# 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                                                                          # needs sage.rings.real_mpfr
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 homogeneous coordinate ring. Using the method objgens() you can obtain both the space and the generators as ready to use variables.

sage: P2, vars = ProjectiveSpace(10, QQ, 't').objgens()
sage: vars
(t0, t1, t2, t3, t4, t5, t6, t7, t8, t9, t10)


You can alternatively use the special syntax with < and >.

sage: P2.<x,y,z> = ProjectiveSpace(2, QQ)
sage: P2
Projective Space of dimension 2 over Rational Field
sage: P2.coordinate_ring()
Multivariate Polynomial Ring in x, y, z over Rational Field


The first of the three lines above is just equivalent to the two lines:

sage: P2 = ProjectiveSpace(2, QQ, 'xyz')
sage: x,y,z = P2.gens()


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()                                                                 # needs sage.libs.singular
0


AUTHORS:

• Ben Hutz: (June 2012): support for rings

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

• Rebecca Lauren Miller (March 2016) : added point_transformation_matrix

sage.schemes.projective.projective_space.ProjectiveSpace(n, R=None, names=None)#

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 a 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')                                            # needs sage.rings.padics
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); PS                                     # needs sage.rings.real_mpfr
Projective Space of dimension 1 over Complex Field with 53 bits of precision

sage: R.<x,y,z> = QQ[]
sage: ProjectiveSpace(R).variable_names()
('x', 'y', 'z')


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

sage: R.<x> = QQ[]
sage: ProjectiveSpace(R)
Projective Space of dimension 0 over Rational Field

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

Return a curve defined by F in this projective space.

INPUT:

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

EXAMPLES:

sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: P.curve([y^2 - x*z])                                                  # needs sage.schemes
Projective Plane Curve over Rational Field defined by y^2 - x*z

line_through(p, q)#

Return the line through p and q.

INPUT:

• p, q – distinct rational points of the projective space

EXAMPLES:

sage: P3.<x0,x1,x2,x3> = ProjectiveSpace(3, QQ)
sage: p1 = P3(1, 2, 3, 4)
sage: p2 = P3(4, 3, 2, 1)
sage: P3.line_through(p1, p2)                                               # needs sage.libs.singular sage.schemes
Projective Curve over Rational Field defined by
-5/4*x0 + 5/2*x1 - 5/4*x2,        -5/2*x0 + 15/4*x1 - 5/4*x3,
-5/4*x0 + 15/4*x2 - 5/2*x3,       -5/4*x1 + 5/2*x2 - 5/4*x3
sage: p3 = P3(2,4,6,8)
sage: P3.line_through(p1, p3)
Traceback (most recent call last):
...
ValueError: not distinct points

subscheme_from_Chow_form(Ch, dim)#

Returns the subscheme defined by the Chow equations associated to the Chow form Ch.

These equations define the subscheme set-theoretically, but only for smooth subschemes and hypersurfaces do they define the subscheme as a scheme.

ALGORITHM:

The Chow form is a polynomial in the Plucker coordinates. The Plucker coordinates are the bracket polynomials. We first re-write the Chow form in terms of the dual Plucker coordinates. Then we expand $$Ch(span(p,L)$$ for a generic point $$p$$ and a generic linear subspace $$L$$. The coefficients as polynomials in the coordinates of $$p$$ are the equations defining the subscheme. [DalbecSturmfels].

INPUT:

• Ch - a homogeneous polynomial.

• dim - the dimension of the associated scheme.

OUTPUT: a projective subscheme.

