Projective \(n\) space over a ring

EXAMPLES: We construct projective space over various rings of various dimensions.

The simplest projective space:

sage: ProjectiveSpace(0)
Projective Space of dimension 0 over Integer Ring

A slightly bigger projective space over \(\QQ\):

sage: X = ProjectiveSpace(1000, QQ); X
Projective Space of dimension 1000 over Rational Field
sage: X.dimension()
1000

We can use “over” notation to create projective spaces over various base rings.

sage: X = ProjectiveSpace(5)/QQ; X
Projective Space of dimension 5 over Rational Field
sage: X/CC
Projective Space of dimension 5 over Complex Field with 53 bits of precision

The third argument specifies the printing names of the generators of the homogenous coordinate ring. Using objgens() you can obtain both the space and the generators as ready to use variables.

sage: P2, (x,y,z) = ProjectiveSpace(2, QQ, 'xyz').objgens()
sage: P2
Projective Space of dimension 2 over Rational Field
sage: x.parent()
Multivariate Polynomial Ring in x, y, z over Rational Field

For example, we use \(x,y,z\) to define the intersection of two lines.

sage: V = P2.subscheme([x+y+z, x+y-z]); V
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
 x + y + z,
 x + y - z
sage: V.dimension()
0

AUTHORS:

  • Ben Hutz: (June 2012): support for rings
sage.schemes.projective.projective_space.ProjectiveSpace(n, R=None, names='x')

Return projective space of dimension \(n\) over the ring \(R\).

EXAMPLES: The dimension and ring can be given in either order.

sage: ProjectiveSpace(3, QQ)
Projective Space of dimension 3 over Rational Field
sage: ProjectiveSpace(5, QQ)
Projective Space of dimension 5 over Rational Field
sage: P = ProjectiveSpace(2, QQ, names='XYZ'); P
Projective Space of dimension 2 over Rational Field
sage: P.coordinate_ring()
Multivariate Polynomial Ring in X, Y, Z over Rational Field

The divide operator does base extension.

sage: ProjectiveSpace(5)/GF(17)
Projective Space of dimension 5 over Finite Field of size 17

The default base ring is \(\ZZ\).

sage: ProjectiveSpace(5)
Projective Space of dimension 5 over Integer Ring

There is also an projective space associated each polynomial ring.

sage: R = GF(7)['x,y,z']
sage: P = ProjectiveSpace(R); P
Projective Space of dimension 2 over Finite Field of size 7
sage: P.coordinate_ring()
Multivariate Polynomial Ring in x, y, z over Finite Field of size 7
sage: P.coordinate_ring() is R
True
sage: ProjectiveSpace(3, Zp(5), 'y')
Projective Space of dimension 3 over 5-adic Ring with capped relative precision 20
sage: ProjectiveSpace(2,QQ,'x,y,z')
Projective Space of dimension 2 over Rational Field
sage: PS.<x,y>=ProjectiveSpace(1,CC)
sage: PS
Projective Space of dimension 1 over Complex Field with 53 bits of precision

Projective spaces are not cached, i.e., there can be several with the same base ring and dimension (to facilitate gluing constructions).

class sage.schemes.projective.projective_space.ProjectiveSpace_field(n, R=Integer Ring, names=None)

Bases: sage.schemes.projective.projective_space.ProjectiveSpace_ring

EXAMPLES:

sage: ProjectiveSpace(3, Zp(5), 'y')
Projective Space of dimension 3 over 5-adic Ring with capped relative precision 20
class sage.schemes.projective.projective_space.ProjectiveSpace_finite_field(n, R=Integer Ring, names=None)

Bases: sage.schemes.projective.projective_space.ProjectiveSpace_field

EXAMPLES:

sage: ProjectiveSpace(3, Zp(5), 'y')
Projective Space of dimension 3 over 5-adic Ring with capped relative precision 20
rational_points(F=None)

Return the list of \(F\)-rational points on the affine space self, where \(F\) is a given finite field, or the base ring of self.

EXAMPLES:

sage: P = ProjectiveSpace(1, GF(3))
sage: P.rational_points()
[(0 : 1), (1 : 1), (2 : 1), (1 : 0)]
sage: P.rational_points(GF(3^2, 'b'))
[(0 : 1), (b : 1), (b + 1 : 1), (2*b + 1 : 1), (2 : 1), (2*b : 1), (2*b + 2 : 1), (b + 2 : 1), (1 : 1), (1 : 0)]
rational_points_dictionary()

Return dictionary of points.

