Weighted projective curves

Weighted projective curves in Sage are curves in a weighted projective space or a weighted projective plane.

EXAMPLES:

For now, only curves in weighted projective plane is supported:

sage: WP.<x, y, z> = WeightedProjectiveSpace([1, 3, 1], QQ)
sage: C1 = WP.curve(y^2 - x^5 * z - 3 * x^2 * z^4 - 2 * z^6); C1
Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6
sage: C2 = Curve(y^2 - x^5 * z - 3 * x^2 * z^4 - 2 * z^6, WP); C2
Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6
sage: C1 == C2
True

>>> from sage.all import *
>>> WP = WeightedProjectiveSpace([Integer(1), Integer(3), Integer(1)], QQ, names=('x', 'y', 'z',)); (x, y, z,) = WP._first_ngens(3)
>>> C1 = WP.curve(y**Integer(2) - x**Integer(5) * z - Integer(3) * x**Integer(2) * z**Integer(4) - Integer(2) * z**Integer(6)); C1
Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6
>>> C2 = Curve(y**Integer(2) - x**Integer(5) * z - Integer(3) * x**Integer(2) * z**Integer(4) - Integer(2) * z**Integer(6), WP); C2
Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6
>>> C1 == C2
True

AUTHORS:

  • Gareth Ma (2025)

class sage.schemes.curves.weighted_projective_curve.WeightedProjectiveCurve(A, X, *kwargs)[source]

Bases: Curve_generic

Curves in weighted projective spaces.

EXAMPLES:

We construct a hyperelliptic curve manually:

sage: WP.<x, y, z> = WeightedProjectiveSpace([1, 3, 1], QQ)
sage: C = Curve(y^2 - x^5 * z - 3 * x^2 * z^4 - 2 * z^6, WP); C
Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6

>>> from sage.all import *
>>> WP = WeightedProjectiveSpace([Integer(1), Integer(3), Integer(1)], QQ, names=('x', 'y', 'z',)); (x, y, z,) = WP._first_ngens(3)
>>> C = Curve(y**Integer(2) - x**Integer(5) * z - Integer(3) * x**Integer(2) * z**Integer(4) - Integer(2) * z**Integer(6), WP); C
Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6
affine_patch(i, AA=None)[source]

Return the \(i\)-th affine patch of this projective curve.

INPUT:

  • i – affine coordinate chart of the projective ambient space of this curve to compute affine patch with respect to

  • AA – (default: None) ambient affine space, this is constructed if it is not given

OUTPUT: a curve in affine space

EXAMPLES:

sage: WP.<x,y,z> = WeightedProjectiveSpace([1, 1, 1], QQ)
sage: C = WP.curve(x^3 - x^2*y + y^3 - x^2*z)
sage: C.affine_patch(1)
Affine Plane Curve over Rational Field defined by x^3 - x^2*z - x^2 + 1

>>> from sage.all import *
>>> WP = WeightedProjectiveSpace([Integer(1), Integer(1), Integer(1)], QQ, names=('x', 'y', 'z',)); (x, y, z,) = WP._first_ngens(3)
>>> C = WP.curve(x**Integer(3) - x**Integer(2)*y + y**Integer(3) - x**Integer(2)*z)
>>> C.affine_patch(Integer(1))
Affine Plane Curve over Rational Field defined by x^3 - x^2*z - x^2 + 1
plot(*args, **kwds)[source]

Plot the real points of an affine patch of the associated projective plane curve.

INPUT:

  • self – an affine plane curve

  • patch – (optional) the affine patch to be plotted; if not specified, the patch corresponding to the last projective coordinate being nonzero

  • *args – (optional) tuples (variable, minimum, maximum) for plotting dimensions

  • **kwds – optional keyword arguments passed on to implicit_plot

EXAMPLES:

A hyperelliptic curve:

sage: # needs sage.plot
sage: P.<x> = QQ[]
sage: f = 4*x^5 - 30*x^3 + 45*x - 22
sage: C = HyperellipticCurve(f)
sage: C.plot()
Graphics object consisting of 1 graphics primitive
sage: C.plot(patch=0)
Graphics object consisting of 1 graphics primitive
sage: C.plot(patch=1)
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> # needs sage.plot
>>> P = QQ['x']; (x,) = P._first_ngens(1)
>>> f = Integer(4)*x**Integer(5) - Integer(30)*x**Integer(3) + Integer(45)*x - Integer(22)
>>> C = HyperellipticCurve(f)
>>> C.plot()
Graphics object consisting of 1 graphics primitive
>>> C.plot(patch=Integer(0))
Graphics object consisting of 1 graphics primitive
>>> C.plot(patch=Integer(1))
Graphics object consisting of 1 graphics primitive
projective_curve()[source]

Return this weighted projective curve as a projective curve.

A weighted homogeneous polynomial \(f(x_1, \ldots, x_n)\), where \(x_i\) has weight \(w_i\), can be viewed as an unweighted homogeneous polynomial \(f(y_1^{w_1}, \ldots, y_n^{w_n})\). This correspondence extends to varieties.

EXAMPLES:

sage: WP = WeightedProjectiveSpace([1, 3, 1], QQ, "x, y, z")
sage: x, y, z = WP.gens()
sage: C = WP.curve(y^2 - (x^5*z + 3*x^2*z^4 - 2*x*z^5 + 4*z^6)); C
Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 + 2*x*z^5 - 4*z^6
sage: C.projective_curve()
Projective Plane Curve over Rational Field defined by y^6 - x^5*z - 3*x^2*z^4 + 2*x*z^5 - 4*z^6

>>> from sage.all import *
>>> WP = WeightedProjectiveSpace([Integer(1), Integer(3), Integer(1)], QQ, "x, y, z")
>>> x, y, z = WP.gens()
>>> C = WP.curve(y**Integer(2) - (x**Integer(5)*z + Integer(3)*x**Integer(2)*z**Integer(4) - Integer(2)*x*z**Integer(5) + Integer(4)*z**Integer(6))); C
Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 + 2*x*z^5 - 4*z^6
>>> C.projective_curve()
Projective Plane Curve over Rational Field defined by y^6 - x^5*z - 3*x^2*z^4 + 2*x*z^5 - 4*z^6
riemann_surface(**kwargs)[source]

Return the complex Riemann surface determined by this curve.

OUTPUT: a RiemannSurface object

EXAMPLES:

sage: WP.<x,y,z> = WeightedProjectiveSpace([1, 1, 1], QQ)
sage: C = WP.curve(x^3 + 3*y^3 + 5*z^3)
sage: C.riemann_surface()
Riemann surface defined by polynomial f = x^3 + 3*y^3 + 5 = 0,
with 53 bits of precision

>>> from sage.all import *
>>> WP = WeightedProjectiveSpace([Integer(1), Integer(1), Integer(1)], QQ, names=('x', 'y', 'z',)); (x, y, z,) = WP._first_ngens(3)
>>> C = WP.curve(x**Integer(3) + Integer(3)*y**Integer(3) + Integer(5)*z**Integer(3))
>>> C.riemann_surface()
Riemann surface defined by polynomial f = x^3 + 3*y^3 + 5 = 0,
with 53 bits of precision