Base class of curves¶
This module defines the base class of curves in Sage.
Curves in Sage are reduced subschemes of dimension 1 of an ambient space. The ambient space is either an affine space or a projective space.
EXAMPLES:
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: C = Curve([x - y, z - 2])
sage: C
Affine Curve over Rational Field defined by x - y, z - 2
sage: C.dimension()
1
>>> from sage.all import *
>>> A = AffineSpace(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> C = Curve([x - y, z - Integer(2)])
>>> C
Affine Curve over Rational Field defined by x - y, z - 2
>>> C.dimension()
1
AUTHORS:
William Stein (2005)
- class sage.schemes.curves.curve.Curve_generic(A, polynomials, category=None)[source]¶
Bases:
AlgebraicScheme_subscheme
Generic curve class.
EXAMPLES:
sage: A.<x,y,z> = AffineSpace(QQ, 3) sage: C = Curve([x - y, z - 2]) sage: loads(C.dumps()) == C True
>>> from sage.all import * >>> A = AffineSpace(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3) >>> C = Curve([x - y, z - Integer(2)]) >>> loads(C.dumps()) == C True
- change_ring(R)[source]¶
Return a new curve which is this curve coerced to
R
.INPUT:
R
– ring or embedding
OUTPUT: a new curve which is this curve coerced to
R
EXAMPLES:
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3) sage: C = Curve([x^2 - y^2, z*y - 4/5*w^2], P) sage: C.change_ring(QuadraticField(-1)) # needs sage.rings.number_field Projective Curve over Number Field in a with defining polynomial x^2 + 1 with a = 1*I defined by x^2 - y^2, y*z - 4/5*w^2
>>> from sage.all import * >>> P = ProjectiveSpace(QQ, Integer(3), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = P._first_ngens(4) >>> C = Curve([x**Integer(2) - y**Integer(2), z*y - Integer(4)/Integer(5)*w**Integer(2)], P) >>> C.change_ring(QuadraticField(-Integer(1))) # needs sage.rings.number_field Projective Curve over Number Field in a with defining polynomial x^2 + 1 with a = 1*I defined by x^2 - y^2, y*z - 4/5*w^2
sage: # needs sage.rings.number_field sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^3 + a^2 - 1) sage: A.<x,y> = AffineSpace(K, 2) sage: C = Curve([K.0*x^2 - x + y^3 - 11], A) sage: L = K.embeddings(QQbar) sage: set_verbose(-1) # suppress warnings for slow computation sage: C.change_ring(L[0]) Affine Plane Curve over Algebraic Field defined by y^3 + (-0.8774388331233464? - 0.744861766619745?*I)*x^2 - x - 11
>>> from sage.all import * >>> # needs sage.rings.number_field >>> R = QQ['a']; (a,) = R._first_ngens(1) >>> K = NumberField(a**Integer(3) + a**Integer(2) - Integer(1), names=('b',)); (b,) = K._first_ngens(1) >>> A = AffineSpace(K, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> C = Curve([K.gen(0)*x**Integer(2) - x + y**Integer(3) - Integer(11)], A) >>> L = K.embeddings(QQbar) >>> set_verbose(-Integer(1)) # suppress warnings for slow computation >>> C.change_ring(L[Integer(0)]) Affine Plane Curve over Algebraic Field defined by y^3 + (-0.8774388331233464? - 0.744861766619745?*I)*x^2 - x - 11
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = P.curve([y*x - 18*x^2 + 17*z^2]) sage: C.change_ring(GF(17)) Projective Plane Curve over Finite Field of size 17 defined by -x^2 + x*y
>>> from sage.all import * >>> P = ProjectiveSpace(QQ, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> C = P.curve([y*x - Integer(18)*x**Integer(2) + Integer(17)*z**Integer(2)]) >>> C.change_ring(GF(Integer(17))) Projective Plane Curve over Finite Field of size 17 defined by -x^2 + x*y
- defining_polynomial()[source]¶
Return the defining polynomial of the curve.
