def __init__(self, J):
"""
"""
R = J.base_ring()
PP = ProjectiveSpace(3, R, ["X0", "X1", "X2", "X3"])
X0, X1, X2, X3 = PP.gens()
C = J.curve()
f, h = C.hyperelliptic_polynomials()
a12 = f[0]
a10 = f[1]
a8 = f[2]
a6 = f[3]
a4 = f[4]
a2 = f[5]
a0 = f[6]
if h != 0:
c6 = h[0]
c4 = h[1]
c2 = h[2]
c0 = h[3]
a12, a10, a8, a6, a4, a2, a0 = \
(4*a12 + c6**2,
4*a10 + 2*c4*c6,
4*a8 + 2*c2*c6 + c4**2,
4*a6 + 2*c0*c6 + 2*c2*c4,
4*a4 + 2*c0*c4 + c2**2,
4*a2 + 2*c0*c2,
4*a0 + c0**2)
F = \
(-4*a8*a12 + a10**2)*X0**4 + \
-4*a6*a12*X0**3*X1 + \
-2*a6*a10*X0**3*X2 + \
-4*a12*X0**3*X3 + \
-4*a4*a12*X0**2*X1**2 + \
(4*a2*a12 - 4*a4*a10)*X0**2*X1*X2 + \
-2*a10*X0**2*X1*X3 + \
(-4*a0*a12 + 2*a2*a10 - 4*a4*a8 + a6**2)*X0**2*X2**2 + \
-4*a8*X0**2*X2*X3 + \
-4*a2*a12*X0*X1**3 + \
(8*a0*a12 - 4*a2*a10)*X0*X1**2*X2 + \
(4*a0*a10 - 4*a2*a8)*X0*X1*X2**2 + \
-2*a6*X0*X1*X2*X3 + \
-2*a2*a6*X0*X2**3 + \
-4*a4*X0*X2**2*X3 + \
-4*X0*X2*X3**2 + \
-4*a0*a12*X1**4 + \
-4*a0*a10*X1**3*X2 + \
-4*a0*a8*X1**2*X2**2 + \
X1**2*X3**2 + \
-4*a0*a6*X1*X2**3 + \
-2*a2*X1*X2**2*X3 + \
(-4*a0*a4 + a2**2)*X2**4 + \
-4*a0*X2**3*X3
AlgebraicScheme_subscheme_projective.__init__(self, PP, F)
X, Y, Z = C.ambient_space().gens()
if a0 == 0:
a0 = a2
phi = Hom(C, self)([0, Z**2, X * Z, a0 * X**2], Schemes())
C._kummer_morphism = phi
J._kummer_surface = self
def __init__(self,J):
"""
"""
R = J.base_ring()
PP = ProjectiveSpace(3,R,["X0","X1","X2","X3"])
X0, X1, X2, X3 = PP.gens()
C = J.curve()
f, h = C.hyperelliptic_polynomials()
a12 = f[0]; a10 = f[1]; a8 = f[2];
a6 = f[3]; a4 = f[4]; a2 = f[5]; a0 = f[6]
if h != 0:
c6 = h[0]; c4 = h[1]; c2 = h[2]; c0 = h[3]
a12, a10, a8, a6, a4, a2, a0 = \
(4*a12 + c6**2,
4*a10 + 2*c4*c6,
4*a8 + 2*c2*c6 + c4**2,
4*a6 + 2*c0*c6 + 2*c2*c4,
4*a4 + 2*c0*c4 + c2**2,
4*a2 + 2*c0*c2,
4*a0 + c0**2)
F = \
(-4*a8*a12 + a10**2)*X0**4 + \
-4*a6*a12*X0**3*X1 + \
-2*a6*a10*X0**3*X2 + \
-4*a12*X0**3*X3 + \
-4*a4*a12*X0**2*X1**2 + \
(4*a2*a12 - 4*a4*a10)*X0**2*X1*X2 + \
-2*a10*X0**2*X1*X3 + \
(-4*a0*a12 + 2*a2*a10 - 4*a4*a8 + a6**2)*X0**2*X2**2 + \
-4*a8*X0**2*X2*X3 + \
-4*a2*a12*X0*X1**3 + \
(8*a0*a12 - 4*a2*a10)*X0*X1**2*X2 + \
(4*a0*a10 - 4*a2*a8)*X0*X1*X2**2 + \
-2*a6*X0*X1*X2*X3 + \
-2*a2*a6*X0*X2**3 + \
-4*a4*X0*X2**2*X3 + \
-4*X0*X2*X3**2 + \
-4*a0*a12*X1**4 + \
-4*a0*a10*X1**3*X2 + \
-4*a0*a8*X1**2*X2**2 + \
X1**2*X3**2 + \
-4*a0*a6*X1*X2**3 + \
-2*a2*X1*X2**2*X3 + \
(-4*a0*a4 + a2**2)*X2**4 + \
-4*a0*X2**3*X3
AlgebraicScheme_subscheme_projective.__init__(self, PP, F)
X, Y, Z = C.ambient_space().gens()
if a0 ==0:
a0 = a2
phi = Hom(C,self)([0,Z**2,X*Z,a0*X**2],Schemes())
C._kummer_morphism = phi
J._kummer_surface = self
def subscheme(self, X):
"""
Return the closed subscheme defined by ``X``.
