r"""
Toric varieties
This module provides support for (normal) toric varieties, corresponding to
:class:`rational polyhedral fans <sage.geometry.fan.RationalPolyhedralFan>`.
See also :mod:`~sage.schemes.toric.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
:func:`AffineToricVariety` and :func:`ToricVariety`, although you may
also be interested in :func:`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]
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,
:func:`~sage.schemes.affine.affine_space.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
:func:`face fan <sage.geometry.fan.FaceFan>` of the "diamond"::
sage: diamond = lattice_polytope.octahedron(2)
sage: diamond.vertices()
[ 1 0 -1 0]
[ 0 1 0 -1]
sage: fan = FaceFan(diamond)
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1
2-d toric variety covered by 4 affine patches
sage: P1xP1.fan().rays()
N( 1, 0),
N( 0, 1),
N(-1, 0),
N( 0, -1)
in 2-d lattice N
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()
N(1, 0),
N(0, 1)
in 2-d lattice N
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
:meth:`~ToricVariety_field.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.octahedron(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 :meth:`~ToricVariety_field.resolve` method,
but (at least for now) you will have to specify which rays should be inserted
into the fan. See also
:func:`~sage.schemes.toric.fano_variety.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 :class:`cohomology ring
<CohomologyRing>` and the :mod:`Chow group
<sage.schemes.toric.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
.. math::
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!
"""
import sys
from sage.functions.all import factorial
from sage.geometry.cone import Cone, is_Cone
from sage.geometry.fan import Fan
from sage.matrix.all import matrix
from sage.misc.all import latex, prod, uniq, cached_method
from sage.structure.unique_representation import UniqueRepresentation
from sage.modules.free_module_element import vector
from sage.rings.all import Infinity, PolynomialRing, ZZ, QQ
from sage.rings.quotient_ring_element import QuotientRingElement
from sage.rings.quotient_ring import QuotientRing_generic
from sage.schemes.affine.affine_space import AffineSpace
from sage.schemes.generic.ambient_space import AmbientSpace
from sage.schemes.toric.homset import SchemeHomset_points_toric_field
from sage.categories.fields import Fields
from sage.misc.cachefunc import ClearCacheOnPickle
_Fields = Fields()
DEFAULT_PREFIX = "z"
def is_ToricVariety(x):
r"""
Check if ``x`` is a toric variety.
INPUT:
- ``x`` -- anything.
OUTPUT:
- ``True`` if ``x`` is a :class:`toric variety <ToricVariety_field>` 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.octahedron(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
"""
return isinstance(x, ToricVariety_field)
def ToricVariety(fan,
coordinate_names=None,
names=None,
coordinate_indices=None,
base_ring=QQ, base_field=None):
r"""
Construct a toric variety.
INPUT:
- ``fan`` -- :class:`rational polyhedral fan
<sage.geometry.fan.RationalPolyhedralFan>`;
- ``coordinate_names`` -- names of variables for the coordinate ring, see
:func:`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:
- :class:`toric variety <ToricVariety_field>`.
EXAMPLES:
We will create the product of two projective lines::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: fan.rays()
N( 1, 0),
N( 0, 1),
N(-1, 0),
N( 0, -1)
in 2-d lattice N
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
"""
if base_field is not None:
base_ring = base_field
if names is not None:
if coordinate_names is not None:
raise ValueError('You must not specify both coordinate_names and names!')
coordinate_names = names
if base_ring not in _Fields:
raise TypeError("need a field to construct a toric variety!\n Got %s"
% base_ring)
return ToricVariety_field(fan, coordinate_names, coordinate_indices,
base_ring)
def AffineToricVariety(cone, *args, **kwds):
r"""
Construct an affine toric variety.
INPUT:
- ``cone`` -- :class:`strictly convex rational polyhedral cone
<sage.geometry.cone.ConvexRationalPolyhedralCone>`.
This cone will be used to construct a :class:`rational polyhedral fan
<sage.geometry.fan.RationalPolyhedralFan>`, which will be passed to
:func:`ToricVariety` with the rest of positional and keyword arguments.
OUTPUT:
- :class:`toric variety <ToricVariety_field>`.
.. 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.
"""
if not cone.is_strictly_convex():
raise ValueError("affine toric varieties are defined for strictly "
"convex cones only!")
fan = Fan([tuple(range(cone.nrays()))], cone.rays(),
check=False, normalize=False)
return ToricVariety(fan, *args, **kwds)
class ToricVariety_field(ClearCacheOnPickle, AmbientSpace):
r"""
Construct a toric variety associated to a rational polyhedral fan.
.. WARNING::
This class does not perform any checks of correctness of input. Use
:func:`ToricVariety` and :func:`AffineToricVariety` to construct toric
varieties.
INPUT:
- ``fan`` -- :class:`rational polyhedral fan
<sage.geometry.fan.RationalPolyhedralFan>`;
- ``coordinate_names`` -- names of variables, see :func:`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:
- :class:`toric variety <ToricVariety_field>`.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
"""
def __init__(self, fan, coordinate_names, coordinate_indices, base_field):
r"""
See :class:`ToricVariety_field` for documentation.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
"""
self._fan = fan
super(ToricVariety_field, self).__init__(fan.lattice_dim(),
base_field)
self._torus_factor_dim = fan.lattice_dim() - fan.dim()
coordinate_names = normalize_names(coordinate_names,
fan.nrays() + self._torus_factor_dim, DEFAULT_PREFIX,
coordinate_indices, return_prefix=True)
self._coordinate_prefix = coordinate_names.pop()
self._assign_names(names=coordinate_names, normalize=False)
def __cmp__(self, right):
r"""
Compare ``self`` and ``right``.
INPUT:
- ``right`` -- anything.
OUTPUT:
- 0 if ``right`` is of the same type as ``self``, their fans are the
same, names of variables are the same and stored in the same order,
and base fields are the same. 1 or -1 otherwise.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1a = ToricVariety(fan, "x s y t")
sage: P1xP1b = ToricVariety(fan)
sage: cmp(P1xP1, P1xP1a)
1
sage: cmp(P1xP1a, P1xP1)
-1
sage: cmp(P1xP1, P1xP1b)
0
sage: P1xP1 is P1xP1b
False
sage: cmp(P1xP1, 1) * cmp(1, P1xP1)
-1
"""
c = cmp(type(self), type(right))
if c:
return c
return cmp([self.fan(),
self.variable_names(),
self.base_ring()],
[right.fan(),
right.variable_names(),
right.base_ring()])
def _an_element_(self):
r"""
Construct an element of ``self``.