EXAMPLES:
sage: x,y,z = PolynomialRing(QQ, 3, names='x,y,z').gens() sage: C = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2) sage: C.defining_polynomial() -x^3 + y^2*z - 17*x*z^2 + y*z^2
>>> from sage.all import * >>> x,y,z = PolynomialRing(QQ, Integer(3), names='x,y,z').gens() >>> C = Curve(y**Integer(2)*z - x**Integer(3) - Integer(17)*x*z**Integer(2) + y*z**Integer(2)) >>> C.defining_polynomial() -x^3 + y^2*z - 17*x*z^2 + y*z^2
- dimension()[source]¶
Return the dimension of the curve.
Curves have dimension one by definition.
EXAMPLES:
sage: x = polygen(QQ) sage: C = HyperellipticCurve(x^7 + x^4 + x) sage: C.dimension() 1 sage: from sage.schemes.projective.projective_subscheme import AlgebraicScheme_subscheme_projective sage: AlgebraicScheme_subscheme_projective.dimension(C) 1
>>> from sage.all import * >>> x = polygen(QQ) >>> C = HyperellipticCurve(x**Integer(7) + x**Integer(4) + x) >>> C.dimension() 1 >>> from sage.schemes.projective.projective_subscheme import AlgebraicScheme_subscheme_projective >>> AlgebraicScheme_subscheme_projective.dimension(C) 1
- divisor(v, base_ring=None, check=True, reduce=True)[source]¶
Return the divisor specified by
v
.Warning
The coefficients of the divisor must be in the base ring and the terms must be reduced. If you set
check=False
and/orreduce=False
it is your responsibility to pass a valid objectv
.EXAMPLES:
sage: x,y,z = PolynomialRing(QQ, 3, names='x,y,z').gens() sage: C = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2) sage: p1 = C(0, -1, 1) sage: p2 = C(0, 0, 1) sage: p3 = C(0, 1, 0) sage: C.divisor([(1, p1), (-1, p2), (2, p3)]) (x, y + z) - (x, y) + 2*(x, z)
>>> from sage.all import * >>> x,y,z = PolynomialRing(QQ, Integer(3), names='x,y,z').gens() >>> C = Curve(y**Integer(2)*z - x**Integer(3) - Integer(17)*x*z**Integer(2) + y*z**Integer(2)) >>> p1 = C(Integer(0), -Integer(1), Integer(1)) >>> p2 = C(Integer(0), Integer(0), Integer(1)) >>> p3 = C(Integer(0), Integer(1), Integer(0)) >>> C.divisor([(Integer(1), p1), (-Integer(1), p2), (Integer(2), p3)]) (x, y + z) - (x, y) + 2*(x, z)
- divisor_group(base_ring=None)[source]¶
Return the divisor group of the curve.
INPUT:
base_ring
– the base ring of the divisor group; usually, this is \(\ZZ\) (default) or \(\QQ\)
OUTPUT: the divisor group of the curve
EXAMPLES:
sage: x,y,z = PolynomialRing(QQ, 3, names='x,y,z').gens() sage: C = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2) sage: Cp = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2) sage: C.divisor_group() is Cp.divisor_group() True
>>> from sage.all import * >>> x,y,z = PolynomialRing(QQ, Integer(3), names='x,y,z').gens() >>> C = Curve(y**Integer(2)*z - x**Integer(3) - Integer(17)*x*z**Integer(2) + y*z**Integer(2)) >>> Cp = Curve(y**Integer(2)*z - x**Integer(3) - Integer(17)*x*z**Integer(2) + y*z**Integer(2)) >>> C.divisor_group() is Cp.divisor_group() True
- genus()[source]¶
Return the geometric genus of the curve.
EXAMPLES:
sage: x,y,z = PolynomialRing(QQ, 3, names='x,y,z').gens() sage: C = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2) sage: C.genus() 1
>>> from sage.all import * >>> x,y,z = PolynomialRing(QQ, Integer(3), names='x,y,z').gens() >>> C = Curve(y**Integer(2)*z - x**Integer(3) - Integer(17)*x*z**Integer(2) + y*z**Integer(2)) >>> C.genus() 1
- geometric_genus()[source]¶
Return the geometric genus of the curve.
