r"""
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
:meth:`~sage.schemes.toric.variety.ToricVariety_field.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
"""
from sage.structure.sage_object import SageObject
from sage.matrix.all import matrix, identity_matrix
from sage.geometry.fan import Fan
from sage.geometry.toric_lattice import ToricLattice
from sage.geometry.lattice_polytope import LatticePolytope
from sage.rings.all import ZZ, QQ, gcd
from sage.schemes.toric.variety import (DEFAULT_PREFIX,
ToricVariety,
normalize_names)
from sage.schemes.toric.fano_variety import CPRFanoToricVariety
from sage.categories.fields import Fields
_Fields = Fields()
toric_varieties_rays_cones = {
'dP6':[
[(0, 1), (-1, 0), (-1, -1), (0, -1), (1, 0), (1, 1)],
[[0,1],[1,2],[2,3],[3,4],[4,5],[5,0]] ],
'dP7':[
[(0, 1), (-1, 0), (-1, -1), (0, -1), (1, 0)],
[[0,1],[1,2],[2,3],[3,4],[4,0]] ],
'dP8':[
[(1,1), (0, 1), (-1, -1), (1, 0)],
[[0,1],[1,2],[2,3],[3,0]]
],
'P1xP1':[
[(1, 0), (-1, 0), (0, 1), (0, -1)],
[[0,2],[2,1],[1,3],[3,0]] ],
'P1xP1_Z2':[
[(1, 1), (-1, -1), (-1, 1), (1, -1)],
[[0,2],[2,1],[1,3],[3,0]] ],
'P1':[
[(1,), (-1,)],
[[0],[1]] ],
'P2':[
[(1,0), (0, 1), (-1, -1)],
[[0,1],[1,2],[2,0]] ],
'A1':[
[(1,)],
[[0]] ],
'A2':[
[(1, 0), (0, 1)],
[[0,1]] ],
'A2_Z2':[
[(1, 0), (1, 2)],
[[0,1]] ],
'P1xA1':[
[(1, 0), (-1, 0), (0, 1)],
[[0,2],[2,1]] ],
'Conifold':[
[(0, 0, 1), (0, 1, 1), (1, 0, 1), (1, 1, 1)],
[[0,1,2,3]] ],
'dP6xdP6':[
[(0, 1, 0, 0), (-1, 0, 0, 0), (-1, -1, 0, 0),
(0, -1, 0, 0), (1, 0, 0, 0), (1, 1, 0, 0),
(0, 0, 0, 1), (0, 0, -1, 0), (0, 0, -1, -1),
(0, 0, 0, -1), (0, 0, 1, 0), (0, 0, 1, 1)],
[[0, 1, 6, 7], [0, 1, 7, 8], [0, 1, 8, 9], [0, 1, 9, 10],
[0, 1, 10, 11], [0, 1, 6, 11], [1, 2, 6, 7], [1, 2, 7, 8],
[1, 2, 8, 9], [1, 2, 9, 10], [1, 2, 10, 11], [1, 2, 6, 11],
[2, 3, 6, 7], [2, 3, 7, 8], [2, 3, 8, 9], [2, 3, 9, 10],
[2, 3, 10, 11], [2, 3, 6, 11], [3, 4, 6, 7], [3, 4, 7, 8],
[3, 4, 8, 9], [3, 4, 9, 10], [3, 4, 10, 11], [3, 4, 6, 11],
[4, 5, 6, 7], [4, 5, 7, 8], [4, 5, 8, 9], [4, 5, 9, 10],
[4, 5, 10, 11], [4, 5, 6, 11], [0, 5, 6, 7], [0, 5, 7, 8],
[0, 5, 8, 9], [0, 5, 9, 10], [0, 5, 10, 11], [0, 5, 6, 11]] ],
'Cube_face_fan':[
[(1, 1, 1), (1, -1, 1), (-1, 1, 1), (-1, -1, 1),
(-1, -1, -1), (-1, 1, -1), (1, -1, -1), (1, 1, -1)],
[[0,1,2,3], [4,5,6,7], [0,1,7,6], [4,5,3,2], [0,2,5,7], [4,6,1,3]] ],
'Cube_sublattice':[
[(1, 0, 0), (0, 1, 0), (0, 0, 1), (-1, 1, 1),
(-1, 0, 0), (0, -1, 0), (0, 0, -1), (1, -1, -1)],