EXAMPLES:

sage: P = ProjectiveSpace(QQ, 4, 'z')
sage: R.<x0,x1,x2,x3,x4> = PolynomialRing(QQ)
sage: H = x1^2 + x2^2 + 5*x3*x4
sage: P.subscheme_from_Chow_form(H, 3)                                      # needs sage.modules
Closed subscheme of Projective Space of dimension 4 over Rational Field defined by:
-5*z0*z1 + z2^2 + z3^2

sage: P = ProjectiveSpace(QQ, 3, 'z')
sage: R.<x0,x1,x2,x3,x4,x5> = PolynomialRing(QQ)
sage: H = x1 - x2 - x3 + x5 + 2*x0
sage: P.subscheme_from_Chow_form(H, 1)                                      # needs sage.modules
Closed subscheme of Projective Space of dimension 3 over Rational Field
defined by:
-z1 + z3,
z0 + z2 + z3,
-z1 - 2*z3,
-z0 - z1 + 2*z2

sage: # needs sage.libs.singular
sage: P.<x0,x1,x2,x3> = ProjectiveSpace(GF(7), 3)
sage: X = P.subscheme([x3^2 + x1*x2, x2 - x0])
sage: Ch = X.Chow_form(); Ch
t0^2 - 2*t0*t3 + t3^2 - t2*t4 - t4*t5
sage: Y = P.subscheme_from_Chow_form(Ch, 1); Y
Closed subscheme of Projective Space of dimension 3
over Finite Field of size 7 defined by:
x1*x2 + x3^2,
-x0*x2 + x2^2,
-x0*x1 - x1*x2 - 2*x3^2,
x0^2 - x0*x2,
x0*x1 + x3^2,
-2*x0*x3 + 2*x2*x3,
2*x0*x3 - 2*x2*x3,
x0^2 - 2*x0*x2 + x2^2
sage: I = Y.defining_ideal()
sage: I.saturation(I.ring().ideal(list(I.ring().gens())))[0]
Ideal (x0 - x2, x1*x2 + x3^2) of Multivariate Polynomial Ring
in x0, x1, x2, x3 over Finite Field of size 7

class sage.schemes.projective.projective_space.ProjectiveSpace_finite_field(n, R=Integer Ring, names=None)#
rational_points(F=None)#

Return the list of F-rational points on this projective space, where F is a given finite field, or the base ring of this space.

EXAMPLES:

sage: P = ProjectiveSpace(1, GF(3))
sage: P.rational_points()
[(0 : 1), (1 : 1), (2 : 1), (1 : 0)]
sage: sorted(P.rational_points(GF(3^2, 'b')), key=str)                      # needs sage.rings.finite_rings
[(0 : 1), (1 : 0), (1 : 1), (2 : 1),
(2*b + 1 : 1), (2*b + 2 : 1), (2*b : 1),
(b + 1 : 1), (b + 2 : 1), (b : 1)]

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)#
rational_points(bound=0)#

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

ALGORITHM:

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.

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)]


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
sage: P = ProjectiveSpace(ZZ, 1, 'x')
True


Equality and hashing:

sage: ProjectiveSpace(QQ, 3, 'a') == ProjectiveSpace(ZZ, 3, 'a')
False
sage: ProjectiveSpace(ZZ, 1, 'a') == ProjectiveSpace(ZZ, 0, 'a')
False
sage: ProjectiveSpace(ZZ, 2, 'a') == AffineSpace(ZZ, 2, 'a')
False

sage: ProjectiveSpace(QQ, 3, 'a') != ProjectiveSpace(ZZ, 3, 'a')
True
sage: ProjectiveSpace(ZZ, 1, 'a') != ProjectiveSpace(ZZ, 0, 'a')
True
sage: ProjectiveSpace(ZZ, 2, 'a') != AffineSpace(ZZ, 2, 'a')
True

sage: hash(ProjectiveSpace(QQ, 3, 'a')) == hash(ProjectiveSpace(ZZ, 3, 'a'))
False
sage: hash(ProjectiveSpace(ZZ, 1, 'a')) == hash(ProjectiveSpace(ZZ, 0, 'a'))
False
sage: hash(ProjectiveSpace(ZZ, 2, 'a')) == hash(AffineSpace(ZZ, 2, 'a'))
False

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: a dynamical system on this projective space.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: E = EllipticCurve(QQ,[-1, 0])                                         # needs sage.schemes
sage: P.Lattes_map(E, 2)                                                    # needs sage.schemes
Dynamical System of Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x : y) to
(1/4*x^4 + 1/2*x^2*y^2 + 1/4*y^4 : x^3*y - 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, x3, x4, x5) to
(x0 : x1 : 1 : x3 : x4 : x5)
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, x3, x4, x5) to
(1 : x0 : x1 : x3 : x4 : x5)

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 this projective space and other.

