names='z+', base_ring=Rational Field)

    Construct the toric variety defined by a face fan over a 3-dimensional cube, but not the unit cube in the N-lattice. See [FultonP65].

    Its Chow group is \(A_2(X)=\mathbb{Z}^5\), which distinguishes it from the face fan of the unit cube.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: Cube_sublattice = toric_varieties.Cube_sublattice()
        sage: Cube_sublattice
        3-d CPR-Fano toric variety covered by 6 affine patches
        sage: Cube_sublattice.fan().rays()
        N( 1,  0,  0),
        N( 0,  1,  0),
        N( 0,  0,  1),
        N(-1,  1,  1),
        N(-1,  0,  0),
        N( 0, -1,  0),
        N( 0,  0, -1),
        N( 1, -1, -1)
        in 3-d lattice N
        sage: Cube_sublattice.gens()
        (z0, z1, z2, z3, z4, z5, z6, z7)

    REFERENCES:

    [FultonP65] | Page 65, 3rd exercise (Section 3.4) of Wiliam Fulton, “Introduction to Toric Varieties”, Princeton University Press

  `P`(n, names='z+', base_ring=Rational Field)

    Construct the `n`-dimensional projective space \(\mathbb{P}^n\).

    INPUT:

    - `n` – positive integer. The dimension of the projective space.
    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: P3 = toric_varieties.P(3)
        sage: P3
        3-d CPR-Fano toric variety covered by 4 affine patches
        sage: P3.fan().rays()
        N( 1,  0,  0),
        N( 0,  1,  0),
        N( 0,  0,  1),
        N(-1, -1, -1)
        in 3-d lattice N
        sage: P3.gens()
        (z0, z1, z2, z3)

  `P1`(names='s t', base_ring=Rational Field)

    Construct the projective line \(\mathbb{P}^1\) as a toric variety.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: P1 = toric_varieties.P1()
        sage: P1
        1-d CPR-Fano toric variety covered by 2 affine patches
        sage: P1.fan().rays()
        N( 1),
        N(-1)
        in 1-d lattice N
        sage: P1.gens()
        (s, t)

  `P1xA1`(names='s t z', base_ring=Rational Field)

    Construct the cartesian product \(\mathbb{P}^1 \times \mathbb{A}^1\) as a toric variety.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `toric variety`.

    EXAMPLES:

        sage: P1xA1 = toric_varieties.P1xA1()
        sage: P1xA1
        2-d toric variety covered by 2 affine patches
        sage: P1xA1.fan().rays()
        N( 1, 0),
        N(-1, 0),
        N( 0, 1)
        in 2-d lattice N
        sage: P1xA1.gens()
        (s, t, z)

  `P1xP1`(names='s t x y', base_ring=Rational Field)

    Construct the del Pezzo surface \(\mathbb{P}^1 \times \mathbb{P}^1\) as a toric variety.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: P1xP1 = toric_varieties.P1xP1()
        sage: P1xP1
        2-d CPR-Fano toric variety covered by 4 affine patches
        sage: P1xP1.fan().rays()
        N( 1,  0),
        N(-1,  0),
        N( 0,  1),
        N( 0, -1)
        in 2-d lattice N
        sage: P1xP1.gens()
        (s, t, x, y)

  `P1xP1_Z2`(names='s t x y', base_ring=Rational Field)

    Construct the toric \(\mathbb{Z}_2\)-orbifold of the del Pezzo surface \(\mathbb{P}^1 \times \mathbb{P}^1\) as a toric variety.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: P1xP1_Z2 = toric_varieties.P1xP1_Z2()
        sage: P1xP1_Z2
        2-d CPR-Fano toric variety covered by 4 affine patches
        sage: P1xP1_Z2.fan().rays()
        N( 1,  1),
        N(-1, -1),
        N(-1,  1),
        N( 1, -1)
        in 2-d lattice N
        sage: P1xP1_Z2.gens()
        (s, t, x, y)
        sage: P1xP1_Z2.Chow_group().degree(1)
        C2 x Z^2

  `P2`(names='x y z', base_ring=Rational Field)

    Construct the projective plane \(\mathbb{P}^2\) as a toric variety.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: P2 = toric_varieties.P2()
        sage: P2
        2-d CPR-Fano toric variety covered by 3 affine patches
        sage: P2.fan().rays()
        N( 1,  0),
        N( 0,  1),
        N(-1, -1)
        in 2-d lattice N
        sage: P2.gens()
        (x, y, z)

  `P2_112`(names='z+', base_ring=Rational Field)

    Construct the weighted projective space \(\mathbb{P}^2(1,1,2)\).

