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_subschemeGeneric 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
REXAMPLES:
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=Falseand/orreduce=Falseit 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
Pis 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
Pis 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
Pis provided, and ifPlies on this curve, returnsTrueifPis a singular point of this curve, andFalseotherwise. If no point is provided, returnsTrueor 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
selfandother.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