INPUT:

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

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.

INPUT:

• R – commutative ring or morphism.

OUTPUT:

• projective space over R.

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: 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

sage: K.<w> = QuadraticField(2)                                             # needs sage.rings.number_field
sage: P = ProjectiveSpace(K, 2, 't')                                        # needs sage.rings.number_field
sage: P.change_ring(K.embeddings(QQbar)[0])                                 # needs sage.rings.number_field
Projective Space of dimension 2 over Algebraic Field

chebyshev_polynomial(n, kind='first', monic=False)#

Generates an endomorphism of this projective line by a Chebyshev polynomial.

Chebyshev polynomials are a sequence of recursively defined orthogonal polynomials. Chebyshev of the first kind are defined as $$T_0(x) = 1$$, $$T_1(x) = x$$, and $$T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$$. Chebyshev of the second kind are defined as $$U_0(x) = 1$$, $$U_1(x) = 2x$$, and $$U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)$$.

INPUT:

• n – a non-negative integer.

• kindfirst or second specifying which kind of chebyshev the user would like to generate. Defaults to first.

• monicTrue or False specifying if the polynomial defining the system should be monic or not. Defaults to False.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: P.chebyshev_polynomial(5, 'first')                                    # needs sage.symbolic
Dynamical System of Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x : y) to
(16*x^5 - 20*x^3*y^2 + 5*x*y^4 : y^5)

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: P.chebyshev_polynomial(3, 'second')                                   # needs sage.symbolic
Dynamical System of Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x : y) to
(8*x^3 - 4*x*y^2 : y^3)

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: P.chebyshev_polynomial(3, 2)                                          # needs sage.symbolic
Traceback (most recent call last):
...
ValueError: keyword 'kind' must have a value of either 'first' or 'second'

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: P.chebyshev_polynomial(-4, 'second')
Traceback (most recent call last):
...
ValueError: first parameter 'n' must be a non-negative integer

sage: P = ProjectiveSpace(QQ, 2, 'x')
sage: P.chebyshev_polynomial(2)
Traceback (most recent call last):
...
TypeError: projective space must be of dimension 1

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: P.chebyshev_polynomial(3, monic=True)                                 # needs sage.symbolic
Dynamical System of Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x : y) to
(x^3 - 3*x*y^2 : y^3)

sage: F.<t> = FunctionField(QQ)
sage: P.<y,z> = ProjectiveSpace(F, 1)
sage: P.chebyshev_polynomial(4, monic=True)                                 # needs sage.symbolic
Dynamical System of Projective Space of dimension 1
over Rational function field in t over Rational Field
Defn: Defined on coordinates by sending (y : z) to
(y^4 + (-4)*y^2*z^2 + 2*z^4 : z^4)

coordinate_ring()#

Return the coordinate ring of this scheme.

EXAMPLES:

sage: ProjectiveSpace(3, GF(19^2,'alpha'), 'abcd').coordinate_ring()        # needs sage.rings.finite_rings
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

hyperplane_transformation_matrix(plane_1, plane_2)#

Return a PGL element sending plane_1 to plane_2.

plane_1 and plane_2 must be hyperplanes (subschemes of codimension 1, each defined by a single linear homogeneous equation).