This function is needed (in particular) for the test framework.
OUTPUT:
- a point of ``self`` with coordinates [1 : 2: ... : n].
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1._an_element_()
[1 : 2 : 3 : 4]
"""
return self(range(1, self.ngens() + 1))
def _check_satisfies_equations(self, coordinates):
r"""
Check if ``coordinates`` define a valid point of ``self``.
INPUT:
- ``coordinates`` -- list of elements of the base field of ``self``.
OUTPUT:
- ``True`` if ``coordinates`` do define a valid point of ``self``,
otherwise a ``TypeError`` or ``ValueError`` exception is raised.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1._check_satisfies_equations([1,1,1,1])
True
sage: P1xP1._check_satisfies_equations([0,0,1,1])
True
sage: P1xP1._check_satisfies_equations([0,1,0,1])
Traceback (most recent call last):
...
TypeError: coordinates (0, 1, 0, 1)
are in the exceptional set!
sage: P1xP1._check_satisfies_equations([1,1,1])
Traceback (most recent call last):
...
TypeError: coordinates (1, 1, 1) must have 4 components!
sage: P1xP1._check_satisfies_equations(1)
Traceback (most recent call last):
...
TypeError: 1 can not be used as coordinates!
Use a list or a tuple.
sage: P1xP1._check_satisfies_equations([1,1,1,fan])
Traceback (most recent call last):
...
TypeError: coordinate Rational polyhedral fan
in 2-d lattice N is not an element of Rational Field!
"""
try:
coordinates = tuple(coordinates)
except TypeError:
raise TypeError("%s can not be used as coordinates! "
"Use a list or a tuple." % coordinates)
n = self.ngens()
if len(coordinates) != n:
raise TypeError("coordinates %s must have %d components!"
% (coordinates, n))
base_field = self.base_ring()
for coordinate in coordinates:
if coordinate not in base_field:
raise TypeError("coordinate %s is not an element of %s!"
% (coordinate, base_field))
zero_positions = set(position
for position, coordinate in enumerate(coordinates)
if coordinate == 0)
if not zero_positions:
return True
for i in range(n - self._torus_factor_dim, n):
if i in zero_positions:
raise ValueError("coordinates on the torus factor cannot be "
"zero! Got %s" % str(coordinates))
if len(zero_positions) == 1:
return True
fan = self.fan()
possible_charts = set(fan._ray_to_cones(zero_positions.pop()))
for i in zero_positions:
possible_charts.intersection_update(fan._ray_to_cones(i))
if possible_charts:
return True
raise TypeError("coordinates %s are in the exceptional set!"
% str(coordinates))
def _point_homset(self, *args, **kwds):
r"""
Construct a Hom-set for ``self``.
INPUT:
- same as for
:class:`~sage.schemes.generic.homset.SchemeHomset_points_toric_field`.
OUPUT:
-
:class:`~sage.schemes.generic.homset.SchemeHomset_points_toric_field`.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1._point_homset(Spec(QQ), P1xP1)
Set of rational points of 2-d toric variety
covered by 4 affine patches
"""
return SchemeHomset_points_toric_field(*args, **kwds)
def _latex_(self):
r"""
Return a LaTeX representation of ``self``.
OUTPUT:
- string.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1._latex_()
'\\mathbb{X}_{\\Sigma^{2}}'
"""
return r"\mathbb{X}_{%s}" % latex(self.fan())
def _latex_generic_point(self, coordinates=None):
"""
Return a LaTeX representation of a point of ``self``.
INPUT:
- ``coordinates`` -- list of coordinates of a point of ``self``.
If not given, names of coordinates of ``self`` will be used.
OUTPUT:
- string.
EXAMPLES::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1._latex_generic_point()
'\\left[z_{0} : z_{1} : z_{2} : z_{3}\\right]'
sage: P1xP1._latex_generic_point([1,2,3,4])
'\\left[1 : 2 : 3 : 4\\right]'
"""
if coordinates is None:
coordinates = self.gens()
return r"\left[%s\right]" % (" : ".join(str(latex(coord))
for coord in coordinates))
def _point(self, *args, **kwds):
r"""
Construct a point of ``self``.
INPUT:
- same as for
:class:`~sage.schemes.generic.morphism.SchemeMorphism_point_toric_field`.
OUPUT:
:class:`~sage.schemes.generic.morphism.SchemeMorphism_point_toric_field`.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1._point(P1xP1, [1,2,3,4])
[1 : 2 : 3 : 4]
"""
from sage.schemes.toric.morphism import SchemeMorphism_point_toric_field
return SchemeMorphism_point_toric_field(*args, **kwds)
def _homset(self, *args, **kwds):
r"""
Return the homset between two toric varieties.
INPUT:
Same as :class:`sage.schemes.generic.homset.SchemeHomset_generic`.
OUTPUT:
A :class:`sage.schemes.toric.homset.SchemeHomset_toric_variety`.
EXAMPLES::
sage: P1xP1 = toric_varieties.P1xP1()
sage: P1 = toric_varieties.P1()
sage: hom_set = P1xP1.Hom(P1); hom_set
Set of morphisms
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
sage: type(hom_set)
<class 'sage.schemes.toric.homset.SchemeHomset_toric_variety_with_category'>
This is also the Hom-set for algebraic subschemes of toric varieties::
sage: P1xP1.inject_variables()
Defining s, t, x, y
sage: P1 = P1xP1.subscheme(s-t)
sage: hom_set = P1xP1.Hom(P1)
sage: hom_set([s,s,x,y])
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: Closed subscheme of 2-d CPR-Fano toric variety covered by 4 affine patches defined by:
s - t
Defn: Defined on coordinates by sending [s : t : x : y] to
[s : s : x : y]
sage: hom_set = P1.Hom(P1)
sage: hom_set([s,s,x,y])
Scheme endomorphism of Closed subscheme of 2-d CPR-Fano toric
variety covered by 4 affine patches defined by:
s - t
Defn: Defined on coordinates by sending [s : t : x : y] to
[t : t : x : y]
"""
from sage.schemes.toric.homset import SchemeHomset_toric_variety
return SchemeHomset_toric_variety(*args, **kwds)
def _repr_(self):
r"""
Return a string representation of ``self``.
