def affine_algebraic_patch(self, cone=None, names=None):
r"""
Return the affine patch corresponding to ``cone`` as an affine
algebraic scheme.
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:
An :class:`affine algebraic subscheme
<sage.schemes.affine.affine_subscheme.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.
REFERENCES:
..
David A. Cox, "The Homogeneous Coordinate Ring of a Toric
Variety", Lemma 2.2.
:arxiv:`alg-geom/9210008v2`
EXAMPLES::
sage: P2.<x,y,z> = toric_varieties.P2()
sage: cone = P2.fan().generating_cone(0)
sage: V = P2.subscheme(x^3+y^3+z^3)
sage: V.affine_algebraic_patch(cone)
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
z0^3 + z1^3 + 1
sage: cone = Cone([(0,1),(2,1)])
sage: A2Z2.<x,y> = 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: V = A2Z2.subscheme(x^2+y^2-1)
sage: patch = V.affine_algebraic_patch(); patch
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
-z0*z1 + z2^2,
z0 + z1 - 1
sage: nbhd_patch = V.neighborhood([1,0]).affine_algebraic_patch(); nbhd_patch
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
-z0*z1 + z2^2,
z0 + z1 - 1
sage: nbhd_patch.embedding_center()
(0, 1, 0)
Here we got two defining equations. The first one describes
the singularity of the ambient space and the second is the
pull-back of `x^2+y^2-1` ::
sage: lp = LatticePolytope([(1,0,0),(1,1,0),(1,1,1),(1,0,1),(-2,-1,-1)],
....: lattice=ToricLattice(3))
sage: X.<x,y,u,v,t> = CPRFanoToricVariety(Delta_polar=lp)
sage: Y = X.subscheme(x*v+y*u+t)
sage: cone = Cone([(1,0,0),(1,1,0),(1,1,1),(1,0,1)])
sage: Y.affine_algebraic_patch(cone)
Closed subscheme of Affine Space of dimension 4 over Rational Field defined by:
z0*z2 - z1*z3,
z1 + z3 + 1
"""
from sage.modules.free_module_element import vector
from sage.misc.misc_c import prod
ambient = self.ambient_space()
fan = ambient.fan()
if cone is None:
assert ambient.is_affine()
cone = fan.generating_cone(0)
else:
cone = fan.embed(cone)
# R/I = C[sigma^dual cap M]
R, I, dualcone = ambient._semigroup_ring(cone, names)
# inhomogenize the Cox homogeneous polynomial with respect to the given cone
inhomogenize = dict((ambient.coordinate_ring().gen(i), 1)
for i in range(0, fan.nrays())
if i not in cone.ambient_ray_indices())
polynomials = [
p.subs(inhomogenize) for p in self.defining_polynomials()
]
# map the monomial x^{D_m} to m, see reference.
n_rho_matrix = cone.rays().matrix()
def pullback_polynomial(p):
result = R.zero()
for coefficient, monomial in p:
exponent = monomial.exponents()[0]
exponent = [exponent[i] for i in cone.ambient_ray_indices()]
exponent = vector(ZZ, exponent)
m = n_rho_matrix.solve_right(exponent)
assert all(x in ZZ for x in m), \
'The polynomial '+str(p)+' does not define a ZZ-divisor!'
m_coeffs = dualcone.Hilbert_coefficients(m)
result += coefficient * prod(
R.gen(i)**m_coeffs[i] for i in range(0, R.ngens()))
return result
# construct the affine algebraic scheme to use as patch
polynomials = [pullback_polynomial(_) for _ in polynomials]
from sage.schemes.affine.affine_space import AffineSpace
patch_cover = AffineSpace(R)
polynomials = list(I.gens()) + polynomials
polynomials = [x for x in polynomials if not x.is_zero()]
patch = patch_cover.subscheme(polynomials)
# TODO: If the cone is not smooth, then the coordinate_ring()
# of the affine toric variety is wrong; it should be the
# G-invariant part. So we can't construct the embedding
# morphism in that case.
if cone.is_smooth():
x = ambient.coordinate_ring().gens()
phi = []
for i in range(0, fan.nrays()):
if i in cone.ambient_ray_indices():
phi.append(pullback_polynomial(x[i]))
else:
phi.append(1)
patch._embedding_morphism = patch.hom(phi, self)
else:
patch._embedding_morphism = (
NotImplementedError,
'I only know how to construct embedding morphisms for smooth patches'
)
try:
point = self.embedding_center()
except AttributeError:
return patch
# it remains to find the preimage of point
# map m to the monomial x^{D_m}, see reference.
F = ambient.coordinate_ring().fraction_field()
image = []
for m in dualcone.Hilbert_basis():
x_Dm = prod([F.gen(i)**(m * n) for i, n in enumerate(fan.rays())])
image.append(x_Dm)
patch._embedding_center = tuple(f(list(point)) for f in image)
return patch
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_algebraic_patch(self, cone=None, names=None):
r"""
Return the affine patch corresponding to ``cone`` as an affine
algebraic scheme.