INPUT:

• plane_1, plane_2 – hyperplanes of this projective space

OUTPUT: An element of PGL

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: plane1 = P.subscheme(x)
sage: plane2 = P.subscheme(y)
sage: m = P.hyperplane_transformation_matrix(plane1, plane2); m             # needs sage.modules
[0 1]
[1 0]
sage: plane2(m*P((0,1)))                                                    # needs sage.modules
(1 : 0)

sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
sage: plane1 = P.subscheme(x + 2*y + z)
sage: plane2 = P.subscheme(2*x + y + z)
sage: P.hyperplane_transformation_matrix(plane1, plane2)                    # needs sage.modules
[1 0 0 0]
[0 4 0 0]
[0 0 2 0]
[0 0 0 1]

sage: P.<x,y> = ProjectiveSpace(ZZ, 1)
sage: plane1 = P.subscheme(x + y)
sage: plane2 = P.subscheme(y)
sage: P.hyperplane_transformation_matrix(plane1, plane2)                    # needs sage.modules
[-1  0]
[ 1  1]

sage: # needs sage.rings.number_field
sage: K.<v> = CyclotomicField(3)
sage: P.<x,y,z> = ProjectiveSpace(K, 2)
sage: plane1 = P.subscheme(x - 2*v*y + z)
sage: plane2 = P.subscheme(x + v*y + v*z)
sage: m = P.hyperplane_transformation_matrix(plane1, plane2); m             # needs sage.modules
[   v    0    0]
[   0 -2*v    0]
[   0    0    1]

sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: K.<k> = NumberField(x^2 + 1)
sage: P.<x,y,z,w> = ProjectiveSpace(K, 3)
sage: plane1 = P.subscheme(k*x + 2*k*y + z)
sage: plane2 = P.subscheme(7*k*x + y + 9*z)
sage: m = P.hyperplane_transformation_matrix(plane1, plane2); m             # needs sage.modules
[   1    0    0    0]
[   0 14*k    0    0]
[   0    0  7/9    0]
[   0    0    0    1]

sage: # needs sage.rings.number_field
sage: K.<v> = CyclotomicField(3)
sage: R.<t> = K[]
sage: F.<w> = K.extension(t^5 + 2)
sage: G.<u> = F.absolute_field()
sage: P.<x,y,z> = ProjectiveSpace(G, 2)
sage: plane1 = P.subscheme(x - 2*u*y + z)
sage: plane2 = P.subscheme(x + u*y + z)
sage: m = P.hyperplane_transformation_matrix(plane1, plane2)                # needs sage.modules
sage: plane2(m*P((2*u, 1, 0)))                                              # needs sage.modules
(-u : 1 : 0)

sage: P.<x,y,z> = ProjectiveSpace(FiniteField(2), 2)
sage: plane1 = P.subscheme(x + y + z)
sage: plane2 = P.subscheme(z)
sage: P.hyperplane_transformation_matrix(plane1, plane2)                    # needs sage.modules
[1 0 0]
[1 1 0]
[1 1 1]

sage: R.<t> = QQ[]
sage: P.<x,y,z> = ProjectiveSpace(R, 2)
sage: plane1 = P.subscheme(x + 9*t*y + z)
sage: plane2 = P.subscheme(x + z)
sage: P.hyperplane_transformation_matrix(plane1, plane2)                    # needs sage.modules
[  1 9*t   0]
[  1   0   0]
[  0   0   1]

is_linearly_independent(points, n=None)#

Return whether the set of points is linearly independent.

Alternatively, specify n to check if every subset of size n is linearly independent.

INPUT:

• points – a list of points in this projective space.

• n – (Optional) A positive integer less than or equal to the length of points. Specifies the size of the subsets to check for linear independence.

OUTPUT:

• True if points is linearly independent, False otherwise.

