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 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 SchemeHomset_points and its subclasses.

Note

You should not create the Hom-sets manually. Instead, use the 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

>>> from sage.all import *
>>> P1xP1 = toric_varieties.P1xP1()
>>> P1 = toric_varieties.P1()
>>> 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]

>>> from sage.all import *
>>> hom_set(matrix([[Integer(1)],[Integer(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.
>>> _.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.all import *
>>> P1xP1.inject_variables()
Defining s, t, x, y
>>> S = P1xP1.subscheme([s*x - t*y])
>>> type(S.Hom(S))
<class 'sage.schemes.toric.homset.SchemeHomset_toric_variety_with_category'>

Finally, you can have morphisms defined through homogeneous coordinates where the codomain is not implemented as a toric variety:

sage: P2_toric.<x,y,z> = toric_varieties.P2()
sage: P2_native.<u,v,w> = ProjectiveSpace(QQ, 2)
sage: toric_to_native = P2_toric.Hom(P2_native);  toric_to_native
Set of morphisms
  From: 2-d CPR-Fano toric variety covered by 3 affine patches
  To:   Projective Space of dimension 2 over Rational Field
sage: type(toric_to_native)
<class 'sage.schemes.toric.homset.SchemeHomset_toric_variety_with_category'>
sage: toric_to_native([x^2, y^2, z^2])
Scheme morphism:
  From: 2-d CPR-Fano toric variety covered by 3 affine patches
  To:   Projective Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending [x : y : z] to (x^2 : y^2 : z^2)

sage: native_to_toric = P2_native.Hom(P2_toric);  native_to_toric
Set of morphisms
  From: Projective Space of dimension 2 over Rational Field
  To:   2-d CPR-Fano toric variety covered by 3 affine patches
sage: type(native_to_toric)
<class 'sage.schemes.projective.projective_homset.SchemeHomset_polynomial_projective_space_with_category'>
sage: native_to_toric([u^2, v^2, w^2])
Scheme morphism:
  From: Projective Space of dimension 2 over Rational Field
  To:   2-d CPR-Fano toric variety covered by 3 affine patches
  Defn: Defined on coordinates by sending (u : v : w) to [u^2 : v^2 : w^2]

>>> from sage.all import *
>>> P2_toric = toric_varieties.P2(names=('x', 'y', 'z',)); (x, y, z,) = P2_toric._first_ngens(3)
>>> P2_native = ProjectiveSpace(QQ, Integer(2), names=('u', 'v', 'w',)); (u, v, w,) = P2_native._first_ngens(3)
>>> toric_to_native = P2_toric.Hom(P2_native);  toric_to_native
Set of morphisms
  From: 2-d CPR-Fano toric variety covered by 3 affine patches
  To:   Projective Space of dimension 2 over Rational Field
>>> type(toric_to_native)
<class 'sage.schemes.toric.homset.SchemeHomset_toric_variety_with_category'>
>>> toric_to_native([x**Integer(2), y**Integer(2), z**Integer(2)])
Scheme morphism:
  From: 2-d CPR-Fano toric variety covered by 3 affine patches
  To:   Projective Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending [x : y : z] to (x^2 : y^2 : z^2)

>>> native_to_toric = P2_native.Hom(P2_toric);  native_to_toric
Set of morphisms
  From: Projective Space of dimension 2 over Rational Field
  To:   2-d CPR-Fano toric variety covered by 3 affine patches
>>> type(native_to_toric)
<class 'sage.schemes.projective.projective_homset.SchemeHomset_polynomial_projective_space_with_category'>
>>> native_to_toric([u**Integer(2), v**Integer(2), w**Integer(2)])
Scheme morphism:
  From: Projective Space of dimension 2 over Rational Field
  To:   2-d CPR-Fano toric variety covered by 3 affine patches
  Defn: Defined on coordinates by sending (u : v : w) to [u^2 : v^2 : w^2]
class sage.schemes.toric.homset.SchemeHomset_points_subscheme_toric_field(X, Y, category=None, check=True, base=Integer Ring)[source]#

Bases: SchemeHomset_points_toric_base

cardinality()[source]#

Return the number of points of the toric variety.

OUTPUT:

An integer or infinity. The cardinality of the set of points.

EXAMPLES:

sage: # needs sage.libs.singular
sage: P2.<x,y,z> = toric_varieties.P2(base_ring=GF(5))
sage: cubic = P2.subscheme([x^3 + y^3 + z^3])
sage: list(cubic.point_set())
[[0 : 1 : 4], [1 : 0 : 4], [1 : 4 : 0], [1 : 1 : 2], [1 : 2 : 1], [1 : 3 : 3]]
sage: cubic.point_set().cardinality()
6

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> P2 = toric_varieties.P2(base_ring=GF(Integer(5)), names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> cubic = P2.subscheme([x**Integer(3) + y**Integer(3) + z**Integer(3)])
>>> list(cubic.point_set())
[[0 : 1 : 4], [1 : 0 : 4], [1 : 4 : 0], [1 : 1 : 2], [1 : 2 : 1], [1 : 3 : 3]]
>>> cubic.point_set().cardinality()
6
class sage.schemes.toric.homset.SchemeHomset_points_toric_base(X, Y, category=None, check=True, base=Integer Ring)[source]#

Bases: SchemeHomset_points

Base class for homsets with toric ambient spaces.