OUTPUT:
- string.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1._repr_()
'2-d toric variety covered by 4 affine patches'
"""
result = "%d-d" % self.dimension_relative()
if self.fan().ngenerating_cones() == 1:
result += " affine toric variety"
else:
result += (" toric variety covered by %d affine patches"
% self.fan().ngenerating_cones())
return result
def _repr_generic_point(self, coordinates=None):
r"""
Return a string representation of a point of ``self``.
INPUT:
- ``coordinates`` -- list of coordinates of a point of ``self``.
If not given, names of coordinates of ``self`` will be used.
OUTPUT:
- string.
EXAMPLES::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1._repr_generic_point()
'[z0 : z1 : z2 : z3]'
sage: P1xP1._repr_generic_point([1,2,3,4])
'[1 : 2 : 3 : 4]'
"""
if coordinates is None:
coordinates = self.gens()
return "[%s]" % (" : ".join(str(coord) for coord in coordinates))
def _validate(self, polynomials):
"""
Check if ``polynomials`` define valid functions on ``self``.
Since this is a toric variety, polynomials must be homogeneous in the
total coordinate ring of ``self``.
INPUT:
- ``polynomials`` -- list of polynomials in the coordinate ring of
``self`` (this function does not perform any conversions).
OUTPUT:
- ``polynomials`` (the input parameter without any modifications) if
``polynomials`` do define valid polynomial functions on ``self``,
otherwise a ``ValueError`` exception is raised.
TESTS:
We will use the product of two projective lines with coordinates
`(x, y)` for one and `(s, t)` for the other::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: fan.rays()
N( 1, 0),
N( 0, 1),
N(-1, 0),
N( 0, -1)
in 2-d lattice N
sage: P1xP1 = ToricVariety(fan, "x s y t")
sage: P1xP1.inject_variables()
Defining x, s, y, t
sage: P1xP1._validate([x - y, x*s + y*t])
[x - y, x*s + y*t]
sage: P1xP1._validate([x + s])
Traceback (most recent call last):
...
ValueError: x + s is not homogeneous on
2-d toric variety covered by 4 affine patches!
"""
for p in polynomials:
if not self.is_homogeneous(p):
raise ValueError("%s is not homogeneous on %s!" % (p, self))
return polynomials
def affine_patch(self, i):
r"""
Return the ``i``-th affine patch of ``self``.
INPUT:
- ``i`` -- integer, index of a generating cone of the fan of ``self``.
OUTPUT:
- affine :class:`toric variety <ToricVariety_field>` 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 :meth:`affine_algebraic_patch`, which expresses the
patches as subvarieties of affine space instead.
EXAMPLES::
sage: fan = FaceFan(lattice_polytope.octahedron(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
"""
i = int(i)
try:
return self._affine_patches[i]
except AttributeError:
self._affine_patches = dict()
except KeyError:
pass
cone = self.fan().generating_cone(i)
names = self.variable_names()
n = self.fan().nrays()
t = self._torus_factor_dim
names = ([names[ray] for ray in cone.ambient_ray_indices()]
+ list(names[n:]))
patch = AffineToricVariety(cone, names, base_field=self.base_ring())
embedding_coordinates = [1] * n
for k, ray in enumerate(cone.ambient_ray_indices()):
embedding_coordinates[ray] = patch.gen(k)
if t > 0:
embedding_coordinates.extend(patch.gens()[-t:])
patch._embedding_morphism = patch.hom(embedding_coordinates, self)
self._affine_patches[i] = patch
return patch
def change_ring(self, F):
r"""
Return a toric variety over ``F`` and otherwise the same as ``self``.
INPUT:
- ``F`` -- field.
OUTPUT:
- :class:`toric variety <ToricVariety_field>` 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
:meth:`base_extend` instead.
EXAMPLES::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1.base_ring()
Rational Field
sage: P1xP1_RR = P1xP1.change_ring(RR)
sage: P1xP1_RR.base_ring()
Real Field with 53 bits of precision
sage: P1xP1_QQ = P1xP1_RR.change_ring(QQ)
sage: P1xP1_QQ.base_ring()
Rational Field
sage: P1xP1_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)!
"""
if self.base_ring() == F:
return self
else:
return ToricVariety(self.fan(), self.variable_names(),
base_field=F)
def coordinate_ring(self):
r"""
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: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1.coordinate_ring()
Multivariate Polynomial Ring in z0, z1, z2, z3
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'>
"""
if "_coordinate_ring" not in self.__dict__:
names = self.variable_names()
self._coordinate_ring = PolynomialRing(self.base_ring(), len(names), names)
return self._coordinate_ring
def embedding_morphism(self):
r"""
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:
- :class:`scheme morphism
<sage.schemes.generic.morphism.SchemeMorphism_polynomial_toric_variety>`
if the default embedding morphism was defined for ``self``,
otherwise a ``ValueError`` exception is raised.
EXAMPLES::
sage: fan = FaceFan(lattice_polytope.octahedron(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]
"""
try:
return self._embedding_morphism
except AttributeError:
raise ValueError("no default embedding was defined for this "
"toric variety!")
def fan(self, dim=None, codim=None):
r"""
Return the underlying fan of ``self`` or its cones.
INPUT:
- ``dim`` -- dimension of the requested cones;
- ``codim`` -- codimension of the requested cones.
OUTPUT:
- :class:`rational polyhedral fan
<sage.geometry.fan.RationalPolyhedralFan>` if no parameters were
given, :class:`tuple` of :class:`cones
<sage.geometry.cone.ConvexRationalPolyhedralCone>` otherwise.
