AUTHORS:
EXAMPLES:
sage: x,y,z = ProjectiveSpace(2, GF(5), names='x,y,z').gens()
sage: C = Curve(y^2*z^7 - x^9 - x*z^8)
sage: pts = C.rational_points(); pts
[(0 : 0 : 1), (0 : 1 : 0), (2 : 2 : 1), (2 : 3 : 1), (3 : 1 : 1), (3 : 4 : 1)]
sage: D1 = C.divisor(pts[0])*3
sage: D2 = C.divisor(pts[1])
sage: D3 = 10*C.divisor(pts[5])
sage: D1.parent() is D2.parent()
True
sage: D = D1 - D2 + D3; D
3*(x, y) - (x, z) + 10*(x + 2*z, y + z)
sage: D[1][0]
-1
sage: D[1][1]
Ideal (x, z) of Multivariate Polynomial Ring in x, y, z over Finite Field of size 5
sage: C.divisor([(3, pts[0]), (-1, pts[1]), (10,pts[5])])
3*(x, y) - (x, z) + 10*(x + 2*z, y + z)
Return the vanishing ideal of a point on a curve.
EXAMPLES:
sage: x,y = AffineSpace(2, QQ, names='xy').gens()
sage: C = Curve(y^2 - x^9 - x)
sage: from sage.schemes.generic.divisor import CurvePointToIdeal
sage: CurvePointToIdeal(C, (0,0))
Ideal (x, y) of Multivariate Polynomial Ring in x, y over Rational Field
Bases: sage.schemes.generic.divisor.Divisor_generic
For any curve , use C.divisor(v) to
construct a divisor on
. Here
can be either
TODO: Divisors shouldn’t be restricted to rational points. The
problem is that the divisor group is the formal sum of the group of
points on the curve, and there’s no implemented notion of point on
that has coordinates in
. This is what
should be implemented, by adding an appropriate class to
schemes/generic/morphism.py.
EXAMPLES:
sage: E = EllipticCurve([0, 0, 1, -1, 0])
sage: P = E(0,0)
sage: 10*P
(161/16 : -2065/64 : 1)
sage: D = E.divisor(P)
sage: D
(x, y)
sage: 10*D
10*(x, y)
sage: E.divisor([P, P])
2*(x, y)
sage: E.divisor([(3,P), (-4,5*P)])
3*(x, y) - 4*(x - 1/4*z, y + 5/8*z)
Synonym for coefficient()
Warning
This method is deprecated. It will be removed in a future release of Sage. Please use the coefficient(P) method instead.
EXAMPLES:
sage: x,y = AffineSpace(2, GF(5), names='xy').gens()
sage: C = Curve(y^2 - x^9 - x)
sage: pts = C.rational_points(); pts
[(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)]
sage: D = C.divisor(pts[0])
sage: D.coefficient(pts[0])
1
Return the coefficient of a given point P in this divisor.
EXAMPLES:
sage: x,y = AffineSpace(2, GF(5), names='xy').gens()
sage: C = Curve(y^2 - x^9 - x)
sage: pts = C.rational_points(); pts
[(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)]
sage: D = C.divisor(pts[0])
sage: D.coefficient(pts[0])
1
sage: D = C.divisor([(3,pts[0]), (-1,pts[1])]); D
3*(x, y) - (x - 2, y - 2)
sage: D.coefficient(pts[0])
3
sage: D.coefficient(pts[1])
-1
Return the support of this divisor, which is the set of points that occur in this divisor with nonzero coefficients.
EXAMPLES:
sage: x,y = AffineSpace(2, GF(5), names='xy').gens()
sage: C = Curve(y^2 - x^9 - x)
sage: pts = C.rational_points(); pts
[(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)]
sage: D = C.divisor_group()([(3,pts[0]), (-1, pts[1])]); D
3*(x, y) - (x - 2, y - 2)
sage: D.support()
[(0, 0), (2, 2)]
TESTS:
This checks that trac ticket #10732 is fixed:
sage: R.<x, y, z> = GF(5)[]
sage: C = Curve(x^7 + y^7 + z^7)
sage: pts = C.rational_points()
sage: D = C.divisor([(2, pts[0])])
sage: D.support()
[(0 : 4 : 1)]
sage: (D + D).support()
[(0 : 4 : 1)]
sage: E = C.divisor([(-3, pts[1]), (1, pts[2])])
sage: (D - 2*E).support()
[(0 : 4 : 1), (1 : 2 : 1), (2 : 1 : 1)]
sage: (D - D).support()
[]
Bases: sage.structure.formal_sum.FormalSum
A Divisor.
Return the scheme that this divisor is on.
EXAMPLES:
sage: A.<x, y> = AffineSpace(2, GF(5))
sage: C = Curve(y^2 - x^9 - x)
sage: pts = C.rational_points(); pts
[(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)]
sage: D = C.divisor(pts[0])*3 - C.divisor(pts[1]); D
3*(x, y) - (x - 2, y - 2)
sage: D.scheme()
Affine Curve over Finite Field of size 5 defined by -x^9 + y^2 - x
Test whether x is an instance of Divisor_generic
INPUT:
OUTPUT:
True or False.
EXAMPLES:
sage: from sage.schemes.generic.divisor import is_Divisor
sage: x,y = AffineSpace(2, GF(5), names='xy').gens()
sage: C = Curve(y^2 - x^9 - x)
sage: is_Divisor( C.divisor([]) )
True
sage: is_Divisor("Ceci n'est pas un diviseur")
False