def __init__(self, X, v, check=True):
"""
The Python constructor.
See :class:`SchemeMorphism_point_affine` for details.
TESTS::
sage: from sage.schemes.affine.affine_point import SchemeMorphism_point_affine
sage: A3.<x,y,z> = AffineSpace(QQ, 3)
sage: SchemeMorphism_point_affine(A3(QQ), [1,2,3])
(1, 2, 3)
"""
SchemeMorphism.__init__(self, X)
if is_SchemeMorphism(v):
v = list(v)
if check:
# Verify that there are the right number of coords
d = self.codomain().ambient_space().ngens()
if len(v) != d:
raise TypeError("Argument v (=%s) must have %s coordinates." %
(v, d))
if not isinstance(v, (list, tuple)):
raise TypeError(
"Argument v (= %s) must be a scheme point, list, or tuple."
% str(v))
# Make sure the coordinates all lie in the appropriate ring
v = Sequence(v, X.value_ring())
# Verify that the point satisfies the equations of X.
X.extended_codomain()._check_satisfies_equations(v)
self._coords = tuple(v)
def __init__(self, X, coordinates, check=True):
r"""
See :class:`SchemeMorphism_point_toric_field` for documentation.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1(1,2,3,4)
[1 : 2 : 3 : 4]
"""
# Convert scheme to its set of points over the base ring
if is_Scheme(X):
X = X(X.base_ring())
super(SchemeMorphism_point_toric_field, self).__init__(X)
if check:
# Verify that there are the right number of coords
# Why is it not done in the parent?
if is_SchemeMorphism(coordinates):
coordinates = list(coordinates)
if not isinstance(coordinates, (list, tuple)):
raise TypeError("coordinates must be a scheme point, list, "
"or tuple. Got %s" % coordinates)
d = X.codomain().ambient_space().ngens()
if len(coordinates) != d:
raise ValueError("there must be %d coordinates! Got only %d: "
"%s" % (d, len(coordinates), coordinates))
# Make sure the coordinates all lie in the appropriate ring
coordinates = Sequence(coordinates, X.value_ring())
# Verify that the point satisfies the equations of X.
X.codomain()._check_satisfies_equations(coordinates)
self._coords = coordinates
def __init__(self, X, coordinates, check=True):
r"""
See :class:`SchemeMorphism_point_toric_field` for documentation.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1(1,2,3,4)
[1 : 2 : 3 : 4]
"""
# Convert scheme to its set of points over the base ring
if is_Scheme(X):
X = X(X.base_ring())
super(SchemeMorphism_point_toric_field, self).__init__(X)
if check:
# Verify that there are the right number of coords
# Why is it not done in the parent?
if is_SchemeMorphism(coordinates):
coordinates = list(coordinates)
if not isinstance(coordinates, (list, tuple)):
raise TypeError("coordinates must be a scheme point, list, "
"or tuple. Got %s" % coordinates)
d = X.codomain().ambient_space().ngens()
if len(coordinates) != d:
raise ValueError("there must be %d coordinates! Got only %d: "
"%s" % (d, len(coordinates), coordinates))
# Make sure the coordinates all lie in the appropriate ring
coordinates = Sequence(coordinates, X.value_ring())
# Verify that the point satisfies the equations of X.
X.codomain()._check_satisfies_equations(coordinates)
self._coords = coordinates
def __init__(self, X, v, check=True):
"""
The Python constructor.
See :class:`SchemeMorphism_point_affine` for details.
TESTS::
sage: from sage.schemes.affine.affine_point import SchemeMorphism_point_affine
sage: A3.<x,y,z> = AffineSpace(QQ, 3)
sage: SchemeMorphism_point_affine(A3(QQ), [1,2,3])
(1, 2, 3)
"""
SchemeMorphism.__init__(self, X)
if is_SchemeMorphism(v):
v = list(v)
if check:
# Verify that there are the right number of coords
d = self.codomain().ambient_space().ngens()
if len(v) != d:
raise TypeError("Argument v (=%s) must have %s coordinates." % (v, d))
if not isinstance(v, (list, tuple)):
raise TypeError("Argument v (= %s) must be a scheme point, list, or tuple." % str(v))
# Make sure the coordinates all lie in the appropriate ring
v = Sequence(v, X.value_ring())
# Verify that the point satisfies the equations of X.
X.extended_codomain()._check_satisfies_equations(v)
self._coords = tuple(v)
def __init__(self, X=None, category=None):
"""
Construct a scheme.
