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 :
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 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
Return projective space of dimension over the ring
.
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 .
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
Projective spaces are not cached, i.e., there can be several with the same base ring and dimension (to facilitate gluing constructions).
Return iterator over the elements of this projective space.
Note that iteration is over the decomposition
, whereEXAMPLES:
sage: FF = FiniteField(3)
sage: PP = ProjectiveSpace(0,FF)
sage: [ x for x in PP ]
[(1)]
sage: PP = ProjectiveSpace(1,FF)
sage: [ x for x in PP ]
[(0 : 1), (1 : 1), (2 : 1), (1 : 0)]
sage: PP = ProjectiveSpace(2,FF)
sage: [ x for x in PP ]
[(0 : 0 : 1),
(1 : 0 : 1),
(2 : 0 : 1),
(0 : 1 : 1),
(1 : 1 : 1),
(2 : 1 : 1),
(0 : 2 : 1),
(1 : 2 : 1),
(2 : 2 : 1),
(0 : 1 : 0),
(1 : 1 : 0),
(2 : 1 : 0),
(1 : 0 : 0)]
AUTHORS:
TODO: Iteration for point sets over finite fields, and return of iter of point set over base field. Note that the point set does not know whether this is a projective space or subscheme.
Return the list of -rational points on the affine space self,
where
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), (2*b : 1), (b + 1 : 1), (b + 2 : 1), (2 : 1), (b : 1), (2*b + 2 : 1), (2*b + 1 : 1), (1 : 1), (1 : 0)]
Returns the projective points over
with
bound.
INPUT:
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
in projective space has a unique
largest index
for which
is not
zero. The algorithm then iterates downward on this
index. We normalize by choosing
positive. Then,
the points
are the points of
affine
-space that are relatively prime to
. We access these by using the Tuples method.
AUTHORS:
Projective space of dimension over the ring
.
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
EXAMPLES:
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: loads(AffineSpace(ZZ, 1, 'x').dumps()) == AffineSpace(ZZ, 1, 'x')
True
EXAMPLES:
sage: ProjectiveSpace(3, Zp(5), 'y')
Projective Space of dimension 3 over 5-adic Ring with capped relative precision 20
Return True if defines a point on the scheme self; raise a
TypeError otherwise.
EXAMPLES:
sage: P = ProjectiveSpace(2, ZZ)
sage: P._check_satisfies_equations([1, 1, 0])
True
sage: P._check_satisfies_equations((0, 1, 0))
True
sage: P._check_satisfies_equations([0, 0, 0])
...
TypeError: The zero vector is not a point in projective space
sage: P._check_satisfies_equations([1, 2, 3, 4, 5])
...
TypeError: The list v=[1, 2, 3, 4, 5] must have 3 components
sage: P._check_satisfies_equations([1/2, 1, 1])
...
TypeError: The components of v=[1/2, 1, 1] must be elements of Integer Ring
sage: P._check_satisfies_equations(5)
...
TypeError: The argument v=5 must be a list or tuple
Return a LaTeX representation of this projective space.
EXAMPLES:
sage: print latex(ProjectiveSpace(1, ZZ, 'x'))
{\mathbf P}_{\Bold{Z}}^1
TESTS:
sage: ProjectiveSpace(3, Zp(5), 'y')._latex_()
'{\\mathbf P}_{\\ZZ_{5}}^3'
Return a LaTeX representation of the generic point corresponding to the list of polys on this projective space.
If polys is None, the representation of the generic point of the projective space is returned.
EXAMPLES:
sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: P._latex_generic_point([z*y-x^2])
'\left(- x^{2} + y z\right)'
sage: P._latex_generic_point()
'\left(x : y : z\right)'
Return a string representation of this projective space.
EXAMPLES:
sage: ProjectiveSpace(1, ZZ, 'x')
Projective Space of dimension 1 over Integer Ring
TESTS:
sage: ProjectiveSpace(3, Zp(5), 'y')._repr_()
'Projective Space of dimension 3 over 5-adic Ring with capped relative precision 20'
Return a string representation of the generic point corresponding to the list of polys on this projective space.
If polys is None, the representation of the generic point of the projective space is returned.
EXAMPLES:
sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: P._repr_generic_point([z*y-x^2])
'(-x^2 + y*z)'
sage: P._repr_generic_point()
'(x : y : z)'
Return a valid tuple of polynomial functions on self given by
. Raise an error if
does not consist of valid
functions (in particular, if they are not homogeneous).
EXAMPLES:
sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: P._validate([x*y-z^2, 1])
(x*y - z^2, 1)
sage: P._validate([x, y, 1/3*z])
...
ValueError: The arguments [x, y, 1/3*z] are not valid polynomial functions on this projective space
sage: P._validate([x*y-z])
...
TypeError: The polynomial(s) [x*y - z] must be homogeneous
Return the affine patch of this projective space.
This is an ambient affine space
where
is the base ring of self, whose “projective embedding”
map is
in the
factor.
INPUT:
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)
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
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
Return the closed subscheme defined by X.
INPUT:
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
Return True if is a projective space, i.e., an ambient space
, where
is a ring and
is an
integer.
EXAMPLES:
sage: from sage.schemes.generic.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