EXAMPLES::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1.fan()
Rational polyhedral fan in 2-d lattice N
sage: P1xP1.fan() is fan
True
sage: P1xP1.fan(1)[0]
1-d cone of Rational polyhedral fan in 2-d lattice N
"""
return self._fan(dim, codim)
def inject_coefficients(self, scope=None, verbose=True):
r"""
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.octahedron(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
"""
if scope is None:
depth = 0
while True:
scope = sys._getframe(depth).f_globals
if (scope["__name__"] == "__main__"
and scope["__package__"] is None):
break
depth += 1
try:
self.base_ring().inject_variables(scope, verbose)
except AttributeError:
pass
@cached_method
def dimension_singularities(self):
r"""
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
"""
for codim in range(0,self.dimension()+1):
if any(not cone.is_smooth() for cone in self.fan(codim)):
return self.dimension() - codim
return -1
def is_homogeneous(self, polynomial):
r"""
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: fan = FaceFan(lattice_polytope.octahedron(2))
sage: fan.rays()
N( 1, 0),
N( 0, 1),
N(-1, 0),
N( 0, -1)
in 2-d lattice N
sage: P1xP1 = ToricVariety(fan, "x s y t")
sage: P1xP1.inject_variables()
Defining x, s, y, t
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
"""
if '_homogeneous_degrees_group' not in self.__dict__:
fan = self.fan()
from sage.modules.free_module import FreeModule
degrees_group = FreeModule(ZZ, fan.nrays()).quotient(
fan.rays().matrix().columns())
self._homogeneous_degrees_group = degrees_group
degrees_group = self._homogeneous_degrees_group
S = self.coordinate_ring()
try:
polynomial = S(polynomial)
except TypeError:
polynomial = S(polynomial.lift())
monomials = polynomial.monomials()
if not monomials:
return True
degree = degrees_group(vector(ZZ,monomials[0].degrees()))
for monomial in monomials:
if degrees_group(vector(ZZ,monomial.degrees())) != degree:
return False
return True
def is_isomorphic(self, another):
r"""
Check if ``self`` is isomorphic to ``another``.
INPUT:
- ``another`` - :class:`toric variety <ToricVariety_field>`.
OUTPUT:
- ``True`` if ``self`` and ``another`` are isomorphic,
``False`` otherwise.
EXAMPLES::
sage: fan1 = FaceFan(lattice_polytope.octahedron(2))
sage: TV1 = ToricVariety(fan1)
sage: fan2 = NormalFan(lattice_polytope.octahedron(2))
sage: TV2 = ToricVariety(fan2)
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!
"""
if self is another:
return True
if not is_ToricVariety(another):
raise TypeError(
"only another toric variety can be checked for isomorphism! "
"Got %s" % another)
raise NotImplementedError("isomorphism check is not yet implemented!")
def is_affine(self):
r"""
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
:func:`AffineToricVariety`.
OUTPUT:
Boolean.
EXAMPLES::
sage: toric_varieties.A2().is_affine()
True
sage: toric_varieties.P1xA1().is_affine()
False
"""
return self.fan().ngenerating_cones() == 1
def is_complete(self):
r"""
Check if ``self`` is complete.
OUTPUT:
- ``True`` if ``self`` is complete and ``False`` otherwise.
EXAMPLES::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1.is_complete()
True
sage: P1xP1.affine_patch(0).is_complete()
False
"""
return self.fan().is_complete()
def is_orbifold(self):
r"""
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.octahedron(2))
sage: P1xP1 = ToricVariety(fan1)
sage: P1xP1.is_orbifold()
True
sage: fan2 = NormalFan(lattice_polytope.octahedron(3))
sage: TV = ToricVariety(fan2)
sage: TV.is_orbifold()
False
"""
return self.fan().is_simplicial()
def is_smooth(self):
r"""
Check if ``self`` is smooth.
OUTPUT:
- ``True`` if ``self`` is smooth and ``False`` otherwise.
EXAMPLES::
sage: diamond = lattice_polytope.octahedron(2)
sage: fan1 = FaceFan(diamond)
sage: P1xP1 = ToricVariety(fan1)
sage: P1xP1.is_smooth()
True
sage: fan2 = NormalFan(lattice_polytope.octahedron(2))
sage: TV = ToricVariety(fan2)
sage: TV.is_smooth()
False
"""
return self.fan().is_smooth()
@cached_method
def Kaehler_cone(self):
r"""
Return the closure of the Kähler cone of ``self``.
OUTPUT:
- :class:`cone <sage.geometry.cone.ConvexRationalPolyhedralCone>`.
.. NOTE::
This cone sits in the rational divisor class group of ``self`` and
the choice of coordinates agrees with
:meth:`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
"""
fan = self.fan()
GT = fan.Gale_transform().columns()
from sage.schemes.toric.divisor import \
ToricRationalDivisorClassGroup_basis_lattice
L = ToricRationalDivisorClassGroup_basis_lattice(
self.rational_class_group())
n = fan.nrays()
K = None
for cone in fan:
sigma = Cone([GT[i] for i in range(n)
if i not in cone.ambient_ray_indices()],
lattice=L)
K = K.intersection(sigma) if K is not None else sigma
return K
@cached_method
def Mori_cone(self):
r"""
Returns the Mori cone of ``self``.
OUTPUT:
- :class:`cone <sage.geometry.cone.ConvexRationalPolyhedralCone>`.
.. 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()
(0, 0, 1, 1, 1, -3, 0),
(3, 2, 0, 0, 0, 1, -6)
in Ambient free module of rank 7
over the principal ideal domain Integer Ring
"""
rays = (ray * self._fan.Gale_transform()
for ray in self.Kaehler_cone().dual().rays())
return Cone(rays, lattice=ZZ**(self._fan.nrays()+1))
def plot(self, **options):
r"""
Plot ``self``, i.e. the corresponding fan.
INPUT:
- any options for toric plots (see :func:`toric_plotter.options
<sage.geometry.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()
"""
if "ray_label" not in options:
gens = self.coordinate_ring().gens()
if self.fan().lattice().degree() <= 2:
options["ray_label"] = ["$%s$" % latex(z) for z in gens]
else:
options["ray_label"] = [str(z) for z in gens]
return self.fan().plot(**options)
def rational_class_group(self):
r"""
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:
- :class:`rational divisor class group
<sage.schemes.toric.divisor.ToricRationalDivisorClassGroup>`.
.. NOTE::
* Coordinates correspond to the rows of
``self.fan().gale_transform()``.
* :meth:`Kaehler_cone` yields a cone in this group.
EXAMPLES::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1.rational_class_group()
The toric rational divisor class group
of a 2-d toric variety covered by 4 affine patches
"""
from sage.schemes.toric.divisor import ToricRationalDivisorClassGroup
return ToricRationalDivisorClassGroup(self)
def Chow_group(self, base_ring=ZZ):
r"""
Return the toric Chow group.
INPUT:
- ``base_ring`` -- either ``ZZ`` (default) or ``QQ``. The
coefficient ring of the Chow group.
OUTPUT:
A :class:`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 ))
"""
from sage.schemes.toric.chow_group import ChowGroup
return ChowGroup(self,base_ring)
def cartesian_product(self, other,
coordinate_names=None, coordinate_indices=None):
r"""
Return the Cartesian product of ``self`` with ``other``.