INPUT:

OUTPUT: A scheme morphism of type SchemeHomset_points_toric_base.

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

>>> from sage.all import *
>>> P1xP1 = toric_varieties.P1xP1()
>>> P1xP1(QQ)
Set of rational points of 2-d CPR-Fano toric variety
covered by 4 affine patches
is_finite()[source]#

Return whether there are finitely many points.

OUTPUT: A boolean.

EXAMPLES:

sage: P2 = toric_varieties.P2()
sage: P2.point_set().is_finite()
False
sage: P2.change_ring(GF(7)).point_set().is_finite()
True

>>> from sage.all import *
>>> P2 = toric_varieties.P2()
>>> P2.point_set().is_finite()
False
>>> P2.change_ring(GF(Integer(7))).point_set().is_finite()
True
class sage.schemes.toric.homset.SchemeHomset_points_toric_field(X, Y, category=None, check=True, base=Integer Ring)[source]#

Bases: SchemeHomset_points_toric_base

Set of rational points of a toric variety.

You should not use this class directly. Instead, use the point_set() method to construct the point set of a toric variety.

INPUT:

OUTPUT: A scheme morphism of type SchemeHomset_points_toric_field.

EXAMPLES:

sage: P1xP1 = toric_varieties.P1xP1()
sage: P1xP1.point_set()
Set of rational points of 2-d CPR-Fano toric variety
covered by 4 affine patches
sage: P1xP1(QQ)
Set of rational points of 2-d CPR-Fano toric variety
covered by 4 affine patches

>>> from sage.all import *
>>> P1xP1 = toric_varieties.P1xP1()
>>> P1xP1.point_set()
Set of rational points of 2-d CPR-Fano toric variety
covered by 4 affine patches
>>> P1xP1(QQ)
Set of rational points of 2-d CPR-Fano toric variety
covered by 4 affine patches

The quotient \(\mathbb{P}^2 / \ZZ_3\) over \(GF(7)\) by the diagonal action. This is tricky because the base field has a 3-rd root of unity:

sage: fan = NormalFan(ReflexivePolytope(2, 0))
sage: X = ToricVariety(fan, base_field=GF(7))
sage: point_set = X.point_set()
sage: point_set.cardinality()
21
sage: sorted(X.point_set().list())
[[0 : 0 : 1], [0 : 1 : 0], [0 : 1 : 1], [0 : 1 : 3],
 [1 : 0 : 0], [1 : 0 : 1], [1 : 0 : 3], [1 : 1 : 0],
 [1 : 1 : 1], [1 : 1 : 2], [1 : 1 : 3], [1 : 1 : 4],
 [1 : 1 : 5], [1 : 1 : 6], [1 : 3 : 0], [1 : 3 : 1],
 [1 : 3 : 2], [1 : 3 : 3], [1 : 3 : 4], [1 : 3 : 5],
 [1 : 3 : 6]]

>>> from sage.all import *
>>> fan = NormalFan(ReflexivePolytope(Integer(2), Integer(0)))
>>> X = ToricVariety(fan, base_field=GF(Integer(7)))
>>> point_set = X.point_set()
>>> point_set.cardinality()
21
>>> sorted(X.point_set().list())
[[0 : 0 : 1], [0 : 1 : 0], [0 : 1 : 1], [0 : 1 : 3],
 [1 : 0 : 0], [1 : 0 : 1], [1 : 0 : 3], [1 : 1 : 0],
 [1 : 1 : 1], [1 : 1 : 2], [1 : 1 : 3], [1 : 1 : 4],
 [1 : 1 : 5], [1 : 1 : 6], [1 : 3 : 0], [1 : 3 : 1],
 [1 : 3 : 2], [1 : 3 : 3], [1 : 3 : 4], [1 : 3 : 5],
 [1 : 3 : 6]]

As for a non-compact example, the blow-up of the plane is the line bundle \(O_{\mathbf{P}^1}(-1)\). Its point set is the Cartesian product of the points on the base \(\mathbf{P}^1\) with the points on the fiber:

sage: fan = Fan([Cone([(1,0), (1,1)]), Cone([(1,1), (0,1)])])
sage: blowup_plane = ToricVariety(fan, base_ring=GF(3))
sage: point_set = blowup_plane.point_set()
sage: sorted(point_set.list())
[[0 : 1 : 0], [0 : 1 : 1], [0 : 1 : 2],
 [1 : 0 : 0], [1 : 0 : 1], [1 : 0 : 2],
 [1 : 1 : 0], [1 : 1 : 1], [1 : 1 : 2],
 [1 : 2 : 0], [1 : 2 : 1], [1 : 2 : 2]]

