# 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
• Ben Hutz (9/2014): added support for cartesian products
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)

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)

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()
{(0 : 1): 0,
(1 : 0): 7,
(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)

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)

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


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

Lattes_map(E, m)

Given an elliptic curve $$E$$ and an integer $$m$$ return the Lattes map associated to multiplication by $$m$$. In other words, the rational map on the quotient $$E/\{\pm 1\} \cong \mathbb{P}^1$$ associated to $$[m]:E \to E$$.

INPUT:

• E – an elliptic curve
• m – an integer

OUTPUT: an endomorphism of self.

Examples:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: E = EllipticCurve(QQ,[-1, 0])
sage: P.Lattes_map(E,2)
Scheme endomorphism of Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x : y) to
(x^4 + 2*x^2*y^2 + y^4 : 4*x^3*y - 4*x*y^3)

affine_patch(i, AA=None)

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.

• AA – (default: None) ambient affine space, this is constructed

if it is not given.

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)

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: P.affine_patch(0).projective_embedding(0).codomain() == P
True

cartesian_product(other)

Return the cartesian product of the projective spaces self and other.

INPUT:

• other - A projective space with the same base ring as self

OUTPUT:

• A cartesian product of projective spaces

EXAMPLES:

sage: P1 = ProjectiveSpace(QQ,1,'x')
sage: P2 = ProjectiveSpace(QQ,2,'y')
sage: PP = P1.cartesian_product(P2); PP
Product of projective spaces P^1 x P^2 over Rational Field
sage: PP.gens()
(x0, x1, y0, y1, y2)

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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