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:
An :class:`affine algebraic subscheme
<sage.schemes.affine.affine_subscheme.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.
REFERENCES:
..
David A. Cox, "The Homogeneous Coordinate Ring of a Toric
Variety", Lemma 2.2.
:arxiv:`alg-geom/9210008v2`
EXAMPLES::
sage: P2.<x,y,z> = toric_varieties.P2()
sage: cone = P2.fan().generating_cone(0)
sage: V = P2.subscheme(x^3+y^3+z^3)
sage: V.affine_algebraic_patch(cone)
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
z0^3 + z1^3 + 1
sage: cone = Cone([(0,1),(2,1)])
sage: A2Z2.<x,y> = 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: V = A2Z2.subscheme(x^2+y^2-1)
sage: patch = V.affine_algebraic_patch(); patch
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
-z0*z1 + z2^2,
z0 + z1 - 1
sage: nbhd_patch = V.neighborhood([1,0]).affine_algebraic_patch(); nbhd_patch
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
-z0*z1 + z2^2,
z0 + z1 - 1
sage: nbhd_patch.embedding_center()
(0, 1, 0)
Here we got two defining equations. The first one describes
the singularity of the ambient space and the second is the
pull-back of `x^2+y^2-1` ::
sage: lp = LatticePolytope([(1,0,0),(1,1,0),(1,1,1),(1,0,1),(-2,-1,-1)],
....: lattice=ToricLattice(3))
sage: X.<x,y,u,v,t> = CPRFanoToricVariety(Delta_polar=lp)
sage: Y = X.subscheme(x*v+y*u+t)
sage: cone = Cone([(1,0,0),(1,1,0),(1,1,1),(1,0,1)])
sage: Y.affine_algebraic_patch(cone)
Closed subscheme of Affine Space of dimension 4 over Rational Field defined by:
z0*z2 - z1*z3,
z1 + z3 + 1
"""
from sage.modules.all import vector
from sage.misc.all import prod
ambient = self.ambient_space()
fan = ambient.fan()
if cone is None:
assert ambient.is_affine()
cone = fan.generating_cone(0)
else:
cone = fan.embed(cone)
# R/I = C[sigma^dual cap M]
R, I, dualcone = ambient._semigroup_ring(cone, names)
# inhomogenize the Cox homogeneous polynomial with respect to the given cone
inhomogenize = dict( (ambient.coordinate_ring().gen(i), 1)
for i in range(0,fan.nrays())
if not i in cone.ambient_ray_indices() )
polynomials = [ p.subs(inhomogenize) for p in self.defining_polynomials() ]
# map the monomial x^{D_m} to m, see reference.
n_rho_matrix = cone.rays().matrix()
def pullback_polynomial(p):
result = R.zero()
for coefficient, monomial in p:
exponent = monomial.exponents()[0]
exponent = [ exponent[i] for i in cone.ambient_ray_indices() ]
exponent = vector(ZZ,exponent)
m = n_rho_matrix.solve_right(exponent)
assert all(x in ZZ for x in m), \
'The polynomial '+str(p)+' does not define a ZZ-divisor!'
m_coeffs = dualcone.Hilbert_coefficients(m)
result += coefficient * prod(R.gen(i)**m_coeffs[i]
for i in range(0,R.ngens()))
return result
# construct the affine algebraic scheme to use as patch
polynomials = [pullback_polynomial(_) for _ in polynomials]
from sage.schemes.affine.affine_space import AffineSpace
patch_cover = AffineSpace(R)
polynomials = list(I.gens()) + polynomials
polynomials = [x for x in polynomials if not x.is_zero()]
patch = patch_cover.subscheme(polynomials)
# TODO: If the cone is not smooth, then the coordinate_ring()
# of the affine toric variety is wrong; it should be the
# G-invariant part. So we can't construct the embedding
# morphism in that case.
if cone.is_smooth():
x = ambient.coordinate_ring().gens()
phi = []
for i in range(0,fan.nrays()):
if i in cone.ambient_ray_indices():
phi.append(pullback_polynomial(x[i]))
else:
phi.append(1)
patch._embedding_morphism = patch.hom(phi, self)
else:
patch._embedding_morphism = (NotImplementedError,
'I only know how to construct embedding morphisms for smooth patches')
try:
point = self.embedding_center()
except AttributeError:
return patch
# it remains to find the preimage of point
# map m to the monomial x^{D_m}, see reference.
F = ambient.coordinate_ring().fraction_field()
image = []
for m in dualcone.Hilbert_basis():
x_Dm = prod([ F.gen(i)**(m*n) for i,n in enumerate(fan.rays()) ])
image.append(x_Dm)
patch._embedding_center = tuple( f(list(point)) for f in image )
return patch