TESTS::
sage: R.<x, y> = QQ[]
sage: I = (x^2 - y^2)*R
sage: RmodI = R.quotient(I)
sage: X = Spec(RmodI)
sage: TestSuite(X).run(skip = ["_test_an_element", "_test_elements",
... "_test_some_elements", "_test_category"]) # See #7946
"""
from sage.schemes.generic.spec import is_Spec
from sage.schemes.generic.morphism import is_SchemeMorphism
if X is None:
try:
from sage.schemes.generic.spec import SpecZ
self._base_scheme = SpecZ
except ImportError: # we are currently constructing SpecZ
self._base_ring = ZZ
elif is_Scheme(X):
self._base_scheme = X
elif is_SchemeMorphism(X):
self._base_morphism = X
elif is_CommutativeRing(X):
self._base_ring = X
elif is_RingHomomorphism(X):
self._base_ring = X.codomain()
else:
raise ValueError("The base must be define by a scheme, " "scheme morphism, or commutative ring.")
from sage.categories.schemes import Schemes
if not X:
default_category = Schemes()
else:
default_category = Schemes(self.base_scheme())
if category is None:
category = default_category
else:
assert category.is_subcategory(default_category), "%s is not a subcategory of %s" % (
category,
default_category,
)
Parent.__init__(self, self.base_ring(), category=category)
def __init__(self, X=None, category=None):
"""
Construct a scheme.
TESTS::
sage: R.<x, y> = QQ[]
sage: I = (x^2 - y^2)*R
sage: RmodI = R.quotient(I)
sage: X = Spec(RmodI)
sage: TestSuite(X).run(skip = ["_test_an_element", "_test_elements",
... "_test_some_elements", "_test_category"]) # See #7946
"""
from sage.schemes.generic.spec import is_Spec
from sage.schemes.generic.morphism import is_SchemeMorphism
if X is None:
try:
from sage.schemes.generic.spec import SpecZ
self._base_scheme = SpecZ
except ImportError: # we are currently constructing SpecZ
self._base_ring = ZZ
elif is_Scheme(X):
self._base_scheme = X
elif is_SchemeMorphism(X):
self._base_morphism = X
elif is_CommutativeRing(X):
self._base_ring = X
elif is_RingHomomorphism(X):
self._base_ring = X.codomain()
else:
raise ValueError('The base must be define by a scheme, '
'scheme morphism, or commutative ring.')
from sage.categories.schemes import Schemes
if not X:
default_category = Schemes()
else:
default_category = Schemes(self.base_scheme())
if category is None:
category = default_category
else:
assert category.is_subcategory(default_category), \
"%s is not a subcategory of %s"%(category, default_category)
Parent.__init__(self, self.base_ring(), category=category)
def __init__(self, X=None, category=None):
"""
Construct a scheme.
TESTS:
The full test suite works since :trac:`7946`::
sage: R.<x, y> = QQ[]
sage: I = (x^2 - y^2)*R
sage: RmodI = R.quotient(I)
sage: X = Spec(RmodI)
sage: TestSuite(X).run()
"""
from sage.schemes.generic.morphism import is_SchemeMorphism
from sage.categories.map import Map
from sage.categories.all import Rings
if X is None:
self._base_ring = ZZ
elif is_Scheme(X):
self._base_scheme = X
elif is_SchemeMorphism(X):
self._base_morphism = X
elif isinstance(X, CommutativeRing):
self._base_ring = X
elif isinstance(X, Map) and X.category_for().is_subcategory(Rings()):
# X is a morphism of Rings
self._base_ring = X.codomain()
else:
raise ValueError('The base must be define by a scheme, '
'scheme morphism, or commutative ring.')
from sage.categories.schemes import Schemes
if X is None:
default_category = Schemes()
else:
default_category = Schemes(self.base_scheme())
if category is None:
category = default_category
else:
assert category.is_subcategory(default_category), \
"%s is not a subcategory of %s"%(category, default_category)
Parent.__init__(self, self.base_ring(), category = category)
def __init__(self, X=None, category=None):
"""
Construct a scheme.