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: P2_112 = toric_varieties.P2_112()
        sage: P2_112
        2-d CPR-Fano toric variety covered by 3 affine patches
        sage: P2_112.fan().rays()
        N( 1,  0),
        N( 0,  1),
        N(-1, -2)
        in 2-d lattice N
        sage: P2_112.gens()
        (z0, z1, z2)

  `P2_123`(names='z+', base_ring=Rational Field)

    Construct the weighted projective space \(\mathbb{P}^2(1,2,3)\).

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: P2_123 = toric_varieties.P2_123()
        sage: P2_123
        2-d CPR-Fano toric variety covered by 3 affine patches
        sage: P2_123.fan().rays()
        N( 1,  0),
        N( 0,  1),
        N(-2, -3)
        in 2-d lattice N
        sage: P2_123.gens()
        (z0, z1, z2)

  `P4_11133`(names='z+', base_ring=Rational Field)

    Construct the weighted projective space \(\mathbb{P}^4(1,1,1,3,3)\).

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: P4_11133 = toric_varieties.P4_11133()
        sage: P4_11133
        4-d CPR-Fano toric variety covered by 5 affine patches
        sage: P4_11133.fan().rays()
        N( 1,  0,  0,  0),
        N( 0,  1,  0,  0),
        N( 0,  0,  1,  0),
        N( 0,  0,  0,  1),
        N(-3, -3, -1, -1)
        in 4-d lattice N
        sage: P4_11133.gens()
        (z0, z1, z2, z3, z4)

  `P4_11133_resolved`(names='z+', base_ring=Rational Field)

    Construct the weighted projective space \(\mathbb{P}^4(1,1,1,3,3)\).

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: P4_11133_resolved = toric_varieties.P4_11133_resolved()
        sage: P4_11133_resolved
        4-d CPR-Fano toric variety covered by 9 affine patches
        sage: P4_11133_resolved.fan().rays()
        N( 1,  0,  0,  0),
        N( 0,  1,  0,  0),
        N( 0,  0,  1,  0),
        N( 0,  0,  0,  1),
        N(-3, -3, -1, -1),
        N(-1, -1,  0,  0)
        in 4-d lattice N
        sage: P4_11133_resolved.gens()
        (z0, z1, z2, z3, z4, z5)

  `P4_11169`(names='z+', base_ring=Rational Field)

    Construct the weighted projective space \(\mathbb{P}^4(1,1,1,6,9)\).

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: P4_11169 = toric_varieties.P4_11169()
        sage: P4_11169
        4-d CPR-Fano toric variety covered by 5 affine patches
        sage: P4_11169.fan().rays()
        N( 1,  0,  0,  0),
        N( 0,  1,  0,  0),
        N( 0,  0,  1,  0),
        N( 0,  0,  0,  1),
        N(-9, -6, -1, -1)
        in 4-d lattice N
        sage: P4_11169.gens()
        (z0, z1, z2, z3, z4)

  `P4_11169_resolved`(names='z+', base_ring=Rational Field)

    Construct the blow-up of the weighted projective space \(\mathbb{P}^4(1,1,1,6,9)\) at its curve of \(\ZZ_3\) orbifold fixed points.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: P4_11169_resolved = toric_varieties.P4_11169_resolved()
        sage: P4_11169_resolved
        4-d CPR-Fano toric variety covered by 9 affine patches
        sage: P4_11169_resolved.fan().rays()
        N( 1,  0,  0,  0),
        N( 0,  1,  0,  0),
        N( 0,  0,  1,  0),
        N( 0,  0,  0,  1),
        N(-9, -6, -1, -1),
        N(-3, -2,  0,  0)
        in 4-d lattice N
        sage: P4_11169_resolved.gens()
        (z0, z1, z2, z3, z4, z5)

  `WP`(*q, **kw)

    Construct weighted projective \(n\)-space over a field.

    INPUT:

    - `q` – a sequence of positive integers relatively prime to one another. The weights `q` can be given either as a list or tuple, or as positional arguments.

    Two keyword arguments:

    - `base_ring` – a field (default: \(\QQ\)).
    - `names` – string or list (tuple) of strings (default ‘z+’). See `normalize_names()` for acceptable formats.

    OUTPUT:

    - A `toric variety`. If \(q=(q_0,\dots,q_n)\), then the output is the weighted projective space \(\mathbb{P}(q_0,\dots,q_n)\) over `base_ring`. `names` are the names of the generators of the homogeneous coordinate ring.

