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
-(x, z) + 3*(x, y) + 10*(x + 2*z, y + z)
sage: D[1][0]
3
sage: D[1][1]
Ideal (x, y) 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])])
-(x, z) + 3*(x, y) + 10*(x + 2*z, y + z)
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)
INPUT:
To create the 0 divisor use [(0, P)], so as to give the curve.
TODO: Include an extension field in the definition of the divisor group.
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.coeff(pts[0])
1
sage: D = C.divisor([(3,pts[0]), (-1,pts[1])]); D
3*(x, y) - (x - 2, y - 2)
sage: D.coeff(pts[0])
3
sage: D.coeff(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([(3,pts[0]), (-1, pts[1])]); D
3*(x, y) - (x - 2, y - 2)
sage: D.support()
[(0, 0), (2, 2)]
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