INPUT:
- ``other`` -- a :class:`toric variety <ToricVariety_field>`;
- ``coordinate_names`` -- names of variables for the coordinate ring,
see :func:`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 :class:`toric variety <ToricVariety_field>`.
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
"""
return ToricVariety(self.fan().cartesian_product(other.fan()),
coordinate_names, coordinate_indices,
base_field=self.base_ring())
def resolve(self, **kwds):
r"""
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 :func:`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
:meth:`~sage.geometry.fan.RationalPolyhedralFan.subdivide` method of
the underlying :class:`rational polyhedral fan
<sage.geometry.fan.RationalPolyhedralFan>`, see its documentation
for the available options.
OUTPUT:
- :class:`toric variety <ToricVariety_field>`.
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.octahedron(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)
"""
coordinate_names = kwds.pop("coordinate_names", None)
coordinate_indices = kwds.pop("coordinate_indices", None)
fan = self.fan()
if fan.dim() != fan.lattice_dim():
raise NotImplementedError("resolution of toric varieties with "
"torus factors is not yet implemented!")
rfan = fan.subdivide(**kwds)
if coordinate_names is None:
coordinate_names = list(self.variable_names())
if coordinate_indices is None:
coordinate_indices = range(fan.nrays(), rfan.nrays())
else:
coordinate_indices = coordinate_indices[fan.nrays():]
coordinate_names.extend(normalize_names(
ngens=rfan.nrays() - fan.nrays(),
indices=coordinate_indices,
prefix=self._coordinate_prefix))
coordinate_names.append(self._coordinate_prefix + "+")
resolution = ToricVariety(rfan, coordinate_names=coordinate_names,
coordinate_indices=coordinate_indices,
base_field=self.base_ring())
R = self.coordinate_ring()
R_res = resolution.coordinate_ring()
resolution_map = resolution.hom(R.hom(R_res.gens()[:R.ngens()]), self)
resolution._resolution_map = resolution_map
return resolution
def resolve_to_orbifold(self, **kwds):
r"""
Construct an orbifold whose fan subdivides the fan of ``self``.
It is a synonym for :meth:`resolve` with ``make_simplicial=True``
option.
INPUT:
- this function accepts only keyword arguments. See :meth:`resolve`
for documentation.
OUTPUT:
- :class:`toric variety <ToricVariety_field>`.
EXAMPLES::
sage: fan = NormalFan(lattice_polytope.octahedron(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
"""
return self.resolve(make_simplicial=True, **kwds)
def subscheme(self, polynomials):
r"""
Return the subscheme of ``self`` defined by ``polynomials``.
INPUT:
- ``polynomials`` -- list of polynomials in the coordinate ring of
``self``.
OUTPUT:
- :class:`subscheme of a toric variety
<sage.schemes.generic.algebraic_scheme.AlgebraicScheme_subscheme_toric>`.
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: fan = FaceFan(lattice_polytope.octahedron(2))
sage: fan.rays()
N( 1, 0),
N( 0, 1),
N(-1, 0),
N( 0, -1)
in 2-d lattice N
sage: P1xP1 = ToricVariety(fan, "x s y t")
sage: P1xP1.inject_variables()
Defining x, s, y, t
sage: X = P1xP1.subscheme([x*s + y*t, x^3+y^3])
sage: X
Closed subscheme of 2-d 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, s, y, 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 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
"""
from sage.schemes.generic.algebraic_scheme import \
AlgebraicScheme_subscheme_toric, AlgebraicScheme_subscheme_affine_toric
if self.is_affine():
return AlgebraicScheme_subscheme_affine_toric(self, polynomials)
else:
return AlgebraicScheme_subscheme_toric(self, polynomials)
def Stanley_Reisner_ideal(self):
"""
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
"""
if "_SR" not in self.__dict__:
R = PolynomialRing(QQ, self.variable_names())
self._SR = self._fan.Stanley_Reisner_ideal(R)
return self._SR
def linear_equivalence_ideal(self):
"""
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
"""
if "_linear_equivalence_ideal" not in self.__dict__:
R = PolynomialRing(QQ, self.variable_names())
self._linear_equivalence_ideal = self._fan.linear_equivalence_ideal(R)
return self._linear_equivalence_ideal
@cached_method
def cohomology_ring(self):
r"""
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:`15050`
and :trac:`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
"""
if self.base_ring().characteristic()>0:
raise NotImplementedError('Only characteristic 0 base fields '
'are implemented.')
return CohomologyRing(self)
@cached_method
def cohomology_basis(self, d=None):
r"""
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
"""
if d!=None:
return self.cohomology_basis()[d]
H = self.cohomology_ring()
basis = [[] for d in range(self.dimension() + 1)]
for x in H.defining_ideal().normal_basis():
basis[x.total_degree()].append(x)
return tuple(tuple(H(x) for x in dbasis)
for dbasis in basis)
@cached_method
def volume_class(self):
r"""
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 :class:`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.
"""
if not self.is_orbifold():
raise NotImplementedError('Cohomology computations are only '
'implemented for orbifolds.')
HH = self.cohomology_ring()
dim = self.dimension_relative()
dVol = HH(self.fan().generating_cone(0)).part_of_degree(dim)
if dVol.is_zero():
raise ValueError, 'Volume class does not exist.'
return dVol
def integrate(self, 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 :meth:`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]
"""
assert self.is_complete(), "Can only integrate over compact varieties."
top_form = cohomology_class.part_of_degree(self.dimension())
if top_form.is_zero(): return 0
return top_form.lc() / self.volume_class().lc()
@cached_method
def Chern_class(self, 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
"""
assert self.is_orbifold(), "Requires the toric variety to be an orbifold."
c = prod([ 1+self.cohomology_ring().gen(i) for i in range(0,self._fan.nrays()) ])
if deg==None:
return c
else:
return c.part_of_degree(deg)
@cached_method
def Chern_character(self, 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
"""
assert self.is_orbifold(), "Requires the toric variety to be an orbifold."
n_rels = self._fan.nrays() - self.dimension()
ch = sum([ self.cohomology_ring().gen(i).exp()
for i in range(0,self._fan.nrays()) ]) - n_rels
if deg==None:
return ch
else:
return ch.part_of_degree(deg)
@cached_method
def Todd_class(self, 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
"""
Td = QQ(1)
if self.dimension() >= 1:
c1 = self.Chern_class(1)
Td += QQ(1)/2 * c1
if self.dimension() >= 2:
c2 = self.Chern_class(2)
Td += QQ(1)/12 * (c1**2 + c2)
if self.dimension() >= 3:
Td += QQ(1)/24 * c1*c2
if self.dimension() >= 4:
c3 = self.Chern_class(3)
c4 = self.Chern_class(4)
Td += -QQ(1)/720 * (c1**4 -4*c1**2*c2 -3*c2**2 -c1*c3 +c4)
if self.dimension() >= 5:
raise NotImplementedError('Todd class is currently only implemented up to degree 4')
if deg==None:
return Td
else:
return Td.part_of_degree(deg)
c = Chern_class
ch = Chern_character
Td = Todd_class
def Euler_number(self):
"""
Return the topological Euler number of the toric variety.