[[0,1,2,3],[4,5,6,7],[0,1,7,6],[4,5,3,2],[0,2,5,7],[4,6,1,3]] ],
'Cube_nonpolyhedral':[
[(1, 2, 3), (1, -1, 1), (-1, 1, 1), (-1, -1, 1),
(-1, -1, -1), (-1, 1, -1), (1, -1, -1), (1, 1, -1)],
[[0,1,2,3],[4,5,6,7],[0,1,7,6],[4,5,3,2],[0,2,5,7],[4,6,1,3]] ],
'BCdlOG':[
[(-1, 0, 0, 2, 3),
( 0,-1, 0, 2, 3),
( 0, 0,-1, 2, 3),
( 0, 0,-1, 1, 2),
( 0, 0, 0,-1, 0),
( 0, 0, 0, 0,-1),
( 0, 0, 0, 2, 3),
( 0, 0, 1, 2, 3),
( 0, 0, 2, 2, 3),
( 0, 0, 1, 1, 1),
( 0, 1, 2, 2, 3),
( 0, 1, 3, 2, 3),
( 1, 0, 4, 2, 3)],
[ [0,6,7,1,4], [0,6,10,2,4], [0,6,1,2,4], [0,9,7,1,5], [0,6,7,1,5],
[0,6,10,2,5], [0,6,1,2,5], [0,9,1,4,5], [0,6,10,4,11],[0,6,7,4,11],
[0,6,10,5,11], [0,9,7,5,11], [0,6,7,5,11], [0,9,4,5,11], [0,10,4,5,11],
[0,9,7,1,8], [0,9,1,4,8], [0,7,1,4,8], [0,9,7,11,8], [0,9,4,11,8],
[0,7,4,11,8], [0,10,2,4,3], [0,1,2,4,3], [0,10,2,5,3], [0,1,2,5,3],
[0,10,4,5,3], [0,1,4,5,3], [12,6,7,1,4], [12,6,10,2,4],[12,6,1,2,4],
[12,9,7,1,5], [12,6,7,1,5], [12,6,10,2,5], [12,6,1,2,5], [12,9,1,4,5],
[12,6,10,4,11],[12,6,7,4,11], [12,6,10,5,11],[12,9,7,5,11],[12,6,7,5,11],
[12,9,4,5,11], [12,10,4,5,11],[12,9,7,1,8], [12,9,1,4,8], [12,7,1,4,8],
[12,9,7,11,8], [12,9,4,11,8], [12,7,4,11,8], [12,10,2,4,3],[12,1,2,4,3],
[12,10,2,5,3], [12,1,2,5,3], [12,10,4,5,3], [12,1,4,5,3] ] ],
'BCdlOG_base':[
[(-1, 0, 0),
( 0,-1, 0),
( 0, 0,-1),
( 0, 0, 1),
( 0, 1, 2),
( 0, 1, 3),
( 1, 0, 4)],
[[0,4,2],[0,4,5],[0,5,3],[0,1,3],[0,1,2],
[6,4,2],[6,4,5],[6,5,3],[6,1,3],[6,1,2]] ],
'P2_112':[
[(1,0), (0, 1), (-1, -2)],
[[0,1],[1,2],[2,0]] ],
'P2_123':[
[(1,0), (0, 1), (-2, -3)],
[[0,1],[1,2],[2,0]] ],
'P4_11169':[
[(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (-9, -6, -1, -1)],
[[0,1,2,3],[0,1,2,4],[0,1,3,4],[0,2,3,4],[1,2,3,4]] ],
'P4_11169_resolved':[
[(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (-9, -6, -1, -1), (-3, -2, 0, 0)],
[[0, 1, 2, 3], [0, 1, 3, 4], [0, 1, 2, 4], [1, 3, 4, 5], [0, 3, 4, 5],
[1, 2, 4, 5], [0, 2, 4, 5], [1, 2, 3, 5], [0, 2, 3, 5]] ],
'P4_11133':[
[(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (-3, -3, -1, -1)],
[[0,1,2,3],[0,1,2,4],[0,1,3,4],[0,2,3,4],[1,2,3,4]] ],
'P4_11133_resolved':[
[(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (-3, -3, -1, -1), (-1, -1, 0, 0)],
[[0, 1, 2, 3], [0, 1, 3, 4], [0, 1, 2, 4], [1, 3, 4, 5], [0, 3, 4, 5],
[1, 2, 4, 5], [0, 2, 4, 5], [1, 2, 3, 5], [0, 2, 3, 5]] ]
}
class ToricVarietyFactory(SageObject):
r"""
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.
