def P(self, n, names='z+', base_ring=QQ):
r"""
Construct the ``n``-dimensional projective space `\mathbb{P}^n`.
INPUT:
- ``n`` -- positive integer. The dimension of the projective space.
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
EXAMPLES::
sage: P3 = toric_varieties.P(3)
sage: P3
3-d CPR-Fano toric variety covered by 4 affine patches
sage: P3.fan().rays()
N( 1, 0, 0),
N( 0, 1, 0),
N( 0, 0, 1),
N(-1, -1, -1)
in 3-d lattice N
sage: P3.gens()
(z0, z1, z2, z3)
"""
# We are going to eventually switch off consistency checks, so we need
# to be sure that the input is acceptable.
try:
n = ZZ(n) # make sure that we got a "mathematical" integer
except TypeError:
raise TypeError("dimension of the projective space must be a "
"positive integer!\nGot: %s" % n)
if n <= 0:
raise ValueError("only projective spaces of positive dimension "
"can be constructed!\nGot: %s" % n)
m = identity_matrix(n).augment(matrix(n, 1, [-1]*n))
charts = [list(range(i)) + list(range(i + 1, n + 1))
for i in range(n + 1)]
return CPRFanoToricVariety(
Delta_polar=LatticePolytope(m.columns(), lattice=ToricLattice(n)),
charts=charts, check=self._check, coordinate_names=names,
base_ring=base_ring)
def _make_CPRFanoToricVariety(self, name, coordinate_names, base_ring):
r"""
Construct a (crepant partially resolved) Fano toric variety
and cache the result.
INPUT:
- ``name`` -- string. One of the pre-defined names in the
``toric_varieties_rays_cones`` data structure.
- ``coordinate_names`` -- A string describing the names of the
homogeneous coordinates of the toric variety.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
EXAMPLES::
sage: toric_varieties.P2() # indirect doctest
2-d CPR-Fano toric variety covered by 3 affine patches
"""
rays, cones = toric_varieties_rays_cones[name]
if coordinate_names is None:
dict_key = (name, base_ring)
else:
coordinate_names = normalize_names(coordinate_names, len(rays),
DEFAULT_PREFIX)
dict_key = (name, base_ring) + tuple(coordinate_names)
if dict_key not in self.__dict__:
polytope = LatticePolytope(rays,
lattice=ToricLattice(len(rays[0])))
points = [tuple(_) for _ in polytope.points()]
ray2point = [points.index(r) for r in rays]
charts = [[ray2point[i] for i in c] for c in cones]
self.__dict__[dict_key] = \
CPRFanoToricVariety(Delta_polar=polytope,
coordinate_points=ray2point,
charts=charts,
coordinate_names=coordinate_names,
base_ring=base_ring,
check=self._check)
return self.__dict__[dict_key]
def _make_CPRFanoToricVariety(self, name, coordinate_names):
"""
Construct a (crepant partially resolved) Fano toric variety
and cache the result.
INPUT:
- ``name`` -- string. One of the pre-defined names in the
``toric_varieties_rays_cones`` data structure.
- ``coordinate_names`` -- A string describing the names of the
homogeneous coordinates of the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
EXAMPLES::
sage: toric_varieties.P2() # indirect doctest
2-d CPR-Fano toric variety covered by 3 affine patches
"""
rays, cones = toric_varieties_rays_cones[name]
if coordinate_names is None:
dict_key = '_cached_'+name
else:
coordinate_names = normalize_names(coordinate_names, len(rays),
DEFAULT_PREFIX)
dict_key = '_cached_'+name+'_'+'_'.join(coordinate_names)
if dict_key not in self.__dict__:
polytope = LatticePolytope( matrix(rays).transpose() )
points = map(tuple, polytope.points().columns())
ray2point = [points.index(r) for r in rays]
charts = [ [ray2point[i] for i in c] for c in cones ]
self.__dict__[dict_key] = \
CPRFanoToricVariety(Delta_polar=polytope,
coordinate_points=ray2point,
charts=charts,
coordinate_names=coordinate_names,
check=self._check)
return self.__dict__[dict_key]