>>> from sage.all import *
>>> fan = Fan([Cone([(Integer(1),Integer(0)), (Integer(1),Integer(1))]), Cone([(Integer(1),Integer(1)), (Integer(0),Integer(1))])])
>>> blowup_plane = ToricVariety(fan, base_ring=GF(Integer(3)))
>>> point_set = blowup_plane.point_set()
>>> sorted(point_set.list())
[[0 : 1 : 0], [0 : 1 : 1], [0 : 1 : 2],
 [1 : 0 : 0], [1 : 0 : 1], [1 : 0 : 2],
 [1 : 1 : 0], [1 : 1 : 1], [1 : 1 : 2],
 [1 : 2 : 0], [1 : 2 : 1], [1 : 2 : 2]]

Toric varieties with torus factors (that is, where the fan is not full-dimensional) also work:

sage: F_times_Fstar = ToricVariety(Fan([Cone([(1,0)])]), base_field=GF(3))
sage: sorted(F_times_Fstar.point_set().list())
[[0 : 1], [0 : 2], [1 : 1], [1 : 2], [2 : 1], [2 : 2]]

>>> from sage.all import *
>>> F_times_Fstar = ToricVariety(Fan([Cone([(Integer(1),Integer(0))])]), base_field=GF(Integer(3)))
>>> sorted(F_times_Fstar.point_set().list())
[[0 : 1], [0 : 2], [1 : 1], [1 : 2], [2 : 1], [2 : 2]]
cardinality()[source]#

Return the number of points of the toric variety.

OUTPUT:

An integer or infinity. The cardinality of the set of points.

EXAMPLES:

sage: o = lattice_polytope.cross_polytope(3)
sage: V = ToricVariety(FaceFan(o))
sage: V.change_ring(GF(2)).point_set().cardinality()
27
sage: V.change_ring(GF(8, "a")).point_set().cardinality()                   # needs sage.rings.finite_rings
729
sage: V.change_ring(GF(101)).point_set().cardinality()
1061208

>>> from sage.all import *
>>> o = lattice_polytope.cross_polytope(Integer(3))
>>> V = ToricVariety(FaceFan(o))
>>> V.change_ring(GF(Integer(2))).point_set().cardinality()
27
>>> V.change_ring(GF(Integer(8), "a")).point_set().cardinality()                   # needs sage.rings.finite_rings
729
>>> V.change_ring(GF(Integer(101))).point_set().cardinality()
1061208

For non-smooth varieties over finite fields, the homogeneous rescalings are solved. This is somewhat slower:

sage: fan = NormalFan(ReflexivePolytope(2, 0))
sage: X = ToricVariety(fan, base_field=GF(7))
sage: X.point_set().cardinality()
21

>>> from sage.all import *
>>> fan = NormalFan(ReflexivePolytope(Integer(2), Integer(0)))
>>> X = ToricVariety(fan, base_field=GF(Integer(7)))
>>> X.point_set().cardinality()
21

Fulton’s formula does not apply since the variety is not smooth. And, indeed, naive application gives a different result:

sage: q = X.base_ring().order()
sage: n = X.dimension()
sage: d = map(len, fan().cones())
sage: sum(dk * (q-1)**(n-k) for k, dk in enumerate(d))
57

>>> from sage.all import *
>>> q = X.base_ring().order()
>>> n = X.dimension()
>>> d = map(len, fan().cones())
>>> sum(dk * (q-Integer(1))**(n-k) for k, dk in enumerate(d))
57

Over infinite fields the number of points is not very tricky:

sage: V.count_points()
+Infinity

>>> from sage.all import *
>>> V.count_points()
+Infinity

ALGORITHM:

Uses the formula in Fulton [Ful1993], section 4.5.

AUTHORS:

  • Beth Malmskog (2013-07-14)

  • Adriana Salerno (2013-07-14)

  • Yiwei She (2013-07-14)

  • Christelle Vincent (2013-07-14)

  • Ursula Whitcher (2013-07-14)

class sage.schemes.toric.homset.SchemeHomset_toric_variety(X, Y, category=None, check=True, base=Integer Ring)[source]#

Bases: 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.

>>> from sage.all import *
>>> P1xP1 = toric_varieties.P1xP1()
>>> P1 = toric_varieties.P1()
>>> 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
>>> type(hom_set)
<class 'sage.schemes.toric.homset.SchemeHomset_toric_variety_with_category'>

>>> hom_set(matrix([[Integer(1)],[Integer(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.