EXAMPLES:

sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: points = [P((1, 0, 1)), P((1, 2, 1)), P((1, 3, 4))]
sage: P.is_linearly_independent(points)                                     # needs sage.modules
True

sage: P.<x,y,z> = ProjectiveSpace(GF(5), 2)
sage: points = [P((1, 0, 1)), P((1, 2, 1)), P((1, 3, 4)), P((0, 0, 1))]
sage: P.is_linearly_independent(points, 2)                                  # needs sage.modules
True

sage: R.<c> = QQ[]
sage: P.<x,y,z> = ProjectiveSpace(R, 2)
sage: points = [P((c, 0, 1)), P((0, c, 1)), P((1, 0, 4)), P((0, 0, 1))]
sage: P.is_linearly_independent(points, 3)                                  # needs sage.modules
False

sage: R.<c> = QQ[]
sage: P.<x,y,z> = ProjectiveSpace(FractionField(R), 2)
sage: points = [P((c, 0, 1)), P((0, c, 1)), P((1, 3, 4)), P((0, 0, 1))]
sage: P.is_linearly_independent(points, 3)                                  # needs sage.modules
True

sage: # needs sage.rings.number_field
sage: K.<k> = CyclotomicField(3)
sage: P.<x,y,z> = ProjectiveSpace(K, 2)
sage: points = [P((k, k^2, 1)), P((0, k, 1)), P((1, 0, 4)), P((0, 0, 1))]
sage: P.is_linearly_independent(points, 3)                                  # needs sage.modules
True

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: points = [P((1, 0)), P((1, 1))]
sage: P.is_linearly_independent(points)                                     # needs sage.modules
True

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 this projective space.

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

EXAMPLES:

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

point(v, check=True)#

Create a point on this projective space.

INPUT:

• v – anything that defines a point

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

OUTPUT: A point of this projective space.

EXAMPLES:

sage: P2 = ProjectiveSpace(QQ, 2)
sage: P2.point([4,5])
(4 : 5 : 1)

sage: P = ProjectiveSpace(QQ, 1)
sage: P.point(infinity)
(1 : 0)

sage: P = ProjectiveSpace(QQ, 2)
sage: P.point(infinity)
Traceback (most recent call last):
...
ValueError: +Infinity not well defined in dimension > 1

sage: P = ProjectiveSpace(ZZ, 2)
sage: P.point([infinity])
Traceback (most recent call last):
...
ValueError: [+Infinity] not well defined in dimension > 1

point_transformation_matrix(points_source, points_target, normalize=True)#

Returns a unique element of PGL that transforms one set of points to another.

Given a projective space of dimension n and a set of n+2 source points and a set of n+2 target points in the same projective space, such that no n+1 points of each set are linearly dependent find the unique element of PGL that translates the source points to the target points.

Warning

over non-exact rings such as the ComplexField, the returned matrix could be very far from correct.

INPUT:

• points_source – points in source projective space.

• points_target – points in target projective space.

• normalize – (default: True) If the returned matrix should be normalized. Only works over exact rings. If the base ring is a field, the matrix is normalized so that the last nonzero entry in the last row is 1. If the base ring is a ring, then the matrix is normalized so that the entries are elements of the base ring.

OUTPUT: Transformation matrix - element of PGL.

ALGORITHM:

See [Hutz2007], Proposition 2.16 for details.

EXAMPLES:

sage: P1.<a,b,c> = ProjectiveSpace(QQ, 2)
sage: points_source = [P1([1, 4, 1]), P1([1, 2, 2]), P1([3, 5, 1]), P1([1, -1, 1])]
sage: points_target = [P1([5, -2, 7]), P1([3, -2, 3]), P1([6, -5, 9]), P1([3, 6, 7])]
sage: m = P1.point_transformation_matrix(points_source, points_target); m   # needs sage.modules
[ -13/59 -128/59  -25/59]
[538/177    8/59  26/177]
[ -45/59 -196/59       1]
sage: [m*points_source[i] == points_target[i] for i in range(4)]            # needs sage.modules
[True, True, True, True]