OUTPUT:

  • dictionary

EXAMPLES:

sage: P1=ProjectiveSpace(GF(7),1,'x')
sage: P1.rational_points_dictionary()
{(1 : 0): 7, (0 : 1): 0, (1 : 1): 1, (2 : 1): 2, (3 : 1): 3, (4 : 1): 4,
(5 : 1): 5, (6 : 1): 6}
class sage.schemes.projective.projective_space.ProjectiveSpace_rational_field(n, R=Integer Ring, names=None)

Bases: sage.schemes.projective.projective_space.ProjectiveSpace_field

EXAMPLES:

sage: ProjectiveSpace(3, Zp(5), 'y')
Projective Space of dimension 3 over 5-adic Ring with capped relative precision 20
rational_points(bound=0)

Returns the projective points \((x_0:\cdots:x_n)\) over \(\QQ\) with \(|x_i| \leq\) bound.

INPUT:

  • bound - integer

EXAMPLES:

sage: PP = ProjectiveSpace(0,QQ)
sage: PP.rational_points(1)
[(1)]
sage: PP = ProjectiveSpace(1,QQ)
sage: PP.rational_points(2)
[(-2 : 1), (-1 : 1), (0 : 1), (1 : 1), (2 : 1), (-1/2 : 1), (1/2 : 1), (1 : 0)]
sage: PP = ProjectiveSpace(2,QQ)
sage: PP.rational_points(2)
[(-2 : -2 : 1), (-1 : -2 : 1), (0 : -2 : 1), (1 : -2 : 1), (2 : -2 : 1),
(-2 : -1 : 1), (-1 : -1 : 1), (0 : -1 : 1), (1 : -1 : 1), (2 : -1 : 1),
(-2 : 0 : 1), (-1 : 0 : 1), (0 : 0 : 1), (1 : 0 : 1), (2 : 0 : 1), (-2 :
1 : 1), (-1 : 1 : 1), (0 : 1 : 1), (1 : 1 : 1), (2 : 1 : 1), (-2 : 2 :
1), (-1 : 2 : 1), (0 : 2 : 1), (1 : 2 : 1), (2 : 2 : 1), (-1/2 : -1 :
1), (1/2 : -1 : 1), (-1 : -1/2 : 1), (-1/2 : -1/2 : 1), (0 : -1/2 : 1),
(1/2 : -1/2 : 1), (1 : -1/2 : 1), (-1/2 : 0 : 1), (1/2 : 0 : 1), (-1 :
1/2 : 1), (-1/2 : 1/2 : 1), (0 : 1/2 : 1), (1/2 : 1/2 : 1), (1 : 1/2 :
1), (-1/2 : 1 : 1), (1/2 : 1 : 1), (-2 : 1 : 0), (-1 : 1 : 0), (0 : 1 :
0), (1 : 1 : 0), (2 : 1 : 0), (-1/2 : 1 : 0), (1/2 : 1 : 0), (1 : 0 :
0)]

Note

The very simple algorithm works as follows: every point \((x_0:\cdots:x_n)\) in projective space has a unique largest index \(i\) for which \(x_i\) is not zero. The algorithm then iterates downward on this index. We normalize by choosing \(x_i\) positive. Then, the points \(x_0,\ldots,x_{i-1}\) are the points of affine \(i\)-space that are relatively prime to \(x_i\). We access these by using the Tuples method.

AUTHORS:

  • Benjamin Antieau (2008-01-12)
class sage.schemes.projective.projective_space.ProjectiveSpace_ring(n, R=Integer Ring, names=None)

Bases: sage.schemes.generic.ambient_space.AmbientSpace

Projective space of dimension \(n\) over the ring \(R\).

EXAMPLES:

sage: X.<x,y,z,w> = ProjectiveSpace(3, QQ)
sage: X.base_scheme()
Spectrum of Rational Field
sage: X.base_ring()
Rational Field
sage: X.structure_morphism()
Scheme morphism:
  From: Projective Space of dimension 3 over Rational Field
  To:   Spectrum of Rational Field
  Defn: Structure map
sage: X.coordinate_ring()
Multivariate Polynomial Ring in x, y, z, w over Rational Field

Loading and saving:

sage: loads(X.dumps()) == X
True
affine_patch(i)

Return the \(i^{th}\) affine patch of this projective space. This is an ambient affine space \(\mathbb{A}^n_R,\) where \(R\) is the base ring of self, whose “projective embedding” map is \(1\) in the \(i^{th}\) factor.