EXAMPLES:
Examples of projective curves:
sage: P2 = ProjectiveSpace(2, GF(5), names=['x','y','z']) sage: x, y, z = P2.coordinate_ring().gens() sage: C = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2) sage: C.geometric_genus() 1 sage: C = Curve(y^2*z - x^3) sage: C.geometric_genus() 0 sage: C = Curve(x^10 + y^7*z^3 + z^10) sage: C.geometric_genus() 3
>>> from sage.all import * >>> P2 = ProjectiveSpace(Integer(2), GF(Integer(5)), names=['x','y','z']) >>> x, y, z = P2.coordinate_ring().gens() >>> C = Curve(y**Integer(2)*z - x**Integer(3) - Integer(17)*x*z**Integer(2) + y*z**Integer(2)) >>> C.geometric_genus() 1 >>> C = Curve(y**Integer(2)*z - x**Integer(3)) >>> C.geometric_genus() 0 >>> C = Curve(x**Integer(10) + y**Integer(7)*z**Integer(3) + z**Integer(10)) >>> C.geometric_genus() 3
Examples of affine curves:
sage: x, y = PolynomialRing(GF(5), 2, 'xy').gens() sage: C = Curve(y^2 - x^3 - 17*x + y) sage: C.geometric_genus() 1 sage: C = Curve(y^2 - x^3) sage: C.geometric_genus() 0 sage: C = Curve(x^10 + y^7 + 1) sage: C.geometric_genus() 3
>>> from sage.all import * >>> x, y = PolynomialRing(GF(Integer(5)), Integer(2), 'xy').gens() >>> C = Curve(y**Integer(2) - x**Integer(3) - Integer(17)*x + y) >>> C.geometric_genus() 1 >>> C = Curve(y**Integer(2) - x**Integer(3)) >>> C.geometric_genus() 0 >>> C = Curve(x**Integer(10) + y**Integer(7) + Integer(1)) >>> C.geometric_genus() 3
Warning
Geometric genus is only defined for geometrically irreducible curve. This method does not check the condition. You may get a nonsensical result if the curve is not geometrically irreducible:
sage: P2.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve(x^2 + y^2, P2) sage: C.geometric_genus() # nonsense! -1
>>> from sage.all import * >>> P2 = ProjectiveSpace(QQ, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3) >>> C = Curve(x**Integer(2) + y**Integer(2), P2) >>> C.geometric_genus() # nonsense! -1
- intersection_points(C, F=None)[source]¶
Return the points in the intersection of this curve and the curve
C
.If the intersection of these two curves has dimension greater than zero, and if the base ring of this curve is not a finite field, then an error is returned.
INPUT:
C
– a curve in the same ambient space as this curveF
– (default:None
) field over which to compute the intersection points; if not specified, the base ring of this curve is used
OUTPUT: list of points in the ambient space of this curve
EXAMPLES:
sage: # needs sage.rings.number_field sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^2 + a + 1) sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3) sage: C = Curve([y^2 - w*z, w^3 - y^3], P) sage: D = Curve([x*y - w*z, z^3 - y^3], P) sage: C.intersection_points(D, F=K) [(-b - 1 : -b - 1 : b : 1), (b : b : -b - 1 : 1), (1 : 0 : 0 : 0), (1 : 1 : 1 : 1)]
>>> from sage.all import * >>> # needs sage.rings.number_field >>> R = QQ['a']; (a,) = R._first_ngens(1) >>> K = NumberField(a**Integer(2) + a + Integer(1), names=('b',)); (b,) = K._first_ngens(1) >>> P = ProjectiveSpace(QQ, Integer(3), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = P._first_ngens(4) >>> C = Curve([y**Integer(2) - w*z, w**Integer(3) - y**Integer(3)], P) >>> D = Curve([x*y - w*z, z**Integer(3) - y**Integer(3)], P) >>> C.intersection_points(D, F=K) [(-b - 1 : -b - 1 : b : 1), (b : b : -b - 1 : 1), (1 : 0 : 0 : 0), (1 : 1 : 1 : 1)]