sage: P.<a,b> = ProjectiveSpace(GF(13),  1)
sage: points_source = [P([-6, 7]), P([1, 4]), P([3, 2])]
sage: points_target = [P([-1, 2]), P([0, 2]), P([-1, 6])]
sage: P.point_transformation_matrix(points_source, points_target)           # needs sage.modules
[10  4]
[10  1]

sage: P.<a,b> = ProjectiveSpace(QQ, 1)
sage: points_source = [P([-6, -4]), P([1, 4]), P([3, 2])]
sage: points_target = [P([-1, 2]), P([0, 2]), P([-7, -3])]
sage: P.point_transformation_matrix(points_source, points_target)           # needs sage.modules
Traceback (most recent call last):
...
ValueError: source points not independent

sage: R.<t> = FunctionField(QQ)
sage: P.<a,b> = ProjectiveSpace(R, 1)
sage: points_source = [P([-6*t, 7]), P([1, 4]), P([3, 2])]
sage: points_target = [P([-1, 2*t]), P([0, 2]), P([-1, 6])]
sage: P.point_transformation_matrix(points_source, points_target)           # needs sage.modules
[             (1/3*t + 7/12)/(t^2 - 53/24*t)       (-1/12*t - 7/48)/(t^2 - 53/24*t)]
[(-2/3*t^2 - 7/36*t - 35/12)/(t^2 - 53/24*t)                                      1]

sage: P1.<a,b,c> = ProjectiveSpace(RR, 2)
sage: points_source = [P1([1, 4, 1]), P1([1, 2, 2]), P1([3, 5, 1]), P1([1, -1, 1])]
sage: points_target = [P1([5, -2, 7]), P1([3, -2, 3]), P1([6, -5, 9]), P1([3, 6, 7])]
sage: P1.point_transformation_matrix(points_source,        # abs tol 1e-13  # needs sage.modules
....:                                points_target)
[-0.0619047619047597  -0.609523809523810  -0.119047619047621]
[  0.853968253968253  0.0380952380952380  0.0412698412698421]
[ -0.214285714285712  -0.933333333333333   0.280952380952379]

sage: P1.<a,b,c> = ProjectiveSpace(ZZ, 2)
sage: points_source = [P1([1, 4, 1]), P1([1, 2, 2]), P1([3, 5, 1]), P1([1, -1, 1])]
sage: points_target = [P1([5, -2, 7]), P1([3, -2, 3]), P1([6, -5, 9]), P1([3, 6, 7])]
sage: P1.point_transformation_matrix(points_source, points_target)          # needs sage.modules
[ -39 -384  -75]
[ 538   24   26]
[-135 -588  177]

sage: P1.<a,b,c> = ProjectiveSpace(ZZ, 2)
sage: points_source = [P1([1, 4, 1]), P1([1, 2, 2]), P1([3, 5, 1]), P1([1, -1, 1])]
sage: points_target = [P1([5, -2, 7]), P1([3, -2, 3]), P1([6, -5, 9]), P1([3, 6, 7])]
sage: P1.point_transformation_matrix(points_source, points_target,          # needs sage.modules
....:                                normalize=False)
[-13/30 -64/15   -5/6]
[269/45   4/15  13/45]
[  -3/2 -98/15  59/30]

sage: R.<t> = ZZ[]
sage: P.<a,b> = ProjectiveSpace(R, 1)
sage: points_source = [P([-6*t, 7]), P([1, 4]), P([3, 2])]
sage: points_target = [P([-1, 2*t]), P([0, 2]), P([-1, 6])]
sage: P.point_transformation_matrix(points_source, points_target)           # needs sage.modules
[         -48*t - 84           12*t + 21]
[96*t^2 + 28*t + 420    -144*t^2 + 318*t]

points_of_bounded_height(**kwds)#

Return an iterator of the points in self of absolute multiplicative height of at most the given bound.

ALGORITHM:

This is an implementation of Algorithm 6 in [Krumm2016].