INPUT:

  • i – integer between 0 and dimension of self, inclusive.

OUTPUT:

  • An ambient affine space with fixed projective_embedding map.

EXAMPLES:

sage: PP = ProjectiveSpace(5) / QQ
sage: AA = PP.affine_patch(2)
sage: AA
Affine Space of dimension 5 over Rational Field
sage: AA.projective_embedding()
Scheme morphism:
  From: Affine Space of dimension 5 over Rational Field
  To:   Projective Space of dimension 5 over Rational Field
  Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4) to
        (x0 : x1 : 1 : x2 : x3 : x4)
sage: AA.projective_embedding(0)
Scheme morphism:
  From: Affine Space of dimension 5 over Rational Field
  To:   Projective Space of dimension 5 over Rational Field
  Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4) to
        (1 : x0 : x1 : x2 : x3 : x4)
change_ring(R)

Return a projective space over ring \(R\) and otherwise the same as self.

INPUT:

  • R – commutative ring

OUTPUT:

  • projective space over R

Note

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

EXAMPLES:

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: PQ = P.change_ring(QQ); PQ
Projective Space of dimension 2 over Rational Field
sage: PQ.change_ring(GF(5))
Projective Space of dimension 2 over Finite Field of size 5
coordinate_ring()

Return the coordinate ring of this scheme.

EXAMPLES:

sage: ProjectiveSpace(3, GF(19^2,'alpha'), 'abcd').coordinate_ring()
Multivariate Polynomial Ring in a, b, c, d over Finite Field in alpha of size 19^2
sage: ProjectiveSpace(3).coordinate_ring()
Multivariate Polynomial Ring in x0, x1, x2, x3 over Integer Ring
sage: ProjectiveSpace(2, QQ, ['alpha', 'beta', 'gamma']).coordinate_ring()
Multivariate Polynomial Ring in alpha, beta, gamma over Rational Field
is_projective()

Return that this ambient space is projective n-space.

EXAMPLES:

sage: ProjectiveSpace(3,QQ).is_projective()
True
ngens()

Return the number of generators of self, i.e. the number of variables in the coordinate ring of self.

EXAMPLES:

sage: ProjectiveSpace(3, QQ).ngens()
4
sage: ProjectiveSpace(7, ZZ).ngens()
8
subscheme(X)

Return the closed subscheme defined by X.

INPUT:

  • X - a list or tuple of equations

EXAMPLES:

sage: A.<x,y,z> = ProjectiveSpace(2, QQ)
sage: X = A.subscheme([x*z^2, y^2*z, x*y^2]); X
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
  x*z^2,
  y^2*z,
  x*y^2
sage: X.defining_polynomials ()
(x*z^2, y^2*z, x*y^2)
sage: I = X.defining_ideal(); I
Ideal (x*z^2, y^2*z, x*y^2) of Multivariate Polynomial Ring in x, y, z over Rational Field
sage: I.groebner_basis()
[x*y^2, y^2*z,  x*z^2]
sage: X.dimension()
0
sage: X.base_ring()
Rational Field
sage: X.base_scheme()
Spectrum of Rational Field
sage: X.structure_morphism()
Scheme morphism:
  From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
  x*z^2,
  y^2*z,
  x*y^2
  To:   Spectrum of Rational Field
  Defn: Structure map

sage: TestSuite(X).run(skip=["_test_an_element", "_test_elements", "_test_elements_eq", "_test_some_elements", "_test_elements_eq_reflexive",  "_test_elements_eq_symmetric", "_test_elements_eq_transitive", "_test_elements_neq"])
sage.schemes.projective.projective_space.is_ProjectiveSpace(x)

Return True if \(x\) is a projective space, i.e., an ambient space \(\mathbb{P}^n_R\), where \(R\) is a ring and \(n\geq 0\) is an integer.

EXAMPLES:

sage: from sage.schemes.projective.projective_space import is_ProjectiveSpace
sage: is_ProjectiveSpace(ProjectiveSpace(5, names='x'))
True
sage: is_ProjectiveSpace(ProjectiveSpace(5, GF(9,'alpha'), names='x'))
True
sage: is_ProjectiveSpace(Spec(ZZ))
False

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