TESTS:
The full test suite works since :trac:`7946`::
sage: R.<x, y> = QQ[]
sage: I = (x^2 - y^2)*R
sage: RmodI = R.quotient(I)
sage: X = Spec(RmodI)
sage: TestSuite(X).run()
"""
from sage.schemes.generic.morphism import is_SchemeMorphism
from sage.categories.map import Map
from sage.categories.all import Rings
if X is None:
self._base_ring = ZZ
elif is_Scheme(X):
self._base_scheme = X
elif is_SchemeMorphism(X):
self._base_morphism = X
elif isinstance(X, CommutativeRing):
self._base_ring = X
elif isinstance(X, Map) and X.category_for().is_subcategory(Rings()):
# X is a morphism of Rings
self._base_ring = X.codomain()
else:
raise ValueError('The base must be define by a scheme, '
'scheme morphism, or commutative ring.')
from sage.categories.schemes import Schemes
if X is None:
default_category = Schemes()
else:
default_category = Schemes(self.base_scheme())
if category is None:
category = default_category
else:
assert category.is_subcategory(default_category), \
"%s is not a subcategory of %s"%(category, default_category)
Parent.__init__(self, self.base_ring(), category=category)
def _call_(self, x):
"""
Construct a scheme from the data in ``x``
EXAMPLES:
Let us first construct the category of schemes::
sage: S = Schemes(); S
Category of schemes
We create a scheme from a ring::
sage: X = S(ZZ); X # indirect doctest
Spectrum of Integer Ring
We create a scheme from a scheme (do nothing)::
sage: S(X)
Spectrum of Integer Ring
We create a scheme morphism from a ring homomorphism.x::
sage: phi = ZZ.hom(QQ); phi
Ring Coercion morphism:
From: Integer Ring
To: Rational Field
sage: f = S(phi); f # indirect doctest
Affine Scheme morphism:
From: Spectrum of Rational Field
To: Spectrum of Integer Ring
Defn: Ring Coercion morphism:
From: Integer Ring
To: Rational Field
sage: f.domain()
Spectrum of Rational Field
sage: f.codomain()
Spectrum of Integer Ring
sage: S(f) # indirect doctest
Affine Scheme morphism:
From: Spectrum of Rational Field
To: Spectrum of Integer Ring
Defn: Ring Coercion morphism:
From: Integer Ring
To: Rational Field
"""
from sage.schemes.generic.scheme import is_Scheme
if is_Scheme(x):
return x
from sage.schemes.generic.morphism import is_SchemeMorphism
if is_SchemeMorphism(x):
return x
from sage.rings.morphism import is_RingHomomorphism
from sage.rings.commutative_ring import is_CommutativeRing
from sage.schemes.generic.spec import Spec
if is_CommutativeRing(x):
return Spec(x)
elif is_RingHomomorphism(x):
A = Spec(x.codomain())
return A.hom(x)
else:
raise TypeError(
"No way to create an object or morphism in %s from %s" %
(self, x))
def __init__(self, X, v, check=True):
"""
The Python constructor.
See :class:`SchemeMorphism_point_projective_ring` for details.
This function still normalizes points so that the rightmost non-zero coordinate is 1.
This is to maintain functionality with current
implementations of curves in projectives space (plane, conic, elliptic, etc).
The :class:`SchemeMorphism_point_projective_ring` is for general use.
EXAMPLES::
sage: P = ProjectiveSpace(2, QQ)
sage: P(2, 3/5, 4)
(1/2 : 3/20 : 1)
::
sage: P = ProjectiveSpace(3, QQ)
sage: P(0, 0, 0, 0)
Traceback (most recent call last):
...
ValueError: [0, 0, 0, 0] does not define a valid point since all entries are 0
::
sage: P.<x, y, z> = ProjectiveSpace(2, QQ)
sage: X = P.subscheme([x^2-y*z])
sage: X([2, 2, 2])
(1 : 1 : 1)
::
sage: P = ProjectiveSpace(1, GF(7))
sage: Q=P([2, 1])
sage: Q[0].parent()
Finite Field of size 7
::
sage: P = ProjectiveSpace(QQ,1)
sage: P.point(Infinity)
(1 : 0)
sage: P(infinity)
(1 : 0)
::
sage: P = ProjectiveSpace(QQ,2)
sage: P(infinity)
Traceback (most recent call last):
...
ValueError: +Infinity not well defined in dimension > 1
sage: P.point(infinity)
Traceback (most recent call last):
...
