def __init__(self, base, ideal=None):
"""
The Python constructor.
EXAMPLES::
sage: Berkovich_Cp_Projective(3)
Projective Berkovich line over Cp(3) of precision 20
"""
if base in ZZ:
if base.is_prime():
base = ProjectiveSpace(Qp(base), 1)
else:
raise ValueError("non-prime passed into Berkovich space")
if base in NumberFields() or isinstance(base,
sage.rings.abc.pAdicField):
base = ProjectiveSpace(base, 1)
if not is_ProjectiveSpace(base):
try:
base = ProjectiveSpace(base)
except:
raise ValueError(
"base of projective Berkovich space must be projective space"
)
if not isinstance(base.base_ring(), sage.rings.abc.pAdicField):
if base.base_ring() not in NumberFields():
raise ValueError("base of projective Berkovich space must be " + \
"projective space over Qp or a number field")
else:
if ideal is None:
raise ValueError('passed a number field but not an ideal')
if base.base_ring() is not QQ:
if not isinstance(ideal, NumberFieldFractionalIdeal):
raise ValueError('ideal was not a number field ideal')
if ideal.number_field() != base.base_ring():
raise ValueError('passed number field ' + \
'%s but ideal was an ideal of %s' %(base.base_ring(), ideal.number_field()))
prime = ideal.smallest_integer()
else:
if ideal not in QQ:
raise ValueError('ideal was not an element of QQ')
prime = ideal
if not ideal.is_prime():
raise ValueError('passed non prime ideal')
self._base_type = 'number field'
else:
prime = base.base_ring().prime()
ideal = None
self._base_type = 'padic field'
if base.dimension_relative() != 1:
raise ValueError("base of projective Berkovich space must be " + \
"projective space of dimension 1 over Qp or a number field")
self._p = prime
self._ideal = ideal
Parent.__init__(self, base=base, category=TopologicalSpaces())
def projective_embedding(self, i=None, PP=None):
"""
Returns a morphism from this space into an ambient projective space
of the same dimension.
INPUT:
- ``i`` -- integer (default: dimension of self = last
coordinate) determines which projective embedding to compute. The
embedding is that which has a 1 in the i-th coordinate, numbered
from 0.
- ``PP`` -- (default: None) ambient projective space, i.e.,
codomain of morphism; this is constructed if it is not
given.
EXAMPLES::
sage: AA = AffineSpace(2, QQ, 'x')
sage: pi = AA.projective_embedding(0); pi
Scheme morphism:
From: Affine Space of dimension 2 over Rational Field
To: Projective Space of dimension 2 over Rational Field
Defn: Defined on coordinates by sending (x0, x1) to
(1 : x0 : x1)
sage: z = AA(3, 4)
sage: pi(z)
(1/4 : 3/4 : 1)
sage: pi(AA(0,2))
(1/2 : 0 : 1)
sage: pi = AA.projective_embedding(1); pi
Scheme morphism:
From: Affine Space of dimension 2 over Rational Field
To: Projective Space of dimension 2 over Rational Field
Defn: Defined on coordinates by sending (x0, x1) to
(x0 : 1 : x1)
sage: pi(z)
(3/4 : 1/4 : 1)
sage: pi = AA.projective_embedding(2)
sage: pi(z)
(3 : 4 : 1)
::
sage: A.<x,y> = AffineSpace(ZZ, 2)
sage: A.projective_embedding(2).codomain().affine_patch(2) == A
True
"""
n = self.dimension_relative()
if i is None:
try:
i = self._default_embedding_index
except AttributeError:
i = int(n)
else:
i = int(i)
try:
phi = self.__projective_embedding[i]
#assume that if you've passed in a new codomain you want to override
#the existing embedding
if PP is None or phi.codomain() == PP:
return(phi)
except AttributeError:
self.__projective_embedding = {}
except KeyError:
pass
#if no i-th embedding exists, we may still be here with PP==None
if PP is None:
from sage.schemes.projective.projective_space import ProjectiveSpace
PP = ProjectiveSpace(n, self.base_ring())
elif PP.dimension_relative() != n:
raise ValueError("projective Space must be of dimension %s"%(n))
R = self.coordinate_ring()
v = list(R.gens())
if n < 0 or n >self.dimension_relative():
raise ValueError("argument i (=%s) must be between 0 and %s, inclusive"%(i,n))
v.insert(i, R(1))
phi = self.hom(v, PP)
self.__projective_embedding[i] = phi
#make affine patch and projective embedding match
PP.affine_patch(i,self)
return phi
def projective_embedding(self, i=None, PP=None):
"""
Returns a morphism from this space into an ambient projective space
of the same dimension.
INPUT:
- ``i`` -- integer (default: dimension of self = last
coordinate) determines which projective embedding to compute. The
embedding is that which has a 1 in the i-th coordinate, numbered
from 0.
- ``PP`` -- (default: None) ambient projective space, i.e.,
codomain of morphism; this is constructed if it is not
given.
EXAMPLES::
sage: AA = AffineSpace(2, QQ, 'x')
sage: pi = AA.projective_embedding(0); pi
Scheme morphism:
From: Affine Space of dimension 2 over Rational Field
To: Projective Space of dimension 2 over Rational Field
Defn: Defined on coordinates by sending (x0, x1) to
(1 : x0 : x1)
sage: z = AA(3,4)
sage: pi(z)
(1/4 : 3/4 : 1)
sage: pi(AA(0,2))
(1/2 : 0 : 1)
sage: pi = AA.projective_embedding(1); pi
Scheme morphism:
From: Affine Space of dimension 2 over Rational Field
To: Projective Space of dimension 2 over Rational Field
Defn: Defined on coordinates by sending (x0, x1) to
(x0 : 1 : x1)
sage: pi(z)
(3/4 : 1/4 : 1)
sage: pi = AA.projective_embedding(2)
sage: pi(z)
(3 : 4 : 1)
::
sage: A.<x,y> = AffineSpace(ZZ,2)
sage: A.projective_embedding(2).codomain().affine_patch(2) == A
True
"""
n = self.dimension_relative()
if i is None:
try:
i = self._default_embedding_index
except AttributeError:
i = int(n)
else:
i = int(i)
try:
phi = self.__projective_embedding[i]
#assume that if you've passed in a new codomain you want to override
#the existing embedding
if PP is None or phi.codomain() == PP:
return (phi)
except AttributeError:
self.__projective_embedding = {}
except KeyError:
pass
#if no ith embedding exists, we may still be here with PP==None
if PP is None:
from sage.schemes.projective.projective_space import ProjectiveSpace
PP = ProjectiveSpace(n, self.base_ring())
elif PP.dimension_relative() != n:
raise ValueError("Projective Space must be of dimension %s" % (n))
R = self.coordinate_ring()
v = list(R.gens())
if n < 0 or n > self.dimension_relative():
raise ValueError(
"Argument i (=%s) must be between 0 and %s, inclusive" %
(i, n))
v.insert(i, R(1))
phi = self.hom(v, PP)
self.__projective_embedding[i] = phi
#make affine patch and projective embedding match
PP.affine_patch(i, self)
return phi