sage: A.<x,y> = AffineSpace(GF(7), 2) sage: C = Curve([y^3 - x^3], A) sage: D = Curve([-x*y^3 + y^4 - 2*x^3 + 2*x^2*y], A) sage: C.intersection_points(D) [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4), (5, 3), (5, 5), (5, 6), (6, 6)]
>>> from sage.all import * >>> A = AffineSpace(GF(Integer(7)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> C = Curve([y**Integer(3) - x**Integer(3)], A) >>> D = Curve([-x*y**Integer(3) + y**Integer(4) - Integer(2)*x**Integer(3) + Integer(2)*x**Integer(2)*y], A) >>> C.intersection_points(D) [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4), (5, 3), (5, 5), (5, 6), (6, 6)]
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([y^3 - x^3], A) sage: D = Curve([-x*y^3 + y^4 - 2*x^3 + 2*x^2*y], A) sage: C.intersection_points(D) Traceback (most recent call last): ... NotImplementedError: the intersection must have dimension zero or (=Rational Field) must be a finite field
>>> from sage.all import * >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> C = Curve([y**Integer(3) - x**Integer(3)], A) >>> D = Curve([-x*y**Integer(3) + y**Integer(4) - Integer(2)*x**Integer(3) + Integer(2)*x**Integer(2)*y], A) >>> C.intersection_points(D) Traceback (most recent call last): ... NotImplementedError: the intersection must have dimension zero or (=Rational Field) must be a finite field
- intersects_at(C, P)[source]¶
Return whether the point
P
is or is not in the intersection of this curve with the curveC
.INPUT:
C
– a curve in the same ambient space as this curveP
– a point in the ambient space of this curve
EXAMPLES:
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3) sage: C = Curve([x^2 - z^2, y^3 - w*x^2], P) sage: D = Curve([w^2 - 2*x*y + z^2, y^2 - w^2], P) sage: Q1 = P([1,1,-1,1]) sage: C.intersects_at(D, Q1) True sage: Q2 = P([0,0,1,-1]) sage: C.intersects_at(D, Q2) False
>>> from sage.all import * >>> P = ProjectiveSpace(QQ, Integer(3), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = P._first_ngens(4) >>> C = Curve([x**Integer(2) - z**Integer(2), y**Integer(3) - w*x**Integer(2)], P) >>> D = Curve([w**Integer(2) - Integer(2)*x*y + z**Integer(2), y**Integer(2) - w**Integer(2)], P) >>> Q1 = P([Integer(1),Integer(1),-Integer(1),Integer(1)]) >>> C.intersects_at(D, Q1) True >>> Q2 = P([Integer(0),Integer(0),Integer(1),-Integer(1)]) >>> C.intersects_at(D, Q2) False
sage: A.<x,y> = AffineSpace(GF(13), 2) sage: C = Curve([y + 12*x^5 + 3*x^3 + 7], A) sage: D = Curve([y^2 + 7*x^2 + 8], A) sage: Q1 = A([9,6]) sage: C.intersects_at(D, Q1) True sage: Q2 = A([3,7]) sage: C.intersects_at(D, Q2) False
>>> from sage.all import * >>> A = AffineSpace(GF(Integer(13)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> C = Curve([y + Integer(12)*x**Integer(5) + Integer(3)*x**Integer(3) + Integer(7)], A) >>> D = Curve([y**Integer(2) + Integer(7)*x**Integer(2) + Integer(8)], A) >>> Q1 = A([Integer(9),Integer(6)]) >>> C.intersects_at(D, Q1) True >>> Q2 = A([Integer(3),Integer(7)]) >>> C.intersects_at(D, Q2) False
- is_singular(P=None)[source]¶
Return whether
P
is a singular point of this curve, or if no point is passed, whether this curve is singular or not.This just uses the is_smooth function for algebraic subschemes.