ValueError: +Infinity not well defined in dimension > 1
"""
SchemeMorphism.__init__(self, X)
if check:
from sage.schemes.elliptic_curves.ell_point import EllipticCurvePoint_field
from sage.rings.ring import CommutativeRing
d = X.codomain().ambient_space().ngens()
if is_SchemeMorphism(v) or isinstance(v, EllipticCurvePoint_field):
v = list(v)
else:
try:
if isinstance(v.parent(), CommutativeRing):
v = [v]
except AttributeError:
pass
if not isinstance(v, (list,tuple)):
raise TypeError("argument v (= %s) must be a scheme point, list, or tuple"%str(v))
if len(v) != d and len(v) != d-1:
raise TypeError("v (=%s) must have %s components"%(v, d))
R = X.value_ring()
v = Sequence(v, R)
if len(v) == d-1: # very common special case
v.append(R(1))
n = len(v)
all_zero = True
for i in range(n):
last = n-1-i
if v[last]:
all_zero = False
c = v[last]
if c == R.one():
break
for j in range(last):
v[j] /= c
v[last] = R.one()
break
if all_zero:
raise ValueError("%s does not define a valid point since all entries are 0"%repr(v))
X.extended_codomain()._check_satisfies_equations(v)
self._coords = tuple(v)
def __init__(self, X, v, check=True):
"""
The Python constructor.
EXAMPLES::
sage: P = ProjectiveSpace(2, QQ)
sage: P(2, 3/5, 4)
(1/2 : 3/20 : 1)
::
sage: P = ProjectiveSpace(1, ZZ)
sage: P([0, 1])
(0 : 1)
::
sage: P = ProjectiveSpace(1, ZZ)
sage: P([0, 0, 1])
Traceback (most recent call last):
...
TypeError: v (=[0, 0, 1]) must have 2 components
::
sage: P = ProjectiveSpace(3, QQ)
sage: P(0,0,0,0)
Traceback (most recent call last):
...
ValueError: [0, 0, 0, 0] does not define a valid point since all entries are 0
It is possible to avoid the possibly time-consuming checks, but be careful!! ::
sage: P = ProjectiveSpace(3, QQ)
sage: P.point([0,0,0,0], check=False)
(0 : 0 : 0 : 0)
::
sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: X = P.subscheme([x^2-y*z])
sage: X([2, 2, 2])
(2 : 2 : 2)
::
sage: R.<t> = PolynomialRing(ZZ)
sage: P = ProjectiveSpace(1, R.quo(t^2+1))
sage: P([2*t, 1])
(2*tbar : 1)
::
sage: P = ProjectiveSpace(ZZ,1)
sage: P.point(Infinity)
(1 : 0)
sage: P(infinity)
(1 : 0)
::
sage: P = ProjectiveSpace(ZZ,2)
sage: P(Infinity)
Traceback (most recent call last):
...
ValueError: +Infinity not well defined in dimension > 1
sage: P.point(infinity)
Traceback (most recent call last):
...
ValueError: +Infinity not well defined in dimension > 1
"""
SchemeMorphism.__init__(self, X)
if check:
from sage.schemes.elliptic_curves.ell_point import EllipticCurvePoint_field
from sage.rings.ring import CommutativeRing
d = X.codomain().ambient_space().ngens()
if is_SchemeMorphism(v) or isinstance(v, EllipticCurvePoint_field):
v = list(v)
else:
try:
if isinstance(v.parent(), CommutativeRing):
v = [v]
except AttributeError:
pass
if not isinstance(v, (list, tuple)):
raise TypeError("argument v (= %s) must be a scheme point, list, or tuple"%str(v))
if len(v) != d and len(v) != d-1:
raise TypeError("v (=%s) must have %s components"%(v, d))
R = X.value_ring()
v = Sequence(v, R)
if len(v) == d-1: # very common special case
v.append(R(1))
n = len(v)
all_zero = True
for i in range(n):
last = n-1-i
if v[last]:
all_zero = False
break
if all_zero:
raise ValueError("%s does not define a valid point since all entries are 0"%repr(v))
X.extended_codomain()._check_satisfies_equations(v)
self._coords = tuple(v)
def _call_(self, x):
"""
Construct a scheme from the data in ``x``
EXAMPLES:
Let us first construct the category of schemes::
