def __init__(self, X, Y, category=None, check=True, base=ZZ):
"""
The Python constructor.
INPUT:
The same as for any homset, see
:mod:`~sage.categories.homset`.
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
An integral matrix defines a fan morphism, since we think of
the matrix as a linear map on the toric lattice. This is why
we need to ``register_conversion`` in the constructor
below. The result is::
sage: hom_set(matrix([[1],[0]]))
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 1-d lattice N.
"""
SchemeHomset_generic.__init__(self, X, Y, category=category, check=check, base=base)
from sage.schemes.toric.variety import is_ToricVariety
if is_ToricVariety(X) and is_ToricVariety(Y):
self.register_conversion(MatrixSpace(ZZ, X.fan().dim(), Y.fan().dim()))
def __init__(self, X, Y, category=None, check=True, base=ZZ):
"""
The Python constructor.
INPUT:
The same as for any homset, see
:mod:`~sage.categories.homset`.
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
An integral matrix defines a fan morphism, since we think of
the matrix as a linear map on the toric lattice. This is why
we need to ``register_conversion`` in the constructor
below. The result is::
sage: hom_set(matrix([[1],[0]]))
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 1-d lattice N.
"""
SchemeHomset_generic.__init__(self, X, Y, category=category, check=check, base=base)
from sage.schemes.toric.variety import is_ToricVariety
if is_ToricVariety(X) and is_ToricVariety(Y):
self.register_conversion(MatrixSpace(ZZ, X.fan().dim(), Y.fan().dim()))
def create_key_and_extra_args(self, toric_variety, base_ring=ZZ, check=True):
"""
Create a key that uniquely determines the :class:`ChowGroup_class`.
INPUT:
- ``toric_variety`` -- a toric variety.
- ``base_ring`` -- either `\ZZ` (default) or `\QQ`. The
coefficient ring of the Chow group.
- ``check`` -- boolean (default: ``True``).
EXAMPLES::
sage: from sage.schemes.toric.chow_group import *
sage: P2 = toric_varieties.P2()
sage: ChowGroup(P2, ZZ, check=True) == ChowGroup(P2, ZZ, check=False) # indirect doctest
True
"""
if not is_ToricVariety(toric_variety):
raise ValueError, 'First argument must be a toric variety.'
if not base_ring in [ZZ,QQ]:
raise ValueError, 'Base ring must be either ZZ or QQ.'
key = tuple([toric_variety, base_ring])
extra = {'check':check}
return key, extra
def TrivialBundle(X, rank=1):
r"""
Return the trivial bundle of rank ``r``.
INPUT:
- ``X`` -- a toric variety. The base space of the bundle.
- ``rank`` -- the rank of the bundle.
OUTPUT:
The trivial bundle as a Klyachko bundle.
EXAMPLES::
sage: P2 = toric_varieties.P2()
sage: from sage.schemes.toric.sheaf.constructor import TrivialBundle
sage: I3 = TrivialBundle(P2, 3); I3
Rank 3 bundle on 2-d CPR-Fano toric variety covered by 3 affine patches.
sage: I3.cohomology(weight=(0,0), dim=True)
(3, 0, 0)
"""
if not is_ToricVariety(X):
raise ValueError('not a toric variety')
from sage.modules.free_module import VectorSpace
base_ring = X.base_ring()
filtrations = dict(
[ray, FilteredVectorSpace(rank, 0, base_ring=base_ring)]
for ray in X.fan().rays())
import klyachko
return klyachko.Bundle(X, filtrations, check=True)
def TangentBundle(X):
r"""
Construct the tangent bundle of a toric variety.
INPUT:
- ``X`` -- a toric variety. The base space of the bundle.
OUTPUT:
The tangent bundle as a Klyachko bundle.
EXAMPLES::
sage: dP7 = toric_varieties.dP7()
sage: from sage.schemes.toric.sheaf.constructor import TangentBundle
sage: TangentBundle(dP7)
Rank 2 bundle on 2-d CPR-Fano toric variety covered by 5 affine patches.