INPUT:
P
– (default:None
) a point on this curve
OUTPUT:
boolean; if a point
P
is provided, and ifP
lies on this curve, returnsTrue
ifP
is a singular point of this curve, andFalse
otherwise. If no point is provided, returnsTrue
or False depending on whether this curve is or is not singular, respectively.EXAMPLES:
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3) sage: C = P.curve([y^2 - x^2 - z^2, z - w]) sage: C.is_singular() False
>>> from sage.all import * >>> P = ProjectiveSpace(QQ, Integer(3), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = P._first_ngens(4) >>> C = P.curve([y**Integer(2) - x**Integer(2) - z**Integer(2), z - w]) >>> C.is_singular() False
sage: A.<x,y,z> = AffineSpace(GF(11), 3) sage: C = A.curve([y^3 - z^5, x^5 - y + 1]) sage: Q = A([7,0,0]) sage: C.is_singular(Q) True
>>> from sage.all import * >>> A = AffineSpace(GF(Integer(11)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3) >>> C = A.curve([y**Integer(3) - z**Integer(5), x**Integer(5) - y + Integer(1)]) >>> Q = A([Integer(7),Integer(0),Integer(0)]) >>> C.is_singular(Q) True
- singular_points(F=None)[source]¶
Return the set of singular points of this curve.
INPUT:
F
– (default:None
) field over which to find the singular points; if not given, the base ring of this curve is used
OUTPUT: list of points in the ambient space of this curve
EXAMPLES:
sage: A.<x,y,z> = AffineSpace(QQ, 3) sage: C = Curve([y^2 - x^5, x - z], A) sage: C.singular_points() [(0, 0, 0)]
>>> from sage.all import * >>> A = AffineSpace(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3) >>> C = Curve([y**Integer(2) - x**Integer(5), x - z], A) >>> C.singular_points() [(0, 0, 0)]
sage: # needs sage.rings.number_field sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^8 - a^4 + 1) sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([359/12*x*y^2*z^2 + 2*y*z^4 + 187/12*y^3*z^2 + x*z^4 ....: + 67/3*x^2*y*z^2 + 117/4*y^5 + 9*x^5 + 6*x^3*z^2 ....: + 393/4*x*y^4 + 145*x^2*y^3 + 115*x^3*y^2 + 49*x^4*y], P) sage: sorted(C.singular_points(K), key=str) [(-1/2*b^5 - 1/2*b^3 + 1/2*b - 1 : 1 : 0), (-2/3*b^4 + 1/3 : 0 : 1), (-b^6 : b^6 : 1), (1/2*b^5 + 1/2*b^3 - 1/2*b - 1 : 1 : 0), (2/3*b^4 - 1/3 : 0 : 1), (b^6 : -b^6 : 1)]
>>> from sage.all import * >>> # needs sage.rings.number_field >>> R = QQ['a']; (a,) = R._first_ngens(1) >>> K = NumberField(a**Integer(8) - a**Integer(4) + Integer(1), names=('b',)); (b,) = K._first_ngens(1) >>> P = ProjectiveSpace(QQ, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> C = Curve([Integer(359)/Integer(12)*x*y**Integer(2)*z**Integer(2) + Integer(2)*y*z**Integer(4) + Integer(187)/Integer(12)*y**Integer(3)*z**Integer(2) + x*z**Integer(4) ... + Integer(67)/Integer(3)*x**Integer(2)*y*z**Integer(2) + Integer(117)/Integer(4)*y**Integer(5) + Integer(9)*x**Integer(5) + Integer(6)*x**Integer(3)*z**Integer(2) ... + Integer(393)/Integer(4)*x*y**Integer(4) + Integer(145)*x**Integer(2)*y**Integer(3) + Integer(115)*x**Integer(3)*y**Integer(2) + Integer(49)*x**Integer(4)*y], P) >>> sorted(C.singular_points(K), key=str) [(-1/2*b^5 - 1/2*b^3 + 1/2*b - 1 : 1 : 0), (-2/3*b^4 + 1/3 : 0 : 1), (-b^6 : b^6 : 1), (1/2*b^5 + 1/2*b^3 - 1/2*b - 1 : 1 : 0), (2/3*b^4 - 1/3 : 0 : 1), (b^6 : -b^6 : 1)]
- singular_subscheme()[source]¶
Return the subscheme of singular points of this curve.