INPUT:
- ``X`` - a list or tuple of equations.
EXAMPLES::
sage: A.<x,y,z> = ProjectiveSpace(2, QQ)
sage: X = A.subscheme([x*z^2, y^2*z, x*y^2]); X
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x*z^2,
y^2*z,
x*y^2
sage: X.defining_polynomials ()
(x*z^2, y^2*z, x*y^2)
sage: I = X.defining_ideal(); I
Ideal (x*z^2, y^2*z, x*y^2) of Multivariate Polynomial Ring in x, y, z over Rational Field
sage: I.groebner_basis()
[x*y^2, y^2*z, x*z^2]
sage: X.dimension()
0
sage: X.base_ring()
Rational Field
sage: X.base_scheme()
Spectrum of Rational Field
sage: X.structure_morphism()
Scheme morphism:
From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x*z^2,
y^2*z,
x*y^2
To: Spectrum of Rational Field
Defn: Structure map
sage: TestSuite(X).run(skip=["_test_an_element", "_test_elements",\
"_test_elements_eq", "_test_some_elements", "_test_elements_eq_reflexive",\
"_test_elements_eq_symmetric", "_test_elements_eq_transitive",\
"_test_elements_neq"])
"""
from sage.schemes.generic.algebraic_scheme import AlgebraicScheme_subscheme_projective
return AlgebraicScheme_subscheme_projective(self, X)
def affine_patch(self, i, AA=None):
r"""
Return the i-th affine patch of this projective curve.
INPUT:
- ``i`` -- affine coordinate chart of the projective ambient space of this curve to compute affine patch
with respect to.
- ``AA`` -- (default: None) ambient affine space, this is constructed if it is not given.
OUTPUT:
- a curve in affine space.
EXAMPLES::
sage: P.<x,y,z,w> = ProjectiveSpace(CC, 3)
sage: C = Curve([y*z - x^2, w^2 - x*y], P)
sage: C.affine_patch(0)
Affine Curve over Complex Field with 53 bits of precision defined by
x0*x1 - 1.00000000000000, x2^2 - x0
::
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: C = Curve(x^3 - x^2*y + y^3 - x^2*z, P)
sage: C.affine_patch(1)
Affine Plane Curve over Rational Field defined by x0^3 - x0^2*x1 - x0^2 + 1
::
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: P.<u,v,w> = ProjectiveSpace(QQ, 2)
sage: C = Curve([u^2 - v^2], P)
sage: C.affine_patch(1, A).ambient_space() == A
True
"""
from .constructor import Curve
return Curve(
AlgebraicScheme_subscheme_projective.affine_patch(self, i, AA))
def affine_patch(self, i, AA=None):
r"""
Return the i-th affine patch of this projective curve.
INPUT:
- ``i`` -- affine coordinate chart of the projective ambient space of this curve to compute affine patch
with respect to.
- ``AA`` -- (default: None) ambient affine space, this is constructed if it is not given.
OUTPUT:
- a curve in affine space.
EXAMPLES::
sage: P.<x,y,z,w> = ProjectiveSpace(CC, 3)
sage: C = Curve([y*z - x^2, w^2 - x*y], P)
sage: C.affine_patch(0)
Affine Curve over Complex Field with 53 bits of precision defined by
x0*x1 - 1.00000000000000, x2^2 - x0
::
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: C = Curve(x^3 - x^2*y + y^3 - x^2*z, P)
sage: C.affine_patch(1)
Affine Plane Curve over Rational Field defined by x0^3 - x0^2*x1 - x0^2 + 1
::
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: P.<u,v,w> = ProjectiveSpace(QQ, 2)
sage: C = Curve([u^2 - v^2], P)
sage: C.affine_patch(1, A).ambient_space() == A
True
"""
from .constructor import Curve
return Curve(AlgebraicScheme_subscheme_projective.affine_patch(self, i, AA))