"""
if not is_ToricVariety(X):
raise ValueError('not a toric variety')
base_ring = X.base_ring()
fan = X.fan()
filtrations = dict()
from sage.modules.filtered_vector_space import FilteredVectorSpace
for i, ray in enumerate(fan.rays()):
F = FilteredVectorSpace(fan.rays(), {0: range(fan.nrays()), 1: [i]})
filtrations[ray] = F
import klyachko
return klyachko.Bundle(X, filtrations, check=True)
def LineBundle(X, D):
"""
Construct the rank-1 bundle `O(D)`.
INPUT:
- ``X`` -- a toric variety. The base space of the bundle.
- ``D`` -- a toric divisor.
OUTPUT:
The line bundle `O(D)` as a Klyachko bundle of rank 1.
EXAMPLES::
sage: X = toric_varieties.dP8()
sage: D = X.divisor(0)
sage: from sage.schemes.toric.sheaf.constructor import LineBundle
sage: O_D = LineBundle(X, D)
sage: O_D.cohomology(dim=True, weight=(0,0))
(1, 0, 0)
"""
if not is_ToricVariety(X):
raise ValueError('not a toric variety')
from sage.modules.free_module import VectorSpace
base_ring = X.base_ring()
filtrations = dict([
X.fan().ray(i),
FilteredVectorSpace(1, D.function_value(i), base_ring=base_ring)
] for i in range(X.fan().nrays()))
import klyachko
return klyachko.Bundle(X, filtrations, check=True)
def TrivialBundle(X, rank=1):
r"""
Return the trivial bundle of rank ``r``.
INPUT:
- ``X`` -- a toric variety. The base space of the bundle.
- ``rank`` -- the rank of the bundle.
OUTPUT:
The trivial bundle as a Klyachko bundle.
EXAMPLES::
sage: P2 = toric_varieties.P2()
sage: from sage.schemes.toric.sheaf.constructor import TrivialBundle
sage: I3 = TrivialBundle(P2, 3); I3
Rank 3 bundle on 2-d CPR-Fano toric variety covered by 3 affine patches.
sage: I3.cohomology(weight=(0,0), dim=True)
(3, 0, 0)
"""
if not is_ToricVariety(X):
raise ValueError('not a toric variety')
from sage.modules.free_module import VectorSpace
base_ring = X.base_ring()
filtrations = dict([ray, FilteredVectorSpace(rank, 0, base_ring=base_ring)]
for ray in X.fan().rays())
import klyachko
return klyachko.Bundle(X, filtrations, check=True)
def TangentBundle(X):
r"""
Construct the tangent bundle of a toric variety.
INPUT:
- ``X`` -- a toric variety. The base space of the bundle.
OUTPUT:
The tangent bundle as a Klyachko bundle.
EXAMPLES::
sage: dP7 = toric_varieties.dP7()
sage: from sage.schemes.toric.sheaf.constructor import TangentBundle
sage: TangentBundle(dP7)
Rank 2 bundle on 2-d CPR-Fano toric variety covered by 5 affine patches.
"""
if not is_ToricVariety(X):
raise ValueError('not a toric variety')
base_ring = X.base_ring()
fan = X.fan()
filtrations = dict()
from sage.modules.filtered_vector_space import FilteredVectorSpace
for i, ray in enumerate(fan.rays()):
F = FilteredVectorSpace(fan.rays(), {0:range(fan.nrays()), 1:[i]})
filtrations[ray] = F
import klyachko
return klyachko.Bundle(X, filtrations, check=True)
def LineBundle(X, D):
"""
Construct the rank-1 bundle `O(D)`.
INPUT:
- ``X`` -- a toric variety. The base space of the bundle.
- ``D`` -- a toric divisor.
OUTPUT:
The line bundle `O(D)` as a Klyachko bundle of rank 1.
EXAMPLES::
sage: X = toric_varieties.dP8()
sage: D = X.divisor(0)
sage: from sage.schemes.toric.sheaf.constructor import LineBundle
sage: O_D = LineBundle(X, D)
sage: O_D.cohomology(dim=True, weight=(0,0))
(1, 0, 0)
"""
if not is_ToricVariety(X):
raise ValueError('not a toric variety')
from sage.modules.free_module import VectorSpace
base_ring = X.base_ring()
filtrations = dict([X.fan().ray(i),
FilteredVectorSpace(1, D.function_value(i), base_ring=base_ring)]
for i in range(X.fan().nrays()))
import klyachko
return klyachko.Bundle(X, filtrations, check=True)