Sometimes, this is also called the Euler
characteristic. :meth:`chi` is a synonym for
:meth:`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
"""
if "_chi" not in self.__dict__:
if self.is_complete():
chi = self.integrate(self.Chern_class())
else:
chi=0
H = self.cohomology_basis()
for d in range(0, self.dimension()+1):
chi += (-1)**d * len(H[d])
self._chi = chi
return self._chi
chi = Euler_number
def K(self):
r"""
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
"""
from sage.schemes.toric.divisor import ToricDivisor
return ToricDivisor(self, [-1]*self._fan.nrays())
def divisor(self, arg, base_ring=None, check=True, reduce=True):
r"""
Return a divisor.
INPUT:
The arguments are the same as in
:func:`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 :meth:`fan` of ``self``.
OUTPUT:
- A :class:`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:`12812` is resolved::
sage: sum(dP6.divisor(i) for i in range(3))
V(x) + V(u) + V(y)
"""
if arg in ZZ:
arg = [(1, self.gen(arg))]
check = True
reduce = False
from sage.schemes.toric.divisor import ToricDivisor
return ToricDivisor(self, ring=base_ring, arg=arg,
check=check, reduce=reduce)
def divisor_group(self, base_ring=ZZ):
r"""
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
:class:`sage.schemes.generic.divisor_group.DivisorGroup_generic`.
.. WARNING::
You almost certainly want the group of toric divisors, see
:meth:`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)
"""
from sage.schemes.generic.divisor_group import DivisorGroup
return DivisorGroup(self, base_ring)
def toric_divisor_group(self, base_ring=ZZ):
r"""
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
:class:`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
"""
from sage.schemes.toric.divisor import ToricDivisorGroup
return ToricDivisorGroup(self, base_ring);
def _semigroup_ring(self, cone=None, names=None):
r"""
Return a presentation of the semigroup ring for the dual of ``cone``.
INPUT:
See :meth:`Spec`.
OUTPUT:
For the given ``cone`` `\sigma`, return a tuple consisting of
* a polynomial ring `R`,
* an ideal `I\in R`,
* the dual cone `\sigma^\vee`
such that `R/I \sim k[\sigma^\vee \cap M]`, where `k` is the
:meth:`base_ring` of the toric variety.
EXAMPLES::
sage: A2Z2 = Cone([(0,1),(2,1)])
sage: AffineToricVariety(A2Z2)._semigroup_ring()
(Multivariate Polynomial Ring in z0, z1, z2 over Rational Field,
Ideal (-z0*z1 + z2^2) of Multivariate Polynomial Ring in z0, z1, z2 over Rational Field,
2-d cone in 2-d lattice M)
sage: P2 = toric_varieties.P2()
sage: cone = P2.fan().generating_cone(0)
sage: P2._semigroup_ring(cone)
(Multivariate Polynomial Ring in z0, z1 over Rational Field,
Ideal (0) of Multivariate Polynomial Ring in z0, z1 over Rational Field,
2-d cone in 2-d lattice M)
sage: P2.change_ring(GF(101))._semigroup_ring(cone)
(Multivariate Polynomial Ring in z0, z1 over Finite Field of size 101,
Ideal (0) of Multivariate Polynomial Ring in z0, z1 over Finite Field of size 101,
2-d cone in 2-d lattice M)
"""
from sage.schemes.toric.ideal import ToricIdeal
if cone is None:
assert self.is_affine(), \
'You may only omit the cone argument for an affine toric variety!'
cone = self.fan().generating_cone(0)
cone = self.fan().embed(cone)
dual = cone.dual()
basis = dual.Hilbert_basis()
N = len(basis)
names = normalize_names(names, N, DEFAULT_PREFIX)
A = basis.column_matrix()
IA = ToricIdeal(A, names, base_ring=self.base_ring())
return (IA.ring(), IA, dual)
def Spec(self, cone=None, names=None):
r"""
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
.. MATH::
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 :class:`Cone
<sage.geometry.cone.ConvexRationalPolyhedralCone>`. 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 :func:`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)
"""
from sage.schemes.generic.spec import Spec
R, I, dualcone = self._semigroup_ring(cone, names)
return Spec(R.quotient(I))
def affine_algebraic_patch(self, cone=None, names=None):
r"""
Return the patch corresponding to ``cone`` as an affine
algebraic subvariety.
INPUT:
- ``cone`` -- a :class:`Cone
<sage.geometry.cone.ConvexRationalPolyhedralCone>` `\sigma`
of the fan. It can be omitted for an affine toric variety,
in which case the single generating cone is used.
OUTPUT:
A :class:`affine algebraic subscheme
<sage.schemes.generic.algebraic_scheme.AlgebraicScheme_subscheme_affine>`
corresponding to the patch `\mathop{Spec}(\sigma^\vee \cap M)`
associated to the cone `\sigma`.
See also :meth:`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
"""
R, I, dualcone = self._semigroup_ring(cone, names)
patch_cover = AffineSpace(R)
patch = patch_cover.subscheme(I)
return patch
def _orbit_closure_projection(self, cone, x):
r"""
Return the projection of ``x`` onto the quotient lattice of ``cone``.
INPUT:
- ``cone`` -- a :class:`cone
<sage.geometry.cone.ConvexRationalPolyhedralCone>` of the :meth:`fan`
of ``self``;
- ``x`` -- a lattice point or a cone of the :meth:`fan` of ``self``.
OUTPUT:
- the projection of ``x`` onto the quotient lattice of ``cone``, which
is either a lattice point or a cone depending on the type of ``x``.
This quotient lattice is the ambient lattice for the fan of the orbit
closure corresponding to ``cone``.