sage: S = Schemes(); S
Category of schemes
We create a scheme from a ring::
sage: X = S(ZZ); X # indirect doctest
Spectrum of Integer Ring
We create a scheme from a scheme (do nothing)::
sage: S(X)
Spectrum of Integer Ring
We create a scheme morphism from a ring homomorphism.x::
sage: phi = ZZ.hom(QQ); phi
Ring Coercion morphism:
From: Integer Ring
To: Rational Field
sage: f = S(phi); f # indirect doctest
Affine Scheme morphism:
From: Spectrum of Rational Field
To: Spectrum of Integer Ring
Defn: Ring Coercion morphism:
From: Integer Ring
To: Rational Field
sage: f.domain()
Spectrum of Rational Field
sage: f.codomain()
Spectrum of Integer Ring
sage: S(f) # indirect doctest
Affine Scheme morphism:
From: Spectrum of Rational Field
To: Spectrum of Integer Ring
Defn: Ring Coercion morphism:
From: Integer Ring
To: Rational Field
"""
from sage.schemes.generic.scheme import is_Scheme
if is_Scheme(x):
return x
from sage.schemes.generic.morphism import is_SchemeMorphism
if is_SchemeMorphism(x):
return x
from sage.rings.morphism import is_RingHomomorphism
from sage.rings.ring import CommutativeRing
from sage.schemes.generic.spec import Spec
if isinstance(x, CommutativeRing):
return Spec(x)
elif is_RingHomomorphism(x):
A = Spec(x.codomain())
return A.hom(x)
else:
raise TypeError("No way to create an object or morphism in %s from %s"%(self, x))
def __init__(self, X, v, check=True):
"""
The Python constructor.
See :class:`SchemeMorphism_point_projective_ring` for details.
This function still normalized points so that the rightmost non-zero coordinate is 1. The is to maintain current functionality with current
implementations of curves in projectives space (plane, connic, elliptic, etc). The class:`SchemeMorphism_point_projective_ring` is for general use.
EXAMPLES::
sage: P = ProjectiveSpace(2, QQ)
sage: P(2, 3/5, 4)
(1/2 : 3/20 : 1)
::
sage: P = ProjectiveSpace(3, QQ)
sage: P(0,0,0,0)
Traceback (most recent call last):
...
ValueError: [0, 0, 0, 0] does not define a valid point since all entries are 0
::
sage: P.<x, y, z> = ProjectiveSpace(2, QQ)
sage: X=P.subscheme([x^2-y*z])
sage: X([2,2,2])
(1 : 1 : 1)
"""
SchemeMorphism.__init__(self, X)
if check:
from sage.schemes.elliptic_curves.ell_point import EllipticCurvePoint_field
d = X.codomain().ambient_space().ngens()
if is_SchemeMorphism(v) or isinstance(v, EllipticCurvePoint_field):
v = list(v)
elif v is infinity:
v = [0] * (d)
v[1] = 1
if not isinstance(v,(list,tuple)):
raise TypeError("Argument v (= %s) must be a scheme point, list, or tuple."%str(v))
if len(v) != d and len(v) != d-1:
raise TypeError("v (=%s) must have %s components"%(v, d))
R = X.value_ring()
v = Sequence(v, R)
if len(v) == d-1: # very common special case
v.append(1)
n = len(v)
all_zero = True
for i in range(n):
last = n-1-i
if v[last]:
all_zero = False
c = v[last]
if c == R.one():
break
for j in range(last):
v[j] /= c
v[last] = R.one()
break
if all_zero:
raise ValueError("%s does not define a valid point since all entries are 0"%repr(v))
X.extended_codomain()._check_satisfies_equations(v)
self._coords = v
def __call__(self, P):
r"""
Returns a rational point P in the abstract Homset J(K), given:
0. A point P in J = Jac(C), returning P; 1. A point P on the curve
C such that J = Jac(C), where C is an odd degree model, returning
[P - oo]; 2. A pair of points (P, Q) on the curve C such that J =
Jac(C), returning [P-Q]; 2. A list of polynomials (a,b) such that
`b^2 + h*b - f = 0 mod a`, returning [(a(x),y-b(x))].
