def affine_patch(self, I, return_embedding=False):
r"""
Return the `I^{th}` affine patch of this projective space product
where ``I`` is a multi-index.
INPUT:
- ``I`` -- a list or tuple of positive integers.
- ``return_embedding`` -- Boolean, if true the projective embedding is also returned.
OUTPUT:
- An affine space.
- An embedding into a product of projective spaces (optional).
EXAMPLES::
sage: PP = ProductProjectiveSpaces([2, 2, 2], ZZ, 'x')
sage: phi = PP.affine_patch([0, 1, 2], True)
sage: phi.domain()
Affine Space of dimension 6 over Integer Ring
sage: phi
Scheme morphism:
From: Affine Space of dimension 6 over Integer Ring
To: Product of projective spaces P^2 x P^2 x P^2 over Integer Ring
Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4, x5) to
(1 : x0 : x1 , x2 : 1 : x3 , x4 : x5 : 1)
"""
if not isinstance(I, (list, tuple)):
raise TypeError(
'the argument I=%s must be a list or tuple of positive integers'
% I)
PP = self.ambient_space()
N = PP._dims
if len(I) != len(N):
raise ValueError('the argument I=%s must have %s entries' %
(I, len(N)))
I = tuple([int(i) for i in I]) # implicit type checking
for i in range(len(I)):
if I[i] < 0 or I[i] > N[i]:
raise ValueError(
"argument i (= %s) must be between 0 and %s." %
(I[i], N[i]))
try:
if return_embedding:
return self.__affine_patches[I][1]
else:
return self.__affine_patches[I][0]
except AttributeError:
self.__affine_patches = {}
except KeyError:
pass
from sage.schemes.affine.affine_space import AffineSpace
AA = AffineSpace(PP.base_ring(), sum(N), 'x')
v = list(AA.gens())
index = 0
for i in range(len(I)):
v.insert(index + I[i], 1)
index += N[i] + 1
phi = AA.hom(v, self)
self.__affine_patches.update({I: (AA, phi)})
if return_embedding:
return phi
else:
return AA
def affine_patch(self, I, return_embedding=False):
r"""
Return the `I^{th}` affine patch of this projective scheme
where 'I' is a multi-index.
INPUT:
- ``I`` -- a list or tuple of positive integers
- ``return_embedding`` -- Boolean, if true the projective embedding is also returned
OUTPUT:
- An affine algebraic scheme
- An embedding into a product of projective space (optional)
EXAMPLES::
sage: PP.<x,y,z,w,u,v> = ProductProjectiveSpaces([3,1],QQ)
sage: W = PP.subscheme([y^2*z-x^3,z^2-w^2,u^3-v^3])
sage: W.affine_patch([0,1],True)
(Closed subscheme of Affine Space of dimension 4 over Rational Field defined by:
x0^2*x1 - 1,
x1^2 - x2^2,
x3^3 - 1, Scheme morphism:
From: Closed subscheme of Affine Space of dimension 4 over Rational Field defined by:
x0^2*x1 - 1,
x1^2 - x2^2,
x3^3 - 1
To: Closed subscheme of Product of projective spaces P^3 x P^1 over Rational Field defined by:
-x^3 + y^2*z,
z^2 - w^2,
u^3 - v^3
Defn: Defined on coordinates by sending (x0, x1, x2, x3) to
(1 : x0 : x1 : x2 , x3 : 1))
"""
if not isinstance(I, (list, tuple)):
raise TypeError(
'The argument I=%s must be a list or tuple of positive integers'
% I)
PP = self.ambient_space()
N = PP.dimension_relative_components()
if len(I) != len(N):
raise ValueError('The argument I=%s must have %s entries' %
(I, len(N)))
I = tuple([int(i) for i in I]) # implicit type checking
for i in range(len(I)):
if I[i] < 0 or I[i] > N[i]:
raise ValueError(
"Argument i (= %s) must be between 0 and %s." %
(I[i], N[i]))
#see if we've already created this affine patch
try:
if return_embedding:
return self.__affine_patches[I]
else:
return self.__affine_patches[I][0]
except AttributeError:
self.__affine_patches = {}
except KeyError:
pass
AA = AffineSpace(PP.base_ring(), sum(N), 'x')
v = list(AA.gens())
# create the projective embedding
index = 0
for i in range(len(I)):
v.insert(index + I[i], 1)
index += N[i] + 1
phi = AA.hom(v, self)
#find the image of the subscheme
polys = self.defining_polynomials()
xi = phi.defining_polynomials()
U = AA.subscheme([f(xi) for f in polys])
phi = U.hom(v, self)
self.__affine_patches.update({I: (U, phi)})
if return_embedding:
return U, phi
else:
return U
def affine_patch(self, I, return_embedding = False):
r"""
Return the `I^{th}` affine patch of this projective space product
where ``I`` is a multi-index.
INPUT:
- ``I`` -- a list or tuple of positive integers
- ``return_embedding`` -- Boolean, if true the projective embedding is also returned
OUTPUT:
- An affine space
- An embedding into a product of projective spaces (optional)
EXAMPLES::
sage: PP = ProductProjectiveSpaces([2,2,2], ZZ, 'x')
sage: phi = PP.affine_patch([0,1,2], True)
sage: phi.domain()
Affine Space of dimension 6 over Integer Ring
sage: phi
Scheme morphism:
From: Affine Space of dimension 6 over Integer Ring
To: Product of projective spaces P^2 x P^2 x P^2 over Integer Ring
Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4, x5) to
(1 : x0 : x1 , x2 : 1 : x3 , x4 : x5 : 1)
"""
if not isinstance(I, (list, tuple)):
raise TypeError('The argument I=%s must be a list or tuple of positive integers'%I)
PP = self.ambient_space()
N = PP._dims
if len(I) != len(N):
raise ValueError('The argument I=%s must have %s entries'%(I,len(N)))
I = tuple([int(i) for i in I]) # implicit type checking
for i in range(len(I)):
if I[i] < 0 or I[i] > N[i]:
raise ValueError("Argument i (= %s) must be between 0 and %s."%(I[i], N[i]))
try:
if return_embedding:
return self.__affine_patches[I][1]
else:
return self.__affine_patches[I][0]
except AttributeError:
self.__affine_patches = {}
except KeyError:
pass
from sage.schemes.affine.affine_space import AffineSpace
AA = AffineSpace(PP.base_ring(),sum(N),'x')
v = list(AA.gens())
index = 0
for i in range(len(I)):
v.insert(index+I[i],1)
index += N[i]+1
phi = AA.hom(v,self)
self.__affine_patches.update({I:(AA,phi)})
if return_embedding:
return phi
else:
return AA