If ``x`` is a cone not in the star of ``cone``, an ``IndexError`` is
raised.
See :meth:`orbit_closure` for more details.
.. warning::
Due to incomplete support of quotient lattices (as of 12-07-2011),
this function actually operates with a generic toric lattice of the
same dimension as the qppropriate quotient lattice. This behaviour
is likely to change in the future releases of Sage.
EXAMPLES::
sage: P2 = toric_varieties.P2()
sage: H = P2.fan(1)[0]
sage: [P2._orbit_closure_projection(H, p) for p in P2.fan().rays()]
[(0), (1), (-1)]
sage: P2._orbit_closure_projection(H, P2.fan(2)[0])
1-d cone in 1-d lattice N
"""
cone = self.fan().embed(cone)
quot = cone.sublattice_quotient()
if x in cone.lattice():
return vector(quot(x))
assert is_Cone(x)
rays = [ vector(quot(r)) for r in x.rays() ]
return Cone(rays)
def orbit_closure(self, cone):
r"""
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 :class:`cone
<sage.geometry.cone.ConvexRationalPolyhedralCone>` of the fan.
OUTPUT:
- a torus orbit closure associated to ``cone`` as a
:class:`toric variety <ToricVariety_field>`.
EXAMPLES::
sage: P1xP1 = toric_varieties.P1xP1()
sage: H = P1xP1.fan(1)[0]
sage: P1xP1.orbit_closure(H)
1-d toric variety covered by 2 affine patches
"""
cone = self.fan().embed(cone)
cones = []
for star_cone in cone.star_generators():
cones.append( self._orbit_closure_projection(cone, star_cone) )
from sage.geometry.fan import discard_faces
return ToricVariety(Fan(discard_faces(cones), check=False))
def count_points(self):
r"""
Return the number of points of ``self``.
This is an alias for ``point_set().cardinality()``, see
:meth:`~sage.schemes.toric.homset.SchemeHomset_points_toric_field.cardinality`
for details.
EXAMPLES::
sage: o = lattice_polytope.octahedron(3)
sage: V = ToricVariety(FaceFan(o))
sage: V2 = V.change_ring(GF(2))
sage: V2.point_set().cardinality()
27
sage: V2.count_points()
27
"""
return self.point_set().cardinality()
@cached_method
def Demazure_roots(self):
"""
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.
"""
if not self.is_complete():
raise NotImplementedError('Demazure_roots() is only implemented '
'for complete toric varieties.')
antiK = -self.K()
fan_rays = self.fan().rays()
roots = [m for m in antiK.sections()
if [ray*m for ray in fan_rays].count(-1) == 1]
return tuple(roots)
def Aut_dimension(self):
r"""
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.
"""
if not self.is_complete():
raise NotImplementedError('Aut_dimension() is only implemented '
'for complete toric varieties.')
return self.fan().lattice_dim() + len(self.Demazure_roots())
def normalize_names(names=None, ngens=None, prefix=None, indices=None,
return_prefix=False):
r"""
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:
#. If necessary, split ``names`` into separate names.
#. If the last name ends with "+", put it into ``prefix``.
#. 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.
#. Check if constructed names are valid. See :func:`certify_names` for
details.
#. 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']
"""
if names is None:
names = []
elif isinstance(names, str):
names = names.replace(",", " ").split()
else:
try:
names = list(names)
except TypeError:
raise TypeError(
"names must be a string or a list or tuple of them!")
for name in names:
if not isinstance(name, str):
raise TypeError(
"names must be a string or a list or tuple of them!")
if names and names[-1].endswith("+"):
prefix = names.pop()[:-1]
if ngens is None:
ngens = len(names)
if len(names) < ngens:
if prefix is None:
raise IndexError("need %d names but only %d are given!"
% (ngens, len(names)))
if indices is None:
indices = range(ngens)
elif len(indices) != ngens:
raise ValueError("need exactly %d indices, but got %d!"
% (ngens, len(indices)))
names += [prefix + str(i) for i in indices[len(names):]]
if len(names) > ngens:
names = names[:ngens]
certify_names(names)
if return_prefix:
names.append(prefix)
return names
def certify_names(names):
r"""
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']
"""
for name in names:
if not name:
raise ValueError("name must be nonempty!")
if not name.isalnum() and not name.replace("_","").isalnum():
raise ValueError("name must be alphanumeric! Got %s" % name)
if not name[0].isalpha():
raise ValueError("name must start with a letter! Got %s" % name)
if len(set(names)) != len(names):
raise ValueError("names must be distinct! Got: %s" % names)
class CohomologyRing(QuotientRing_generic, UniqueRepresentation):
r"""
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
:meth:`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
:meth:`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
:meth:`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
"""
def __init__(self, variety):
r"""
See :class:`CohomologyRing` for documentation.
TESTS::
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
TESTS::
sage: cone1 = Cone([(1,0)]); cone2 = Cone([(1,0)])
sage: cone1 is cone2
False
sage: fan1 = Fan([cone1]); fan2 = Fan([cone2])
sage: fan1 is fan2
False
sage: X1 = ToricVariety(fan1); X2 = ToricVariety(fan2)
sage: X1 is X2
False
sage: X1.cohomology_ring() is X2.cohomology_ring() # see http://trac.sagemath.org/sage_trac/ticket/10325
True
sage: TDiv = X1.toric_divisor_group()
sage: X1.toric_divisor_group() is TDiv
True
sage: X2.toric_divisor_group() is TDiv
True
sage: TDiv.scheme() is X1 # as you expect
True
sage: TDiv.scheme() is X2 # perhaps less obvious, but toric_divisor_group is unique!
False
sage: TDiv.scheme() == X2 # isomorphic, but not necessarily identical
True
sage: TDiv.scheme().cohomology_ring() is X2.cohomology_ring() # this is where it gets tricky
True
sage: TDiv.gen(0).Chern_character() * X2.cohomology_ring().one()
[1]
"""
self._variety = variety
if not variety.is_orbifold():
raise NotImplementedError, 'Requires an orbifold toric variety.'
R = PolynomialRing(QQ, variety.variable_names())
self._polynomial_ring = R
I = variety._fan.linear_equivalence_ideal(R) + variety._fan.Stanley_Reisner_ideal(R)
super(CohomologyRing, self).__init__(R, I, names=variety.variable_names())
def _repr_(self):
r"""
Return a string representation of the cohomology ring.
OUTPUT:
A string.