INPUT:

kwds:

• bound - a real number

• precision - (default: 53) a positive integer

OUTPUT:

• an iterator of points of bounded height

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: sorted(list(P.points_of_bounded_height(bound=2)))
[(-2 : 1), (-1 : 1), (-1/2 : 1), (0 : 1),
(1/2 : 1), (1 : 0), (1 : 1), (2 : 1)]

sage: u = QQ['u'].0
sage: P.<x,y,z> = ProjectiveSpace(NumberField(u^2 - 2, 'v'), 2)             # needs sage.rings.number_field
sage: len(list(P.points_of_bounded_height(bound=2)))                        # needs sage.rings.number_field
265

sage: # needs sage.rings.number_field
sage: CF.<a> = CyclotomicField(3)
sage: R.<x> = CF[]
sage: L.<l> = CF.extension(x^3 + 2)
sage: Q.<x,y> = ProjectiveSpace(L, 1)
sage: sorted(list(Q.points_of_bounded_height(bound=1)))
[(0 : 1), (1 : 0), (a + 1 : 1), (a : 1),
(-1 : 1), (-a - 1 : 1), (-a : 1), (1 : 1)]

sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: F.<a> = NumberField(x^4 - 8*x^2 + 3)
sage: P.<x,y,z> = ProjectiveSpace(F, 2)
sage: all(exp(p.global_height()) <= 1                                       # needs sage.symbolic
....:     for p in P.points_of_bounded_height(bound=1))
True

sage: K.<a> = CyclotomicField(3)                                            # needs sage.rings.number_field
sage: P.<x,y,z> = ProjectiveSpace(K, 2)                                     # needs sage.rings.number_field
sage: len(list(P.points_of_bounded_height(bound=1)))                        # needs sage.rings.number_field
57

sage: u = QQ['u'].0
sage: K.<k> = NumberField(u^2 - 2)                                          # needs sage.rings.number_field
sage: P.<x,y> = ProjectiveSpace(K, 1)                                       # needs sage.rings.number_field
sage: len(list(P.points_of_bounded_height(bound=2)))                        # needs sage.rings.number_field
24

sage: R.<x> = QQ[]
sage: K.<k> = NumberField(x^4 - 8*x^2 + 3)                                  # needs sage.rings.number_field
sage: P.<x,y> = ProjectiveSpace(K, 1)                                       # needs sage.rings.number_field
sage: len(list(P.points_of_bounded_height(bound=2)))                        # needs sage.rings.number_field
108

sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: K.<v> = NumberField(x^5 + x^3 + 1)
sage: P.<x,y,z> = ProjectiveSpace(K, 2)
sage: L = P.points_of_bounded_height(bound=1.2)
sage: len(list(L))
109

sage: # needs sage.rings.number_field
sage: P.<x,y> = ProjectiveSpace(K, 1)
sage: sorted(list(P.points_of_bounded_height(bound=2)))
[(-v - 2 : 1), (-v - 1 : 1), (-2 : 1), (-1/2*v - 1 : 1), (-v : 1), (-1 : 1),
(-1/2*v : 1), (v - 2 : 1), (-1/2 : 1), (-v + 1 : 1), (1/2*v - 1 : 1), (0 : 1),
(-1/2*v + 1 : 1), (v - 1 : 1), (1/2 : 1), (-v + 2 : 1), (1/2*v : 1), (1 : 0),
(1 : 1), (v : 1), (1/2*v + 1 : 1), (2 : 1), (v + 1 : 1), (v + 2 : 1)]

sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(3*x^2 + 1)
sage: P.<z,w> = ProjectiveSpace(K, 1)
sage: sorted(list(P.points_of_bounded_height(bound=1)))
[(-1 : 1), (-3/2*a - 1/2 : 1), (3/2*a - 1/2 : 1), (0 : 1),
(-3/2*a + 1/2 : 1), (3/2*a + 1/2 : 1), (1 : 0), (1 : 1)]

sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(3*x^2 + 1)
sage: O = K.maximal_order()
sage: P.<z,w> = ProjectiveSpace(O, 1)
sage: len(sorted(list(P.points_of_bounded_height(bound=2))))
44

sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^3 - 7)
sage: O = K.maximal_order()
sage: P.<z,w> = ProjectiveSpace(O, 1)
sage: len(sorted(list(P.points_of_bounded_height(bound=2))))
28

sage: P.<w,z> = ProjectiveSpace(ZZ, 1)
sage: sorted(list(P.points_of_bounded_height(bound=2)))
[(-2 : -1), (-2 : 1), (-1 : -2), (-1 : -1),
(-1 : 0), (-1 : 1), (-1 : 2), (0 : -1)]

sage: R.<x> = QQ[]
sage: P.<z,w> = ProjectiveSpace(R, 1)
sage: P.points_of_bounded_height(bound=2)
Traceback (most recent call last):
...
NotImplementedError: self must be a projective space over
a number field or a ring of integers

sage: # needs sage.rings.number_field
sage: K.<i> = NumberField(x^2 + 1)
sage: PK.<t> = K[]
sage: L.<a> = K.extension(t^4  - i)
sage: P.<z,w> = ProjectiveSpace(L, 1)
sage: sorted(list(P.points_of_bounded_height(bound=1)))
[(0 : 1), (1 : 0), (a : 1), (a^2 : 1), (a^3 : 1), (i : 1),
(i*a : 1), (i*a^2 : 1), (i*a^3 : 1), (-1 : 1), (-a : 1), (-a^2 : 1),
(-a^3 : 1), (-i : 1), (-i*a : 1), (-i*a^2 : 1), (-i*a^3 : 1), (1 : 1)]

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()                                                    # needs sage.libs.singular
[x*y^2, y^2*z,  x*z^2]
sage: X.dimension()                                                         # needs sage.libs.singular
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

veronese_embedding(d, CS=None, order='lex')#

Return the degree d Veronese embedding from this projective space.

INPUT:

• d – a positive integer.

• CS – a projective ambient space to embed into. If this projective space has dimension $$N$$, the dimension of CS must be $$\binom{N + d}{d} - 1$$. This is constructed if not specified. Default: None.

• order – a monomial order to use to arrange the monomials defining the embedding. The monomials will be arranged from greatest to least with respect to this order. Default: 'lex'.

OUTPUT:

• a scheme morphism from this projective space to CS.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: vd = P.veronese_embedding(4, order='invlex')                          # needs sage.combinat
sage: vd                                                                    # needs sage.combinat
Scheme morphism:
From: Projective Space of dimension 1 over Rational Field
To:   Projective Space of dimension 4 over Rational Field
Defn: Defined on coordinates by sending (x : y) to
(y^4 : x*y^3 : x^2*y^2 : x^3*y : x^4)


Veronese surface:

sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: Q.<q,r,s,t,u,v> = ProjectiveSpace(QQ, 5)
sage: vd = P.veronese_embedding(2, Q)                                       # needs sage.combinat
sage: vd                                                                    # needs sage.combinat
Scheme morphism:
From: Projective Space of dimension 2 over Rational Field
To:   Projective Space of dimension 5 over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
(x^2 : x*y : x*z : y^2 : y*z : z^2)
sage: vd(P.subscheme([]))                                                   # needs sage.combinat sage.libs.singular
Closed subscheme of Projective Space of dimension 5 over Rational Field
defined by:
-u^2 + t*v,
-s*u + r*v,
-s*t + r*u,
-s^2 + q*v,
-r*s + q*u,
-r^2 + q*t

sage.schemes.projective.projective_space.is_ProjectiveSpace(x)#

Return True if x is a projective space.

In other words, if x is 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'))         # needs sage.rings.finite_rings
True
sage: is_ProjectiveSpace(Spec(ZZ))
False