"""
_check = True
def _make_ToricVariety(self, name, coordinate_names, base_ring):
"""
Construct a toric variety and cache the result.
INPUT:
- ``name`` -- string. One of the pre-defined names in the
``toric_varieties_rays_cones`` data structure.
- ``coordinate_names`` -- A string describing the names of the
homogeneous coordinates of the toric variety.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`.
EXAMPLES::
sage: toric_varieties.A1() # indirect doctest
1-d affine toric variety
"""
rays, cones = toric_varieties_rays_cones[name]
if coordinate_names is None:
dict_key = (name, base_ring)
else:
coordinate_names = normalize_names(coordinate_names, len(rays),
DEFAULT_PREFIX)
dict_key = (name, base_ring) + tuple(coordinate_names)
if dict_key not in self.__dict__:
fan = Fan(cones, rays, check=self._check)
self.__dict__[dict_key] = \
ToricVariety(fan,
coordinate_names=coordinate_names,
base_ring=base_ring)
return self.__dict__[dict_key]
def _make_CPRFanoToricVariety(self, name, coordinate_names, base_ring):
"""
Construct a (crepant partially resolved) Fano toric variety
and cache the result.
INPUT:
- ``name`` -- string. One of the pre-defined names in the
``toric_varieties_rays_cones`` data structure.
- ``coordinate_names`` -- A string describing the names of the
homogeneous coordinates of the toric variety.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
EXAMPLES::
sage: toric_varieties.P2() # indirect doctest
2-d CPR-Fano toric variety covered by 3 affine patches
"""
rays, cones = toric_varieties_rays_cones[name]
if coordinate_names is None:
dict_key = (name, base_ring)
else:
coordinate_names = normalize_names(coordinate_names, len(rays),
DEFAULT_PREFIX)
dict_key = (name, base_ring) + tuple(coordinate_names)
if dict_key not in self.__dict__:
polytope = LatticePolytope( matrix(rays).transpose() )
points = map(tuple, polytope.points().columns())
ray2point = [points.index(r) for r in rays]
charts = [ [ray2point[i] for i in c] for c in cones ]
self.__dict__[dict_key] = \
CPRFanoToricVariety(Delta_polar=polytope,
coordinate_points=ray2point,
charts=charts,
coordinate_names=coordinate_names,
base_ring=base_ring,
check=self._check)
return self.__dict__[dict_key]
def dP6(self, names='x u y v z w', base_ring=QQ):
r"""
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
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
return self._make_CPRFanoToricVariety('dP6', names, base_ring)
def dP7(self, names='x u y v z', base_ring=QQ):
r"""
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
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
return self._make_CPRFanoToricVariety('dP7', names, base_ring)
def dP8(self, names='t x y z', base_ring=QQ):
r"""
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
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
return self._make_CPRFanoToricVariety('dP8', names, base_ring)
def P1xP1(self, names='s t x y', base_ring=QQ):
r"""
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
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
return self._make_CPRFanoToricVariety('P1xP1', names, base_ring)
def P1xP1_Z2(self, names='s t x y', base_ring=QQ):
r"""
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
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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
"""
return self._make_CPRFanoToricVariety('P1xP1_Z2', names, base_ring)
def P1(self, names='s t', base_ring=QQ):
r"""
Construct the projective line `\mathbb{P}^1` as a toric
variety.
INPUT:
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
return self._make_CPRFanoToricVariety('P1', names, base_ring)
def P2(self, names='x y z', base_ring=QQ):
r"""
Construct the projective plane `\mathbb{P}^2` as a toric
variety.