EXAMPLES::
sage: toric_varieties.P2().cohomology_ring()._repr_()
'Rational cohomology ring of a 2-d CPR-Fano toric variety covered by 3 affine patches'
"""
return 'Rational cohomology ring of a '+self._variety._repr_()
def _latex_(self):
r"""
Return a latex representation of the cohomology ring.
OUTPUT:
A string.
EXAMPLES::
sage: cohomology_ring = toric_varieties.P2().cohomology_ring()
sage: cohomology_ring._latex_()
'H^\\ast\\left(\\mathbb{P}_{\\Delta^{2}},\\QQ\\right)'
"""
return 'H^\\ast\\left('+self._variety._latex_()+',\QQ\\right)'
def _element_constructor_(self,x):
r"""
Construct a :class:`CohomologyClass`.
INPUT::
- ``x`` -- something that defines a cohomology class. Either a
cohomology class, a cone of the fan, or something that can
be converted into a polynomial in the homogeneous
coordinates.
OUTPUT:
The :class:`CohomologyClass` defined by ``x``.
EXAMPLES::
sage: dP6 = toric_varieties.dP6()
sage: H = dP6.cohomology_ring()
sage: cone = dP6.fan().cone_containing(2,3); cone
2-d cone of Rational polyhedral fan in 2-d lattice N
sage: H(cone) # indirect doctest
[-w^2]
sage: H( Cone(cone) )
[-w^2]
sage: H( dP6.fan(0)[0] ) # trivial cone
[1]
Non-smooth cones are a bit tricky. For such a cone, the
intersection of the divisors corresponding to the rays is
still proportional to the product of the variables, but the
coefficient is a multiple depending on the orbifold
singularity. See also [CLS]_, Lemma 12.5.2::
sage: P2_123 = toric_varieties.P2_123()
sage: HH = P2_123.cohomology_ring()
sage: HH(Cone([(1,0)])) * HH(Cone([(-2,-3)]))
[2*z2^2]
sage: HH(Cone([(1,0), (-2,-3)]))
[6*z2^2]
sage: [ HH(c) for c in P2_123.fan().generating_cones() ]
[[6*z2^2], [6*z2^2], [6*z2^2]]
sage: P2_123.volume_class()
[6*z2^2]
sage: [ HH(c.facets()[0]) * HH(c.facets()[1]) for c in P2_123.fan().generating_cones() ]
[[6*z2^2], [3*z2^2], [2*z2^2]]
Numbers will be converted into the ring::
sage: P2 = toric_varieties.P2()
sage: H = P2.cohomology_ring()
sage: H._element_constructor_(1)
[1]
sage: H(1)
[1]
sage: type( H(1) )
<class 'sage.schemes.toric.variety.CohomologyClass'>
sage: P2.inject_variables()
Defining x, y, z
sage: H(1+x*y+z)
[z^2 + z + 1]
"""
fan = self._variety.fan()
if isinstance(x, CohomologyClass) and x.parent()==self:
return x
if isinstance(x, QuotientRingElement):
x = x.lift()
elif is_Cone(x):
cone = fan.embed(x)
assert cone.ambient() is fan
mult = cone.rays().column_matrix().index_in_saturation()
x = prod((self.cover_ring().gen(i) for i in cone.ambient_ray_indices()),
z=self.cover_ring().one()) * mult
else:
try:
return x.cohomology_class()
except AttributeError:
x = self.cover_ring()(x)
return CohomologyClass(self, x)
def __call__(self, x, coerce=True):
r"""
Turn ``x`` into a ``CohomologyClass``.
EXAMPLES::
sage: P2 = toric_varieties.P2()
sage: H = P2.cohomology_ring()
sage: H(1)
[1]
sage: type( H(1) )
<class 'sage.schemes.toric.variety.CohomologyClass'>
"""
return self._element_constructor_(x)
def gens(self):
r"""
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])
"""
if "_gens" not in self.__dict__:
self._gens = tuple( self.gen(i) for i in range(0,self._variety.fan().nrays()) )
return self._gens
def gen(self, i):
r"""
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]
"""
return CohomologyClass(self, self._polynomial_ring.gen(i))
def is_CohomologyClass(x):
r"""
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
:class:`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
"""
return isinstance(x,CohomologyClass)
class CohomologyClass(QuotientRingElement):
r"""
An element of the :class:`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
:meth:`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]
"""
def __init__(self, cohomology_ring, representative):
r"""
Construct the cohomology class.
INPUT:
- ``cohomology_ring`` -- :class:`CohomologyRing`.
- ``representative`` -- a polynomial in the generators of the cohomology ring.
OUTPUT:
An instance of :class:`CohomologyClass`.
EXAMPLES::
sage: P2 = toric_varieties.P2()
sage: H = P2.cohomology_ring()
sage: from sage.schemes.toric.variety import CohomologyClass
sage: CohomologyClass(H, H.defining_ideal().ring().zero() )
[0]
"""
assert representative in cohomology_ring.defining_ideal().ring(), \
'The given representative is not in the parent polynomial ring.'
super(CohomologyClass, self).__init__(cohomology_ring, representative)
def _repr_(self):
r"""
Return a string representation of the cohomology class.
OUTPUT:
A string.
EXAMPLES::
sage: toric_varieties.P2().cohomology_ring().gen(0)._repr_()
'[z]'
"""
return '['+super(CohomologyClass,self)._repr_()+']'
def _latex_(self):
r"""
Return a latex representation of the cohomology class.
OUTPUT:
A string.
EXAMPLES::
sage: cohomology_class = toric_varieties.P2().cohomology_ring().gen(0)^2/2
sage: cohomology_class._latex_()
'\\left[ \\frac{1}{2} z^{2} \\right]'
"""
return r'\left[ %s \right]' % latex(self.lift())
def deg(self):
r"""
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
"""
return self.lift().degree()
def part_of_degree(self, d):
r"""
Project the (mixed-degree) cohomology class to the given degree.
.. MATH::
\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]
"""
Q = self.parent()
p = filter( lambda x: x[1].total_degree() == d, self.lift() )
if len(p)==0:
return Q.zero()
else:
return Q(sum(x[0]*x[1] for x in p))
def exp(self):
"""
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]
"""
if not self.part_of_degree(0).is_zero():
raise ValueError, 'Must not have a constant part.'
exp_x = self.parent().one()
for d in range(1,self.parent()._variety.dimension()+1):
exp_x += self**d / factorial(d)
return exp_x