EXAMPLES::
sage: P.<x> = PolynomialRing(QQ)
sage: f = x^5 - x + 1; h = x
sage: C = HyperellipticCurve(f,h,'u,v')
sage: P = C(0,1,1)
sage: J = C.jacobian()
sage: Q = J(QQ)(P)
sage: for i in range(6): i*Q
(1)
(u, v - 1)
(u^2, v + u - 1)
(u^2, v + 1)
(u, v + 1)
(1)
::
sage: F.<a> = GF(3)
sage: R.<x> = F[]
sage: f = x^5-1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: X = J(F)
sage: a = x^2-x+1
sage: b = -x +1
sage: c = x-1
sage: d = 0
sage: D1 = X([a,b])
sage: D1
(x^2 + 2*x + 1, y + x + 2)
sage: D2 = X([c,d])
sage: D2
(x + 2, y)
sage: D1+D2
(x^2 + 2*x + 2, y + 2*x + 1)
"""
if isinstance(P, (int, long, Integer)) and P == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(self, (R(1), R(0)))
elif isinstance(P, (list, tuple)):
if len(P) == 1 and P[0] == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(self, (R(1), R(0)))
elif len(P) == 2:
P1 = P[0]
P2 = P[1]
if is_Integer(P1) and is_Integer(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
P2 = R(P2)
return JacobianMorphism_divisor_class_field(
self, tuple([P1, P2]))
if is_Integer(P1) and is_Polynomial(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
return JacobianMorphism_divisor_class_field(
self, tuple([P1, P2]))
if is_Integer(P2) and is_Polynomial(P1):
R = PolynomialRing(self.value_ring(), 'x')
P2 = R(P2)
return JacobianMorphism_divisor_class_field(
self, tuple([P1, P2]))
if is_Polynomial(P1) and is_Polynomial(P2):
return JacobianMorphism_divisor_class_field(self, tuple(P))
if is_SchemeMorphism(P1) and is_SchemeMorphism(P2):
return self(P1) - self(P2)
raise TypeError("Argument P (= %s) must have length 2." % P)
elif isinstance(
P,
JacobianMorphism_divisor_class_field) and self == P.parent():
return P
elif is_SchemeMorphism(P):
x0 = P[0]
y0 = P[1]
R, x = PolynomialRing(self.value_ring(), 'x').objgen()
return self((x - x0, R(y0)))
raise TypeError(
"Argument P (= %s) does not determine a divisor class" % P)
def __init__(self, X, v, check=True):
"""
The Python constructor.
EXAMPLES::
sage: P = ProjectiveSpace(2, QQ)
sage: P(2, 3/5, 4)
(1/2 : 3/20 : 1)
::
sage: P = ProjectiveSpace(1, ZZ)
sage: P([0, 1])
(0 : 1)
::
sage: P = ProjectiveSpace(1, ZZ)
sage: P([0, 0, 1])
Traceback (most recent call last):
...
TypeError: v (=[0, 0, 1]) must have 2 components
::
sage: P = ProjectiveSpace(3, QQ)
sage: P(0,0,0,0)
Traceback (most recent call last):
...
ValueError: [0, 0, 0, 0] does not define a valid point since all entries are 0
::
It is possible to avoid the possibly time consuming checks, but be careful!!
sage: P = ProjectiveSpace(3, QQ)
sage: P.point([0,0,0,0],check=False)
(0 : 0 : 0 : 0)
::
sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: X=P.subscheme([x^2-y*z])
sage: X([2,2,2])
(2 : 2 : 2)
"""
SchemeMorphism.__init__(self, X)
if check:
from sage.schemes.elliptic_curves.ell_point import EllipticCurvePoint_field
d = X.codomain().ambient_space().ngens()
if is_SchemeMorphism(v) or isinstance(v, EllipticCurvePoint_field):
v = list(v)
elif v is infinity:
v = [0] * (d)
v[1] = 1
if not isinstance(v,(list,tuple)):
raise TypeError("Argument v (= %s) must be a scheme point, list, or tuple."%str(v))
if len(v) != d and len(v) != d-1:
raise TypeError("v (=%s) must have %s components"%(v, d))
R = X.value_ring()
v = Sequence(v, R)
if len(v) == d-1: # very common special case
v.append(1)
n = len(v)
all_zero = True
for i in range(n):
last = n-1-i
if v[last]:
all_zero = False
break
if all_zero:
raise ValueError("%s does not define a valid point since all entries are 0"%repr(v))
X.extended_codomain()._check_satisfies_equations(v)
if isinstance(X.codomain().base_ring(), QuotientRing_generic):
lift_coords = [P.lift() for P in v]
else:
lift_coords = v
v = Sequence(lift_coords)
self._coords = v
def _call_(self, x):
"""
Construct a scheme from the data in ``x``