INPUT:
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
return self._make_CPRFanoToricVariety('P2', names, base_ring)
def P(self, n, names='z+', base_ring=QQ):
r"""
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
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
try:
n = ZZ(n)
except TypeError:
raise TypeError("dimension of the projective space must be a "
"positive integer!\nGot: %s" % n)
if n <= 0:
raise ValueError("only projective spaces of positive dimension "
"can be constructed!\nGot: %s" % n)
m = identity_matrix(n).augment(matrix(n, 1, [-1]*n))
charts = [ range(0,i)+range(i+1,n+1) for i in range(0,n+1) ]
return CPRFanoToricVariety(Delta_polar=LatticePolytope(m),
charts=charts, check=self._check,
coordinate_names=names, base_ring=base_ring)
def A1(self, names='z', base_ring=QQ):
r"""
Construct the affine line `\mathbb{A}^1` as a toric variety.
INPUT:
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`.
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,)
"""
return self._make_ToricVariety('A1', names, base_ring)
def A2(self, names='x y', base_ring=QQ):
r"""
Construct the affine plane `\mathbb{A}^2` as a toric variety.
INPUT:
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`.
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)
"""
return self._make_ToricVariety('A2', names, base_ring)
def A(self, n, names='z+', base_ring=QQ):
r"""
Construct the ``n``-dimensional affine space.
INPUT:
- ``n`` -- positive integer. The dimension of the affine space.
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`.
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)
"""
try:
n = ZZ(n)
except TypeError:
raise TypeError("dimension of the affine space must be a "
"positive integer!\nGot: %s" % n)
if n <= 0:
raise ValueError("only affine spaces of positive dimension can "
"be constructed!\nGot: %s" % n)
rays = identity_matrix(n).columns()
cones = [ range(0,n) ]
fan = Fan(cones, rays, check=self._check)
return ToricVariety(fan, coordinate_names=names)
def A2_Z2(self, names='x y', base_ring=QQ):
r"""
Construct the orbifold `\mathbb{A}^2 / \ZZ_2` as a toric
variety.
INPUT:
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`.
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)
"""
return self._make_ToricVariety('A2_Z2', names, base_ring)
def P1xA1(self, names='s t z', base_ring=QQ):
r"""
Construct the cartesian product `\mathbb{P}^1 \times \mathbb{A}^1` as
a toric variety.
INPUT:
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`.
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)
"""
return self._make_ToricVariety('P1xA1', names, base_ring)
def Conifold(self, names='u x y v', base_ring=QQ):
r"""
Construct the conifold as a toric variety.
INPUT:
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`.
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)
"""
return self._make_ToricVariety('Conifold', names, base_ring)
def dP6xdP6(self, names='x0 x1 x2 x3 x4 x5 y0 y1 y2 y3 y4 y5', base_ring=QQ):
r"""
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
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
return self._make_CPRFanoToricVariety('dP6xdP6', names, base_ring)
def Cube_face_fan(self, names='z+', base_ring=QQ):
r"""
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
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
return self._make_CPRFanoToricVariety('Cube_face_fan', names, base_ring)
def Cube_sublattice(self, names='z+', base_ring=QQ):
r"""
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
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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
"""
return self._make_CPRFanoToricVariety('Cube_sublattice', names, base_ring)
def Cube_nonpolyhedral(self, names='z+', base_ring=QQ):
r"""
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
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`.
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)
"""
return self._make_ToricVariety('Cube_nonpolyhedral', names, base_ring)
def Cube_deformation(self,k, names=None, base_ring=QQ):
r"""
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
:meth:`Cube_face_fan`.
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`
`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
"""
try:
k = ZZ(k)
except TypeError:
raise TypeError("cube deformations X_k are defined only for "
"non-negative integer k!\nGot: %s" % k)
if k < 0:
raise ValueError("cube deformations X_k are defined only for "
"non-negative k!\nGot: %s" % k)
rays = lambda kappa: matrix([[ 1, 1, 2*kappa+1],[ 1,-1, 1],[-1, 1, 1],[-1,-1, 1],
[-1,-1,-1],[-1, 1,-1],[ 1,-1,-1],[ 1, 1,-1]])
cones = [[0,1,2,3],[4,5,6,7],[0,1,7,6],[4,5,3,2],[0,2,5,7],[4,6,1,3]]
fan = Fan(cones, rays(k))
return ToricVariety(fan, coordinate_names=names)
def BCdlOG(self, names='v1 v2 c1 c2 v4 v5 b e1 e2 e3 f g v6', base_ring=QQ):
r"""
Construct the 5-dimensional toric variety studied in
[BCdlOG]_, [HLY]_
INPUT:
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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
"""
return self._make_CPRFanoToricVariety('BCdlOG', names, base_ring)
def BCdlOG_base(self, names='d4 d3 r2 r1 d2 u d1', base_ring=QQ):
r"""
Construct the base of the `\mathbb{P}^2(1,2,3)` fibration
:meth:`BCdlOG`.
