Products of projective spaces

This class builds on the projective space class and its point and morphism classes.

Products of projective spaces of varying dimension are convenient ambient spaces for complete intersections.

Group actions on them, and the interplay with representation theory, provide many interesting examples of algebraic varieties.

EXAMPLES:

We construct products projective spaces of various dimensions over the same ring:

sage: P1 = ProjectiveSpace(ZZ, 1, 'x')
sage: P2 = ProjectiveSpace(ZZ, 2, 'y')
sage: ProductProjectiveSpaces([P1, P2])
Product of projective spaces P^1 x P^2 over Integer Ring

We can also construct the product by specifying the dimensions and the base ring:

sage: ProductProjectiveSpaces([1, 2, 3], QQ, 'z')
Product of projective spaces P^1 x P^2 x P^3 over Rational Field

sage: P2xP2 = ProductProjectiveSpaces([2, 2], QQ, names=['x', 'y'])
sage: P2xP2.coordinate_ring().inject_variables()
Defining x0, x1, x2, y0, y1, y2

sage.schemes.product_projective.space.ProductProjectiveSpaces(n, R=None, names='x')

Return the Cartesian product of projective spaces.

The input n is either a list of projective space over the same base ring or the list of dimensions, R the base ring, and names the variable names.

INPUT:

• n – a list of integers or a list of projective spaces

• R – a ring

• names – a string or list of strings

EXAMPLES:

sage: P1 = ProjectiveSpace(QQ, 2, 'x')
sage: P2 = ProjectiveSpace(QQ, 3, 'y')
sage: ProductProjectiveSpaces([P1, P2])
Product of projective spaces P^2 x P^3 over Rational Field

sage: ProductProjectiveSpaces([2, 2],GF(7), 'y')
Product of projective spaces P^2 x P^2 over Finite Field of size 7

sage: P1 = ProjectiveSpace(ZZ, 2, 'x')
sage: P2 = ProjectiveSpace(QQ, 3, 'y')
sage: ProductProjectiveSpaces([P1, P2])
Traceback (most recent call last):
...
AttributeError: components must be over the same base ring

class sage.schemes.product_projective.space.ProductProjectiveSpaces_field(N, R=Rational Field, names=None)

Bases: sage.schemes.product_projective.space.ProductProjectiveSpaces_ring

points_of_bounded_height(**kwds)

Returns an iterator of the points in this product of projective spaces with the absolute heights of the components of at most the given bound.

Bound check is strict for the rational field. Requires the base field of this space to be a number field. Uses the Doyle-Krumm algorithm 4 (algorithm 5 for imaginary quadratic) for computing algebraic numbers up to a given height [DK2013].

The algorithm requires floating point arithmetic, so the user is allowed to specify the precision for such calculations. Additionally, due to floating point issues, points slightly larger than the bound may be returned. This can be controlled by lowering the tolerance.

INPUT:

• bound – a real number

• tolerance – a rational number in (0,1] used in doyle-krumm algorithm-4

• precision – the precision to use for computing the elements of bounded height of number fields.

OUTPUT: an iterator of points in this space

EXAMPLES:

sage: PP = ProductProjectiveSpaces(QQ, [1, 2])
sage: sorted(list(PP.points_of_bounded_height(bound=1)))
[(-1 : 1 , -1 : -1 : 1), (-1 : 1 , -1 : 0 : 1), (-1 : 1 , -1 : 1 : 0), (-1 : 1 , -1 : 1 : 1),
 (-1 : 1 , 0 : -1 : 1), (-1 : 1 , 0 : 0 : 1), (-1 : 1 , 0 : 1 : 0), (-1 : 1 , 0 : 1 : 1),
 (-1 : 1 , 1 : -1 : 1), (-1 : 1 , 1 : 0 : 0), (-1 : 1 , 1 : 0 : 1), (-1 : 1 , 1 : 1 : 0),
 (-1 : 1 , 1 : 1 : 1), (0 : 1 , -1 : -1 : 1), (0 : 1 , -1 : 0 : 1), (0 : 1 , -1 : 1 : 0),
 (0 : 1 , -1 : 1 : 1), (0 : 1 , 0 : -1 : 1), (0 : 1 , 0 : 0 : 1), (0 : 1 , 0 : 1 : 0),
 (0 : 1 , 0 : 1 : 1), (0 : 1 , 1 : -1 : 1), (0 : 1 , 1 : 0 : 0), (0 : 1 , 1 : 0 : 1),
 (0 : 1 , 1 : 1 : 0), (0 : 1 , 1 : 1 : 1), (1 : 0 , -1 : -1 : 1), (1 : 0 , -1 : 0 : 1),
 (1 : 0 , -1 : 1 : 0), (1 : 0 , -1 : 1 : 1), (1 : 0 , 0 : -1 : 1), (1 : 0 , 0 : 0 : 1),
 (1 : 0 , 0 : 1 : 0), (1 : 0 , 0 : 1 : 1), (1 : 0 , 1 : -1 : 1), (1 : 0 , 1 : 0 : 0),
 (1 : 0 , 1 : 0 : 1), (1 : 0 , 1 : 1 : 0), (1 : 0 , 1 : 1 : 1), (1 : 1 , -1 : -1 : 1),
 (1 : 1 , -1 : 0 : 1), (1 : 1 , -1 : 1 : 0), (1 : 1 , -1 : 1 : 1), (1 : 1 , 0 : -1 : 1),
 (1 : 1 , 0 : 0 : 1), (1 : 1 , 0 : 1 : 0), (1 : 1 , 0 : 1 : 1), (1 : 1 , 1 : -1 : 1),
 (1 : 1 , 1 : 0 : 0), (1 : 1 , 1 : 0 : 1), (1 : 1 , 1 : 1 : 0), (1 : 1 , 1 : 1 : 1)]

sage: u = QQ['u'].0
sage: P = ProductProjectiveSpaces([1, 1], NumberField(u^2 - 2, 'v'))
sage: sorted(list(P.points_of_bounded_height(bound=1.5)))
[(-v : 1 , -v : 1), (-v : 1 , -1 : 1), (-v : 1 , -1/2*v : 1), (-v : 1 , 0 : 1), (-v : 1 , 1/2*v : 1),
 (-v : 1 , 1 : 0), (-v : 1 , 1 : 1), (-v : 1 , v : 1), (-1 : 1 , -v : 1), (-1 : 1 , -1 : 1),
 (-1 : 1 , -1/2*v : 1), (-1 : 1 , 0 : 1), (-1 : 1 , 1/2*v : 1), (-1 : 1 , 1 : 0), (-1 : 1 , 1 : 1),
 (-1 : 1 , v : 1), (-1/2*v : 1 , -v : 1), (-1/2*v : 1 , -1 : 1), (-1/2*v : 1 , -1/2*v : 1), (-1/2*v : 1 , 0 : 1),
 (-1/2*v : 1 , 1/2*v : 1), (-1/2*v : 1 , 1 : 0), (-1/2*v : 1 , 1 : 1), (-1/2*v : 1 , v : 1), (0 : 1 , -v : 1),
 (0 : 1 , -1 : 1), (0 : 1 , -1/2*v : 1), (0 : 1 , 0 : 1), (0 : 1 , 1/2*v : 1), (0 : 1 , 1 : 0),
 (0 : 1 , 1 : 1), (0 : 1 , v : 1), (1/2*v : 1 , -v : 1), (1/2*v : 1 , -1 : 1), (1/2*v : 1 , -1/2*v : 1),
 (1/2*v : 1 , 0 : 1), (1/2*v : 1 , 1/2*v : 1), (1/2*v : 1 , 1 : 0), (1/2*v : 1 , 1 : 1), (1/2*v : 1 , v : 1),
 (1 : 0 , -v : 1), (1 : 0 , -1 : 1), (1 : 0 , -1/2*v : 1), (1 : 0 , 0 : 1), (1 : 0 , 1/2*v : 1),
 (1 : 0 , 1 : 0), (1 : 0 , 1 : 1), (1 : 0 , v : 1), (1 : 1 , -v : 1), (1 : 1 , -1 : 1),
 (1 : 1 , -1/2*v : 1), (1 : 1 , 0 : 1), (1 : 1 , 1/2*v : 1), (1 : 1 , 1 : 0), (1 : 1 , 1 : 1),
 (1 : 1 , v : 1), (v : 1 , -v : 1), (v : 1 , -1 : 1), (v : 1 , -1/2*v : 1), (v : 1 , 0 : 1),
 (v : 1 , 1/2*v : 1), (v : 1 , 1 : 0), (v : 1 , 1 : 1), (v : 1 , v : 1)]

class sage.schemes.product_projective.space.ProductProjectiveSpaces_finite_field(N, R=Rational Field, names=None)