OUTPUT: a subscheme in the ambient space of this curve
EXAMPLES:
sage: A.<x,y> = AffineSpace(CC, 2) sage: C = Curve([y^4 - 2*x^5 - x^2*y], A) sage: C.singular_subscheme() Closed subscheme of Affine Space of dimension 2 over Complex Field with 53 bits of precision defined by: (-2.00000000000000)*x^5 + y^4 - x^2*y, (-10.0000000000000)*x^4 + (-2.00000000000000)*x*y, 4.00000000000000*y^3 - x^2
>>> from sage.all import * >>> A = AffineSpace(CC, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> C = Curve([y**Integer(4) - Integer(2)*x**Integer(5) - x**Integer(2)*y], A) >>> C.singular_subscheme() Closed subscheme of Affine Space of dimension 2 over Complex Field with 53 bits of precision defined by: (-2.00000000000000)*x^5 + y^4 - x^2*y, (-10.0000000000000)*x^4 + (-2.00000000000000)*x*y, 4.00000000000000*y^3 - x^2
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3) sage: C = Curve([y^8 - x^2*z*w^5, w^2 - 2*y^2 - x*z], P) sage: C.singular_subscheme() Closed subscheme of Projective Space of dimension 3 over Rational Field defined by: y^8 - x^2*z*w^5, -2*y^2 - x*z + w^2, -x^3*y*z^4 + 3*x^2*y*z^3*w^2 - 3*x*y*z^2*w^4 + 8*x*y*z*w^5 + y*z*w^6, x^2*z*w^5, -5*x^2*z^2*w^4 - 4*x*z*w^6, x^4*y*z^3 - 3*x^3*y*z^2*w^2 + 3*x^2*y*z*w^4 - 4*x^2*y*w^5 - x*y*w^6, -2*x^3*y*z^3*w + 6*x^2*y*z^2*w^3 - 20*x^2*y*z*w^4 - 6*x*y*z*w^5 + 2*y*w^7, -5*x^3*z*w^4 - 2*x^2*w^6
>>> from sage.all import * >>> P = ProjectiveSpace(QQ, Integer(3), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = P._first_ngens(4) >>> C = Curve([y**Integer(8) - x**Integer(2)*z*w**Integer(5), w**Integer(2) - Integer(2)*y**Integer(2) - x*z], P) >>> C.singular_subscheme() Closed subscheme of Projective Space of dimension 3 over Rational Field defined by: y^8 - x^2*z*w^5, -2*y^2 - x*z + w^2, -x^3*y*z^4 + 3*x^2*y*z^3*w^2 - 3*x*y*z^2*w^4 + 8*x*y*z*w^5 + y*z*w^6, x^2*z*w^5, -5*x^2*z^2*w^4 - 4*x*z*w^6, x^4*y*z^3 - 3*x^3*y*z^2*w^2 + 3*x^2*y*z*w^4 - 4*x^2*y*w^5 - x*y*w^6, -2*x^3*y*z^3*w + 6*x^2*y*z^2*w^3 - 20*x^2*y*z*w^4 - 6*x*y*z*w^5 + 2*y*w^7, -5*x^3*z*w^4 - 2*x^2*w^6
- union(other)[source]¶
Return the union of
self
andother
.EXAMPLES:
sage: x,y,z = PolynomialRing(QQ, 3, names='x,y,z').gens() sage: C1 = Curve(z - x) sage: C2 = Curve(y - x) sage: C1.union(C2).defining_polynomial() x^2 - x*y - x*z + y*z
>>> from sage.all import * >>> x,y,z = PolynomialRing(QQ, Integer(3), names='x,y,z').gens() >>> C1 = Curve(z - x) >>> C2 = Curve(y - x) >>> C1.union(C2).defining_polynomial() x^2 - x*y - x*z + y*z