INPUT:
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`.
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)
"""
return self._make_ToricVariety('BCdlOG_base', names, base_ring)
def P2_112(self, names='z+', base_ring=QQ):
r"""
Construct the weighted projective space
`\mathbb{P}^2(1,1,2)`.
INPUT:
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
return self._make_CPRFanoToricVariety('P2_112', names, base_ring)
def P2_123(self, names='z+', base_ring=QQ):
r"""
Construct the weighted projective space
`\mathbb{P}^2(1,2,3)`.
INPUT:
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
return self._make_CPRFanoToricVariety('P2_123', names, base_ring)
def P4_11169(self, names='z+', base_ring=QQ):
r"""
Construct the weighted projective space
`\mathbb{P}^4(1,1,1,6,9)`.
INPUT:
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
return self._make_CPRFanoToricVariety('P4_11169', names, base_ring)
def P4_11169_resolved(self, names='z+', base_ring=QQ):
r"""
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
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
return self._make_CPRFanoToricVariety('P4_11169_resolved', names, base_ring)
def P4_11133(self, names='z+', base_ring=QQ):
"""
Construct the weighted projective space
`\mathbb{P}^4(1,1,1,3,3)`.
INPUT:
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
return self._make_CPRFanoToricVariety('P4_11133', names, base_ring)
def P4_11133_resolved(self, names='z+', base_ring=QQ):
"""
Construct the weighted projective space
`\mathbb{P}^4(1,1,1,3,3)`.
INPUT:
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
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)
"""
return self._make_CPRFanoToricVariety('P4_11133_resolved', names, base_ring)
def WP(self, *q, **kw):
r"""
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
:func:`~sage.schemes.toric.variety.normalize_names` for
acceptable formats.
OUTPUT:
- A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`. 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
"""
if len(q)==1:
if isinstance(q[0], (list, tuple)):
q = q[0]
q = list(q)
m = len(q)
if m < 2:
raise ValueError("more than one weight must be provided (got %s)" % q)
for i in range(m):
try:
q[i] = ZZ(q[i])
except(TypeError):
raise TypeError("the weights (=%s) must be integers" % q)
if q[i] <= 0:
raise ValueError("the weights (=%s) must be positive integers" % q)
if not gcd(q) == 1:
raise ValueError("the weights (=%s) must be relatively prime" % q)
base_ring = QQ
names = 'z+'
for key in kw:
if key == 'K':
base_ring = kw['K']
elif key == 'base_ring':
base_ring = kw['base_ring']
elif key == 'names':
names = kw['names']
names = normalize_names(names, m, DEFAULT_PREFIX)
else:
raise TypeError("got an unexpected keyword argument %r" % key)
if base_ring not in _Fields:
raise TypeError("base_ring (=%r) must be a field" % base_ring)
L = ToricLattice(m)
L_sub = L.submodule([L(q)])
Q = L/L_sub
rays = []
cones = []
w = range(m)
L_basis = L.basis()
for i in w:
b = L_basis[i]
v = Q.coordinate_vector(Q(b))
rays = rays + [v]
w_c = w[:i] + w[i+1:]
cones = cones + [tuple(w_c)]
fan = Fan(cones,rays)
return ToricVariety(fan, coordinate_names=names, base_ring=base_ring)
def torus(self, n, names='z+', base_ring=QQ):
r"""
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
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`.
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]]
"""
try:
n = ZZ(n)
except TypeError:
raise TypeError('dimension of the torus must be an integer')
if n < 0:
raise ValueError('dimension must be non-negative')
N = ToricLattice(n)
fan = Fan([], lattice=N)
return ToricVariety(fan, coordinate_names=names, base_field=base_ring)
toric_varieties = ToricVarietyFactory()
