Divisors on schemes#
AUTHORS:
William Stein
David Kohel
David Joyner
Volker Braun (2010-07-16): Documentation, doctests, coercion fixes, bugfixes.
EXAMPLES:
sage: x,y,z = ProjectiveSpace(2, GF(5), names='x,y,z').gens() # optional - sage.rings.finite_rings
sage: C = Curve(y^2*z^7 - x^9 - x*z^8) # optional - sage.rings.finite_rings
sage: pts = C.rational_points(); pts # optional - sage.rings.finite_rings
[(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 # optional - sage.rings.finite_rings
sage: D2 = C.divisor(pts[1]) # optional - sage.rings.finite_rings
sage: D3 = 10*C.divisor(pts[5]) # optional - sage.rings.finite_rings
sage: D1.parent() is D2.parent() # optional - sage.rings.finite_rings
True
sage: D = D1 - D2 + D3; D # optional - sage.rings.finite_rings
3*(x, y) - (x, z) + 10*(x + 2*z, y + z)
sage: D[1][0] # optional - sage.rings.finite_rings
-1
sage: D[1][1] # optional - sage.rings.finite_rings
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])]) # optional - sage.rings.finite_rings
3*(x, y) - (x, z) + 10*(x + 2*z, y + z)
- sage.schemes.generic.divisor.CurvePointToIdeal(C, P)#
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
- class sage.schemes.generic.divisor.Divisor_curve(v, parent=None, check=True, reduce=True)#
Bases:
Divisor_generic
For any curve \(C\), use
C.divisor(v)
to construct a divisor on \(C\). Here \(v\) can be eithera rational point on \(C\)
a list of rational points
a list of 2-tuples \((c,P)\), where \(c\) is an integer and \(P\) is a rational point.
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 \(E/K\) that has coordinates in \(L\). 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)
- coefficient(P)#
Return the coefficient of a given point P in this divisor.
EXAMPLES:
sage: x,y = AffineSpace(2, GF(5), names='xy').gens() # optional - sage.rings.finite_rings sage: C = Curve(y^2 - x^9 - x) # optional - sage.rings.finite_rings sage: pts = C.rational_points(); pts # optional - sage.rings.finite_rings [(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)] sage: D = C.divisor(pts[0]) # optional - sage.rings.finite_rings sage: D.coefficient(pts[0]) # optional - sage.rings.finite_rings 1 sage: D = C.divisor([(3, pts[0]), (-1, pts[1])]); D # optional - sage.rings.finite_rings 3*(x, y) - (x - 2, y - 2) sage: D.coefficient(pts[0]) # optional - sage.rings.finite_rings 3 sage: D.coefficient(pts[1]) # optional - sage.rings.finite_rings -1
- support()#
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() # optional - sage.rings.finite_rings sage: C = Curve(y^2 - x^9 - x) # optional - sage.rings.finite_rings sage: pts = C.rational_points(); pts # optional - sage.rings.finite_rings [(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)] sage: D = C.divisor_group()([(3, pts[0]), (-1, pts[1])]); D # optional - sage.rings.finite_rings 3*(x, y) - (x - 2, y - 2) sage: D.support() # optional - sage.rings.finite_rings [(0, 0), (2, 2)]
- class sage.schemes.generic.divisor.Divisor_generic(v, parent, check=True, reduce=True)#
Bases:
FormalSum
A Divisor.
- scheme()#
Return the scheme that this divisor is on.
EXAMPLES:
sage: A.<x, y> = AffineSpace(2, GF(5)) # optional - sage.rings.finite_rings sage: C = Curve(y^2 - x^9 - x) # optional - sage.rings.finite_rings sage: pts = C.rational_points(); pts # optional - sage.rings.finite_rings [(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)] sage: D = C.divisor(pts[0])*3 - C.divisor(pts[1]); D # optional - sage.rings.finite_rings 3*(x, y) - (x - 2, y - 2) sage: D.scheme() # optional - sage.rings.finite_rings Affine Plane Curve over Finite Field of size 5 defined by -x^9 + y^2 - x
- sage.schemes.generic.divisor.is_Divisor(x)#
Test whether
x
is an instance ofDivisor_generic
INPUT:
x
– anything.
OUTPUT:
True
orFalse
.EXAMPLES:
sage: from sage.schemes.generic.divisor import is_Divisor sage: x,y = AffineSpace(2, GF(5), names='xy').gens() # optional - sage.rings.finite_rings sage: C = Curve(y^2 - x^9 - x) # optional - sage.rings.finite_rings sage: is_Divisor(C.divisor([])) # optional - sage.rings.finite_rings True sage: is_Divisor("Ceci n'est pas un diviseur") False