Bases: sage.schemes.product_projective.space.ProductProjectiveSpaces_field

rational_points(F=None)

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

EXAMPLES:

sage: P = ProductProjectiveSpaces([1, 1], GF(5))
sage: P.rational_points()
[(0 : 1 , 0 : 1), (1 : 1 , 0 : 1), (2 : 1 , 0 : 1), (3 : 1 , 0 : 1), (4 : 1 , 0 : 1), (1 : 0 , 0 : 1),
(0 : 1 , 1 : 1), (1 : 1 , 1 : 1), (2 : 1 , 1 : 1), (3 : 1 , 1 : 1), (4 : 1 , 1 : 1), (1 : 0 , 1 : 1),
(0 : 1 , 2 : 1), (1 : 1 , 2 : 1), (2 : 1 , 2 : 1), (3 : 1 , 2 : 1), (4 : 1 , 2 : 1), (1 : 0 , 2 : 1),
(0 : 1 , 3 : 1), (1 : 1 , 3 : 1), (2 : 1 , 3 : 1), (3 : 1 , 3 : 1), (4 : 1 , 3 : 1), (1 : 0 , 3 : 1),
(0 : 1 , 4 : 1), (1 : 1 , 4 : 1), (2 : 1 , 4 : 1), (3 : 1 , 4 : 1), (4 : 1 , 4 : 1), (1 : 0 , 4 : 1),
(0 : 1 , 1 : 0), (1 : 1 , 1 : 0), (2 : 1 , 1 : 0), (3 : 1 , 1 : 0), (4 : 1 , 1 : 0), (1 : 0 , 1 : 0)]

sage: P = ProductProjectiveSpaces([1, 1], GF(2))
sage: P.rational_points(GF(2^2,'a'))
[(0 : 1 , 0 : 1), (a : 1 , 0 : 1), (a + 1 : 1 , 0 : 1), (1 : 1 , 0 : 1), (1 : 0 , 0 : 1), (0 : 1 , a : 1),
(a : 1 , a : 1), (a + 1 : 1 , a : 1), (1 : 1 , a : 1), (1 : 0 , a : 1), (0 : 1 , a + 1 : 1), (a : 1 , a + 1 : 1),
(a + 1 : 1 , a + 1 : 1), (1 : 1 , a + 1 : 1), (1 : 0 , a + 1 : 1), (0 : 1 , 1 : 1), (a : 1 , 1 : 1),
(a + 1 : 1 , 1 : 1), (1 : 1 , 1 : 1), (1 : 0 , 1 : 1), (0 : 1 , 1 : 0), (a : 1 , 1 : 0), (a + 1 : 1 , 1 : 0),
(1 : 1 , 1 : 0), (1 : 0 , 1 : 0)]

class sage.schemes.product_projective.space.ProductProjectiveSpaces_ring(N, R=Rational Field, names=None)

Bases: sage.schemes.generic.ambient_space.AmbientSpace

Cartesian product of projective spaces \(\mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_r}\).

EXAMPLES:

sage: P.<x0,x1,x2,x3,x4> = ProductProjectiveSpaces([1, 2], QQ); P
Product of projective spaces P^1 x P^2 over Rational Field
sage: P.coordinate_ring()
Multivariate Polynomial Ring in x0, x1, x2, x3, x4 over Rational Field
sage: P[0]
Projective Space of dimension 1 over Rational Field
sage: P[1]
Projective Space of dimension 2 over Rational Field
sage: Q = P(6, 3, 2, 2, 2); Q
(2 : 1 , 1 : 1 : 1)
sage: Q[0]
(2 : 1)
sage: H = Hom(P,P)
sage: f = H([x0^2*x3, x2*x1^2, x2^2, 2*x3^2, x4^2])
sage: f(Q)
(4 : 1 , 1 : 2 : 1)

affine_patch(I, return_embedding=False)

Return the \(I^{th}\) affine patch of this projective space product where I is a multi-index.

INPUT:

• I – a list or tuple of positive integers.

• return_embedding – Boolean, if true the projective embedding is also returned.

OUTPUT:

• An affine space.

• An embedding into a product of projective spaces (optional).