    EXAMPLES:

    A hyperelliptic curve \(C\) of genus 2 as a subscheme of the weighted projective plane \(\mathbb{P}(1,3,1)\):

        sage: X = toric_varieties.WP([1,3,1], names='x y z')
        sage: X.inject_variables()
        Defining x, y, z
        sage: g = y^2-(x^6-z^6)
        sage: C = X.subscheme([g]); C
        Closed subscheme of 2-d toric variety covered by 3 affine patches defined by:
          -x^6 + z^6 + y^2

  `dP6`(names='x u y v z w', base_ring=Rational Field)

    Construct the del Pezzo surface of degree 6 (\(\mathbb{P}^2\) blown up at 3 points) as a toric variety.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: dP6 = toric_varieties.dP6()
        sage: dP6
        2-d CPR-Fano toric variety covered by 6 affine patches
        sage: dP6.fan().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: dP6.gens()
        (x, u, y, v, z, w)

  `dP6xdP6`(names='x0 x1 x2 x3 x4 x5 y0 y1 y2 y3 y4 y5', base_ring=Rational Field)

    Construct the product of two del Pezzo surfaces of degree 6 (\(\mathbb{P}^2\) blown up at 3 points) as a toric variety.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: dP6xdP6 = toric_varieties.dP6xdP6()
        sage: dP6xdP6
        4-d CPR-Fano toric variety covered by 36 affine patches
        sage: dP6xdP6.fan().rays()
        N( 0,  1,  0,  0),
        N(-1,  0,  0,  0),
        N(-1, -1,  0,  0),
        N( 0, -1,  0,  0),
        N( 1,  0,  0,  0),
        N( 1,  1,  0,  0),
        N( 0,  0,  0,  1),
        N( 0,  0, -1,  0),
        N( 0,  0, -1, -1),
        N( 0,  0,  0, -1),
        N( 0,  0,  1,  0),
        N( 0,  0,  1,  1)
        in 4-d lattice N
        sage: dP6xdP6.gens()
        (x0, x1, x2, x3, x4, x5, y0, y1, y2, y3, y4, y5)

  `dP7`(names='x u y v z', base_ring=Rational Field)

    Construct the del Pezzo surface of degree 7 (\(\mathbb{P}^2\) blown up at 2 points) as a toric variety.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: dP7 = toric_varieties.dP7()
        sage: dP7
        2-d CPR-Fano toric variety covered by 5 affine patches
        sage: dP7.fan().rays()
        N( 0,  1),
        N(-1,  0),
        N(-1, -1),
        N( 0, -1),
        N( 1,  0)
        in 2-d lattice N
        sage: dP7.gens()
        (x, u, y, v, z)

  `dP8`(names='t x y z', base_ring=Rational Field)

    Construct the del Pezzo surface of degree 8 (\(\mathbb{P}^2\) blown up at 1 point) as a toric variety.

    INPUT:

    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `CPR-Fano toric variety`.

    EXAMPLES:

        sage: dP8 = toric_varieties.dP8()
        sage: dP8
        2-d CPR-Fano toric variety covered by 4 affine patches
        sage: dP8.fan().rays()
        N( 1,  1),
        N( 0,  1),
        N(-1, -1),
        N( 1,  0)
        in 2-d lattice N
        sage: dP8.gens()
        (t, x, y, z)

  `torus`(n, names='z+', base_ring=Rational Field)

    Construct the `n`-dimensional algebraic torus \((\mathbb{F}^\times)^n\).

    INPUT:

    - `n` – non-negative integer. The dimension of the algebraic torus.
    - `names` – string. Names for the homogeneous coordinates. See `normalize_names()` for acceptable formats.
    - `base_ring` – a ring (default: \(\QQ\)). The base ring for the toric variety.

    OUTPUT:

    A `toric variety`.

    EXAMPLES:

        sage: T3 = toric_varieties.torus(3);  T3
        3-d affine toric variety
        sage: T3.fan().rays()
        Empty collection
        in 3-d lattice N
        sage: T3.fan().virtual_rays()
        N(1, 0, 0),
        N(0, 1, 0),
        N(0, 0, 1)
        in 3-d lattice N
        sage: T3.gens()
        (z0, z1, z2)
        sage: sorted(T3.change_ring(GF(3)).point_set().list())
        [[1 : 1 : 1], [1 : 1 : 2], [1 : 2 : 1], [1 : 2 : 2],
         [2 : 1 : 1], [2 : 1 : 2], [2 : 2 : 1], [2 : 2 : 2]]

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