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

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

Python

>>> 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

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]

Python

>>> 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

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'>

Python

>>> 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

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]

Python

>>> 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: integer or infinity; the cardinality of the set of points

EXAMPLES:

Sage

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

Python

>>> 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:

same as for SchemeHomset_points.

OUTPUT: a scheme morphism of type SchemeHomset_points_toric_base

EXAMPLES:

Sage

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

Python

>>> 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: boolean

EXAMPLES:

Sage

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

Python

>>> 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:

same as for SchemeHomset_points.

OUTPUT: a scheme morphism of type SchemeHomset_points_toric_field

EXAMPLES:

Sage

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

Python

>>> 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

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]]

Python

>>> 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

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]]

Python

>>> 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

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]]

Python

>>> 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: integer or infinity; the cardinality of the set of points

EXAMPLES:

Sage

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

Python

>>> 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

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

Python

>>> 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

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

Python

>>> 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

sage: V.count_points()
+Infinity

Python

>>> 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

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.

Python

>>> 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.