EXAMPLES:

sage: PP = ProductProjectiveSpaces([2, 2, 2], ZZ, 'x')
sage: phi = PP.affine_patch([0, 1, 2], True)
sage: phi.domain()
Affine Space of dimension 6 over Integer Ring
sage: phi
Scheme morphism:
      From: Affine Space of dimension 6 over Integer Ring
      To:   Product of projective spaces P^2 x P^2 x P^2 over Integer Ring
      Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4, x5) to
            (1 : x0 : x1 , x2 : 1 : x3 , x4 : x5 : 1)

change_ring(R)

Return a product of projective spaces over a ring R and otherwise the same as this projective space.

INPUT:

• R – commutative ring or morphism

OUTPUT:

• Product of projective spaces 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: T.<x,y,z,u,v,w> = ProductProjectiveSpaces([2, 2], QQ)
sage: T.change_ring(GF(17))
Product of projective spaces P^2 x P^2 over Finite Field of size 17

components()

Return the components of this product of projective spaces.

OUTPUT: a list of projective spaces

EXAMPLES:

sage: P.<x,y,z,u,v> = ProductProjectiveSpaces(QQ,[2,1])
sage: P.components()
[Projective Space of dimension 2 over Rational Field,
Projective Space of dimension 1 over Rational Field]

dimension()

Return the absolute dimension of the product of projective spaces.

OUTPUT: a positive integer

EXAMPLES:

sage: T.<x,y,z,u,v,w> = ProductProjectiveSpaces([2, 2], GF(17))
sage: T.dimension_absolute()
4
sage: T.dimension()
4

dimension_absolute()

Return the absolute dimension of the product of projective spaces.

OUTPUT: a positive integer

EXAMPLES:

sage: T.<x,y,z,u,v,w> = ProductProjectiveSpaces([2, 2], GF(17))
sage: T.dimension_absolute()
4
sage: T.dimension()
4

dimension_absolute_components()

Return the absolute dimension of the product of projective spaces.

OUTPUT: a list of positive integers

EXAMPLES:

sage: T.<x,y,z,u,v,w> = ProductProjectiveSpaces([2, 2], GF(17))
sage: T.dimension_absolute_components()
[2, 2]
sage: T.dimension_components()
[2, 2]

dimension_components()

Return the absolute dimension of the product of projective spaces.

OUTPUT: a list of positive integers

EXAMPLES:

sage: T.<x,y,z,u,v,w> = ProductProjectiveSpaces([2, 2], GF(17))
sage: T.dimension_absolute_components()
[2, 2]
sage: T.dimension_components()
[2, 2]

dimension_relative()

Return the relative dimension of the product of projective spaces.

OUTPUT: a positive integer

EXAMPLES:

sage: T.<a,x,y,z,u,v,w> = ProductProjectiveSpaces([3,2],QQ)
sage: T.dimension_relative()
5

dimension_relative_components()

Return the relative dimension of the product of projective spaces.

OUTPUT: a list of positive integers

EXAMPLES:

sage: T.<a,x,y,z,u,v,w> = ProductProjectiveSpaces([3, 2], QQ)
sage: T.dimension_relative_components()
[3, 2]

ngens()

Return the number of generators of this space.

This is the number of variables in the coordinate ring of the projective space.

OUTPUT: an integer

EXAMPLES:

sage: T = ProductProjectiveSpaces([1, 1, 1], GF(5), 'x')
sage: T.ngens()
6

num_components()

Returns the number of components of this space.

OUTPUT: an integer

EXAMPLES:

sage: T = ProductProjectiveSpaces([1, 1, 1], GF(5), 'x')
sage: T.num_components()
3

segre_embedding(PP=None, var='u')

Return the Segre embedding of this space into the appropriate projective space.

INPUT:

• PP – (default: None) ambient image projective space;

  this is constructed if it is not given.

• var – string, variable name of the image projective space, default \(u\) (optional).

OUTPUT:

Hom – from this space to the appropriate subscheme of projective space.

Todo

Cartesian products with more than two components.

EXAMPLES:

sage: X.<y0,y1,y2,y3,y4,y5> = ProductProjectiveSpaces(ZZ, [2, 2])
sage: phi = X.segre_embedding(); phi
Scheme morphism:
  From: Product of projective spaces P^2 x P^2 over Integer Ring
  To:   Closed subscheme of Projective Space of dimension 8 over Integer Ring defined by:
  -u5*u7 + u4*u8,
  -u5*u6 + u3*u8,
  -u4*u6 + u3*u7,
  -u2*u7 + u1*u8,
  -u2*u4 + u1*u5,
  -u2*u6 + u0*u8,
  -u1*u6 + u0*u7,
  -u2*u3 + u0*u5,
  -u1*u3 + u0*u4
  Defn: Defined by sending (y0 : y1 : y2 , y3 : y4 : y5) to
        (y0*y3 : y0*y4 : y0*y5 : y1*y3 : y1*y4 : y1*y5 : y2*y3 : y2*y4 : y2*y5).

