r"""
Set of homomorphisms between two toric varieties.
For schemes `X` and `Y`, this module implements the set of morphisms
`Hom(X,Y)`. This is done by :class:`SchemeHomset_generic`.
As a special case, the Hom-sets can also represent the points of a
scheme. Recall that the `K`-rational points of a scheme `X` over `k`
can be identified with the set of morphisms `Spec(K) \to X`. In Sage,
the rational points are implemented by such scheme morphisms. This is
done by :class:`SchemeHomset_points` and its subclasses.
.. note::
You should not create the Hom-sets manually. Instead, use the
:meth:`~sage.structure.parent.Hom` method that is inherited by all
schemes.
AUTHORS:
- Volker Braun (2012-02-18): Initial version
EXAMPLES:
Here is a simple example, the projection of
`\mathbb{P}^1\times\mathbb{P}^1\to \mathbb{P}^1` ::
sage: P1xP1 = toric_varieties.P1xP1()
sage: P1 = toric_varieties.P1()
sage: hom_set = P1xP1.Hom(P1); hom_set
Set of morphisms
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
In terms of the fan, we can define this morphism by the projection
onto the first coordinate. The Hom-set can construct the morphism from
the projection matrix alone::
sage: hom_set(matrix([[1],[0]]))
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 1-d lattice N.
sage: _.as_polynomial_map()
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
Defn: Defined on coordinates by sending [s : t : x : y] to
[s : t]
In the case of toric algebraic schemes (defined by polynomials in
toric varieties), this module defines the underlying morphism of the
ambient toric varieties::
sage: P1xP1.inject_variables()
Defining s, t, x, y
sage: S = P1xP1.subscheme([s*x-t*y])
sage: type(S.Hom(S))
<class 'sage.schemes.toric.homset.SchemeHomset_toric_variety_with_category'>
"""
from sage.rings.all import ZZ, is_RingHomomorphism
from sage.matrix.matrix import is_Matrix
from sage.matrix.matrix_space import MatrixSpace
from sage.geometry.fan_morphism import FanMorphism
from sage.schemes.generic.homset import (SchemeHomset_generic,
SchemeHomset_points)
class SchemeHomset_toric_variety(SchemeHomset_generic):
"""
Set of homomorphisms between two toric varieties.
EXAMPLES::
sage: P1xP1 = toric_varieties.P1xP1()
sage: P1 = toric_varieties.P1()
sage: hom_set = P1xP1.Hom(P1); hom_set
Set of morphisms
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
sage: type(hom_set)
<class 'sage.schemes.toric.homset.SchemeHomset_toric_variety_with_category'>
sage: hom_set(matrix([[1],[0]]))
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 1-d lattice N.
"""
def __init__(self, X, Y, category=None, check=True, base=ZZ):
"""
The Python constructor.
INPUT:
The same as for any homset, see
:mod:`~sage.categories.homset`.
EXAMPLES::
sage: P1xP1 = toric_varieties.P1xP1()
sage: P1 = toric_varieties.P1()
sage: hom_set = P1xP1.Hom(P1); hom_set
Set of morphisms
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
An integral matrix defines a fan morphism, since we think of
the matrix as a linear map on the toric lattice. This is why
we need to ``register_conversion`` in the constructor
below. The result is::
sage: hom_set(matrix([[1],[0]]))
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 1-d lattice N.
"""
SchemeHomset_generic.__init__(self, X, Y, category=category, check=check, base=base)
self.register_conversion(MatrixSpace(ZZ, X.fan().dim(), Y.fan().dim()))
def _element_constructor_(self, x, check=True):
"""
Construct a scheme morphism.
INPUT:
- `x` -- anything that defines a morphism of toric
varieties. A matrix, fan morphism, or a list or tuple of
homogeneous polynomials that define a morphism.
- ``check`` -- boolean (default: ``True``) passed onto
functions called by this to be more careful about input
argument type checking
OUTPUT:
The morphism of toric varieties determined by ``x``.
EXAMPLES:
First, construct from fan morphism::
sage: dP8.<t,x0,x1,x2> = toric_varieties.dP8()
sage: P2.<y0,y1,y2> = toric_varieties.P2()
sage: hom_set = dP8.Hom(P2)
sage: fm = FanMorphism(identity_matrix(2), dP8.fan(), P2.fan())
sage: hom_set(fm) # calls hom_set._element_constructor_()
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 2-d lattice N.
A matrix will automatically be converted to a fan morphism::
sage: hom_set(identity_matrix(2))
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 2-d lattice N.
Alternatively, one can use homogeneous polynomials to define morphisms::
sage: P2.inject_variables()
Defining y0, y1, y2
sage: dP8.inject_variables()
Defining t, x0, x1, x2
sage: hom_set([x0,x1,x2])
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined on coordinates by sending [t : x0 : x1 : x2] to
[x0 : x1 : x2]
A morphism of the coordinate ring will also work::
sage: ring_hom = P2.coordinate_ring().hom([x0,x1,x2], dP8.coordinate_ring())
sage: ring_hom
Ring morphism:
From: Multivariate Polynomial Ring in y0, y1, y2 over Rational Field
To: Multivariate Polynomial Ring in t, x0, x1, x2 over Rational Field
Defn: y0 |--> x0
y1 |--> x1
y2 |--> x2
sage: hom_set(ring_hom)
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined on coordinates by sending [t : x0 : x1 : x2] to
[x0 : x1 : x2]
"""
from sage.schemes.toric.morphism import SchemeMorphism_polynomial_toric_variety
if isinstance(x, (list, tuple)):
return SchemeMorphism_polynomial_toric_variety(self, x, check=check)
if is_RingHomomorphism(x):
assert x.domain() is self.codomain().coordinate_ring()
assert x.codomain() is self.domain().coordinate_ring()
return SchemeMorphism_polynomial_toric_variety(self, x.im_gens(), check=check)
from sage.schemes.toric.morphism import SchemeMorphism_fan_toric_variety
if isinstance(x, FanMorphism):
return SchemeMorphism_fan_toric_variety(self, x, check=check)
if is_Matrix(x):
fm = FanMorphism(x, self.domain().fan(), self.codomain().fan())
return SchemeMorphism_fan_toric_variety(self, fm, check=check)
raise TypeError, "x must be a fan morphism or a list/tuple of polynomials"
class SchemeHomset_points_toric_field(SchemeHomset_points):
"""
Set of rational points of a toric variety.
INPUT:
- same as for :class:`SchemeHomset_points`.
OUPUT:
A scheme morphism of type
:class:`SchemeHomset_points_toric_field`.
EXAMPLES::
sage: P1xP1 = toric_varieties.P1xP1()
sage: P1xP1(QQ)
Set of rational points of 2-d CPR-Fano toric variety
covered by 4 affine patches
TESTS::
sage: import sage.schemes.toric.homset as HOM
sage: HOM.SchemeHomset_points_toric_field(Spec(QQ), P1xP1)
Set of rational points of 2-d CPR-Fano toric variety covered by 4 affine patches
"""
pass