EXAMPLES:
Let us first construct the category of schemes::
sage: S = Schemes(); S
Category of schemes
We create a scheme from a ring::
sage: X = S(ZZ); X # indirect doctest
Spectrum of Integer Ring
We create a scheme from a scheme (do nothing)::
sage: S(X)
Spectrum of Integer Ring
We create a scheme morphism from a ring homomorphism.x::
sage: phi = ZZ.hom(QQ); phi
Natural morphism:
From: Integer Ring
To: Rational Field
sage: f = S(phi); f # indirect doctest
Affine Scheme morphism:
From: Spectrum of Rational Field
To: Spectrum of Integer Ring
Defn: Natural morphism:
From: Integer Ring
To: Rational Field
sage: f.domain()
Spectrum of Rational Field
sage: f.codomain()
Spectrum of Integer Ring
sage: S(f) # indirect doctest
Affine Scheme morphism:
From: Spectrum of Rational Field
To: Spectrum of Integer Ring
Defn: Natural morphism:
From: Integer Ring
To: Rational Field
"""
from sage.schemes.generic.scheme import is_Scheme
if is_Scheme(x):
return x
from sage.schemes.generic.morphism import is_SchemeMorphism
if is_SchemeMorphism(x):
return x
from sage.rings.ring import CommutativeRing
from sage.schemes.generic.spec import Spec
from sage.categories.map import Map
from sage.categories.all import Rings
if isinstance(x, CommutativeRing):
return Spec(x)
elif isinstance(x, Map) and x.category_for().is_subcategory(Rings()):
# x is a morphism of Rings
A = Spec(x.codomain())
return A.hom(x)
else:
raise TypeError("No way to create an object or morphism in %s from %s"%(self, x))
def __call__(self, P):
r"""
Returns a rational point P in the abstract Homset J(K), given:
0. A point P in J = Jac(C), returning P; 1. A point P on the curve
C such that J = Jac(C), where C is an odd degree model, returning
[P - oo]; 2. A pair of points (P, Q) on the curve C such that J =
Jac(C), returning [P-Q]; 2. A list of polynomials (a,b) such that
`b^2 + h*b - f = 0 mod a`, returning [(a(x),y-b(x))].
EXAMPLES::
sage: P.<x> = PolynomialRing(QQ)
sage: f = x^5 - x + 1; h = x
sage: C = HyperellipticCurve(f,h,'u,v')
sage: P = C(0,1,1)
sage: J = C.jacobian()
sage: Q = J(QQ)(P)
sage: for i in range(6): i*Q
(1)
(u, v - 1)
(u^2, v + u - 1)
(u^2, v + 1)
(u, v + 1)
(1)
::
sage: F.<a> = GF(3)
sage: R.<x> = F[]
sage: f = x^5-1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: X = J(F)
sage: a = x^2-x+1
sage: b = -x +1
sage: c = x-1
sage: d = 0
sage: D1 = X([a,b])
sage: D1
(x^2 + 2*x + 1, y + x + 2)
sage: D2 = X([c,d])
sage: D2
(x + 2, y)
sage: D1+D2
(x^2 + 2*x + 2, y + 2*x + 1)
"""
if isinstance(P,(int,long,Integer)) and P == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(self, (R(1),R(0)))
elif isinstance(P,(list,tuple)):
if len(P) == 1 and P[0] == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(self, (R(1),R(0)))
elif len(P) == 2:
P1 = P[0]
P2 = P[1]
if is_Integer(P1) and is_Integer(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
P2 = R(P2)
return JacobianMorphism_divisor_class_field(self, tuple([P1,P2]))
if is_Integer(P1) and is_Polynomial(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
return JacobianMorphism_divisor_class_field(self, tuple([P1,P2]))
if is_Integer(P2) and is_Polynomial(P1):
R = PolynomialRing(self.value_ring(), 'x')
P2 = R(P2)
return JacobianMorphism_divisor_class_field(self, tuple([P1,P2]))
if is_Polynomial(P1) and is_Polynomial(P2):
return JacobianMorphism_divisor_class_field(self, tuple(P))
if is_SchemeMorphism(P1) and is_SchemeMorphism(P2):
return self(P1) - self(P2)
raise TypeError("Argument P (= %s) must have length 2."%P)
elif isinstance(P,JacobianMorphism_divisor_class_field) and self == P.parent():
return P
elif is_SchemeMorphism(P):
x0 = P[0]; y0 = P[1]
R, x = PolynomialRing(self.value_ring(), 'x').objgen()
return self((x-x0,R(y0)))
raise TypeError("Argument P (= %s) does not determine a divisor class"%P)