::

sage: T = ProductProjectiveSpaces([1, 2], CC, 'z')
sage: T.segre_embedding()
Scheme morphism:
  From: Product of projective spaces P^1 x P^2 over Complex Field with 53 bits of precision
  To:   Closed subscheme of Projective Space of dimension 5 over Complex Field with 53 bits of precision defined by:
  -u2*u4 + u1*u5,
  -u2*u3 + u0*u5,
  -u1*u3 + u0*u4
  Defn: Defined by sending (z0 : z1 , z2 : z3 : z4) to
        (z0*z2 : z0*z3 : z0*z4 : z1*z2 : z1*z3 : z1*z4).

::

sage: T = ProductProjectiveSpaces([1, 2, 1], QQ, 'z')
sage: T.segre_embedding()
Scheme morphism:
  From: Product of projective spaces P^1 x P^2 x P^1 over Rational Field
  To:   Closed subscheme of Projective Space of dimension 11 over
Rational Field defined by:
  -u9*u10 + u8*u11,
  -u7*u10 + u6*u11,
  -u7*u8 + u6*u9,
  -u5*u10 + u4*u11,
  -u5*u8 + u4*u9,
  -u5*u6 + u4*u7,
  -u5*u9 + u3*u11,
  -u5*u8 + u3*u10,
  -u5*u8 + u2*u11,
  -u4*u8 + u2*u10,
  -u3*u8 + u2*u9,
  -u3*u6 + u2*u7,
  -u3*u4 + u2*u5,
  -u5*u7 + u1*u11,
  -u5*u6 + u1*u10,
  -u3*u7 + u1*u9,
  -u3*u6 + u1*u8,
  -u5*u6 + u0*u11,
  -u4*u6 + u0*u10,
  -u3*u6 + u0*u9,
  -u2*u6 + u0*u8,
  -u1*u6 + u0*u7,
  -u1*u4 + u0*u5,
  -u1*u2 + u0*u3
  Defn: Defined by sending (z0 : z1 , z2 : z3 : z4 , z5 : z6) to
        (z0*z2*z5 : z0*z2*z6 : z0*z3*z5 : z0*z3*z6 : z0*z4*z5 : z0*z4*z6
: z1*z2*z5 : z1*z2*z6 : z1*z3*z5 : z1*z3*z6 : z1*z4*z5 : z1*z4*z6).

subscheme(X)

Return the closed subscheme defined by X.

INPUT:

• X – a list or tuple of equations

OUTPUT:

AlgebraicScheme_subscheme_projective_cartesian_product.

EXAMPLES:

sage: P.<x,y,z,w> = ProductProjectiveSpaces([1, 1],GF(5))
sage: X = P.subscheme([x-y, z-w]);X
Closed subscheme of Product of projective spaces P^1 x P^1 over Finite Field of size 5 defined by:
      x - y,
      z - w
sage: X.defining_polynomials ()
[x - y, z - w]
sage: I = X.defining_ideal(); I
Ideal (x - y, z - w) of Multivariate Polynomial Ring in x, y, z, w over
Finite Field of size 5
sage: X.dimension()
0
sage: X.base_ring()
Finite Field of size 5
sage: X.base_scheme()
Spectrum of Finite Field of size 5
sage: X.structure_morphism()
Scheme morphism:
      From: Closed subscheme of Product of projective spaces P^1 x P^1 over Finite Field of size 5 defined by:
      x - y,
      z - w
      To:   Spectrum of Finite Field of size 5
      Defn: Structure map

sage.schemes.product_projective.space.is_ProductProjectiveSpaces(x)

Return True if x is a product of projective spaces.

This is an ambient space defined by \(\mathbb{P}^n_R \times \cdots \times \mathbb{P}^m_R\), where \(R\) is a ring and \(n,\ldots, m\geq 0\) are integers.

OUTPUT: Boolean.

EXAMPLES:

sage: is_ProductProjectiveSpaces(ProjectiveSpace(5, names='x'))
False
sage: is_ProductProjectiveSpaces(ProductProjectiveSpaces([1, 2, 3], ZZ, 'x'))
True