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sage.schemes.weighted_projective.weighted_projective_space.WeightedProjectiveSpace(weights, R=None, names=None)[source]

Return a weighted projective space with the given weights over the ring R.

EXAMPLES:

sage: WP = WeightedProjectiveSpace([1, 3, 1]); WP
Weighted Projective Space of dimension 2 with weights (1, 3, 1) over Integer Ring

>>> from sage.all import *
>>> WP = WeightedProjectiveSpace([Integer(1), Integer(3), Integer(1)]); WP
Weighted Projective Space of dimension 2 with weights (1, 3, 1) over Integer Ring
class sage.schemes.weighted_projective.weighted_projective_space.WeightedProjectiveSpace_ring(weights: tuple[~sage.rings.integer.Integer], R=Integer Ring, names=None)[source]

Bases: UniqueRepresentation, AmbientSpace

Weighted projective space with the given weights over the ring \(R\).

EXAMPLES:

sage: WeightedProjectiveSpace(Zp(5), [1, 3, 1], 'y')                        # needs sage.rings.padics
Weighted Projective Space of dimension 2 with weights (1, 3, 1) over 5-adic Ring with
capped relative precision 20
sage: WeightedProjectiveSpace(QQ, 5, 'y')
Projective Space of dimension 5 over Rational Field
sage: _ is ProjectiveSpace(QQ, 5, 'y')
True

>>> from sage.all import *
>>> WeightedProjectiveSpace(Zp(Integer(5)), [Integer(1), Integer(3), Integer(1)], 'y')                        # needs sage.rings.padics
Weighted Projective Space of dimension 2 with weights (1, 3, 1) over 5-adic Ring with
capped relative precision 20
>>> WeightedProjectiveSpace(QQ, Integer(5), 'y')
Projective Space of dimension 5 over Rational Field
>>> _ is ProjectiveSpace(QQ, Integer(5), 'y')
True
change_ring(R)[source]

Return a weighted projective space over ring R.

INPUT:

  • R – commutative ring or morphism

OUTPUT: weighted projective space over R. If R is a morphism, return a weighted projective space over its codomain.

Note

There is no need to have any relation between R and the base ring of this space, if you want to have such a relation, use self.base_extend(R) instead.

EXAMPLES:

sage: WP = WeightedProjectiveSpace([1, 3, 1], ZZ); WP
Weighted Projective Space of dimension 2 with weights (1, 3, 1) over Integer Ring
sage: WP.change_ring(QQ)
Weighted Projective Space of dimension 2 with weights (1, 3, 1) over Rational Field
sage: WP.change_ring(GF(5))
Weighted Projective Space of dimension 2 with weights (1, 3, 1) over Finite Field of size 5

>>> from sage.all import *
>>> WP = WeightedProjectiveSpace([Integer(1), Integer(3), Integer(1)], ZZ); WP
Weighted Projective Space of dimension 2 with weights (1, 3, 1) over Integer Ring
>>> WP.change_ring(QQ)
Weighted Projective Space of dimension 2 with weights (1, 3, 1) over Rational Field
>>> WP.change_ring(GF(Integer(5)))
Weighted Projective Space of dimension 2 with weights (1, 3, 1) over Finite Field of size 5
coordinate_ring()[source]

Return the coordinate ring of this weighted projective space.

EXAMPLES:

sage: WP = WeightedProjectiveSpace(GF(19^2, 'α'), [1, 3, 4, 1], 'abcd')
sage: # needs sage.rings.finite_rings
sage: R = WP.coordinate_ring(); R
Multivariate Polynomial Ring in a, b, c, d over Finite Field in α of size 19^2
sage: R.term_order()
Weighted degree reverse lexicographic term order with weights (1, 3, 4, 1)

>>> from sage.all import *
>>> WP = WeightedProjectiveSpace(GF(Integer(19)**Integer(2), 'α'), [Integer(1), Integer(3), Integer(4), Integer(1)], 'abcd')
>>> # needs sage.rings.finite_rings
>>> R = WP.coordinate_ring(); R
Multivariate Polynomial Ring in a, b, c, d over Finite Field in α of size 19^2
>>> R.term_order()
Weighted degree reverse lexicographic term order with weights (1, 3, 4, 1)


sage: WP = WeightedProjectiveSpace(QQ, [1, 1, 1], ['alpha', 'beta', 'gamma'])
sage: R = WP.coordinate_ring(); R
Multivariate Polynomial Ring in alpha, beta, gamma over Rational Field
sage: R.term_order()
Weighted degree reverse lexicographic term order with weights (1, 1, 1)
[Python] 
>>> from sage.all import *
>>> WP = WeightedProjectiveSpace(QQ, [Integer(1), Integer(1), Integer(1)], ['alpha', 'beta', 'gamma'])
>>> R = WP.coordinate_ring(); R
Multivariate Polynomial Ring in alpha, beta, gamma over Rational Field
>>> R.term_order()
Weighted degree reverse lexicographic term order with weights (1, 1, 1)
curve(F)[source]

Return a curve defined by F in this weighted projective space.

INPUT:

  • F – a polynomial, or a list or tuple of polynomials in the coordinate ring of this weighted projective space

EXAMPLES:

sage: WP.<x, y, z> = WeightedProjectiveSpace([1, 3, 1], QQ)
sage: WP.curve(y^2 - x^5 * z - 3 * x^2 * z^4 - 2 * z^6)                     # needs sage.schemes
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)
>>> WP.curve(y**Integer(2) - x**Integer(5) * z - Integer(3) * x**Integer(2) * z**Integer(4) - Integer(2) * z**Integer(6))                     # needs sage.schemes
Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6
ngens()[source]

Return the number of generators of this weighted projective space.

This is the number of variables in the coordinate ring of self.

EXAMPLES:

sage: WeightedProjectiveSpace(QQ, [1, 3, 1]).ngens()
3
sage: WeightedProjectiveSpace(ZZ, 5).ngens()
6

>>> from sage.all import *
>>> WeightedProjectiveSpace(QQ, [Integer(1), Integer(3), Integer(1)]).ngens()
3
>>> WeightedProjectiveSpace(ZZ, Integer(5)).ngens()
6
point(v, check=True)[source]

Create a point on this weighted projective space.

INPUT:

INPUT:

  • v – anything that defines a point

  • check – boolean (default: True); whether to check the defining data for consistency

OUTPUT: A point of this weighted projective space.

EXAMPLES:

sage: WP = WeightedProjectiveSpace(QQ, [1, 3, 1])
sage: WP.point([2, 3, 1])
(2 : 3 : 1)

>>> from sage.all import *
>>> WP = WeightedProjectiveSpace(QQ, [Integer(1), Integer(3), Integer(1)])
>>> WP.point([Integer(2), Integer(3), Integer(1)])
(2 : 3 : 1)
subscheme(*_, **__)[source]
weights()[source]

Return the tuple of weights of this weighted projective space.

EXAMPLES:

sage: WeightedProjectiveSpace(QQ, [1, 3, 1]).weights()
(1, 3, 1)

>>> from sage.all import *
>>> WeightedProjectiveSpace(QQ, [Integer(1), Integer(3), Integer(1)]).weights()
(1, 3, 1)