Enumerate Points of a Toric Variety

The classes here are not meant to be instatiated manually. Instead, you should always use the methods of the point set of the variety.

In this module, points are always represented by tuples instead of Sage’s class for points of the toric variety. All Sage library code must then convert it to proper point objects before returning it to the user.

EXAMPLES:

sage: P2 = toric_varieties.P2(base_ring=GF(3))
sage: point_set = P2.point_set()
sage: point_set.cardinality()
13
sage: iter(point_set).next()
[0 : 0 : 1]
sage: list(point_set)[0:5]
[[0 : 0 : 1], [1 : 0 : 0], [0 : 1 : 0], [0 : 1 : 1], [0 : 1 : 2]]
class sage.schemes.toric.points.InfinitePointEnumerator(fan, ring)

Bases: object

Point enumerator for infinite fields.

INPUT:

  • fan – fan of the toric variety.
  • ring – infinite base ring over which to enumerate points.

TESTS:

sage: from sage.schemes.toric.points import InfinitePointEnumerator
sage: fan = toric_varieties.P2().fan()
sage: n = InfinitePointEnumerator(fan, QQ)
sage: ni = iter(n)
sage: [ni.next() for k in range(10)]
[(0, 1, 1), (1, 1, 1), (-1, 1, 1), (1/2, 1, 1), (-1/2, 1, 1), 
 (2, 1, 1), (-2, 1, 1), (1/3, 1, 1), (-1/3, 1, 1), (3, 1, 1)]

sage: X = ToricVariety(Fan([], lattice=ZZ^0))
sage: X.point_set().cardinality()
1
sage: X.base_ring().is_finite()
False
sage: X.point_set().list()
([],)
class sage.schemes.toric.points.NaiveFinitePointEnumerator(fan, ring)

Bases: object

The naive point enumerator.

This is very slow.

INPUT:

  • fan – fan of the toric variety.
  • ring – finite base ring over which to enumerate points.

EXAMPLES:

sage: from sage.schemes.toric.points import NaiveFinitePointEnumerator
sage: fan = toric_varieties.P2().fan()
sage: n = NaiveFinitePointEnumerator(fan, GF(3))
sage: iter(n).next()
(0, 0, 1)
cone_iter()

Iterate over all cones of the fan

OUTPUT:

Iterator over the cones, starting with the high-dimensional ones.

EXAMPLES:

sage: ne = toric_varieties.dP6(base_ring=GF(11)).point_set()._naive_enumerator()
sage: for cone in ne.cone_iter(): 
....:     print cone.ambient_ray_indices()
(0, 1)
(1, 2)
(2, 3)
(3, 4)
(4, 5)
(0, 5)
(0,)
(1,)
(2,)
(3,)
(4,)
(5,)
()
coordinate_iter()

Iterate over all distinct homogeneous coordinates.

This method does NOT identify homogeneous coordinates that are equivalent by a homogeneous rescaling.

OUTPUT:

An iterator over the points.

EXAMPLES:

sage: F2 = GF(2)
sage: ni = toric_varieties.P2(base_ring=F2).point_set()._naive_enumerator()
sage: list(ni.coordinate_iter())
[(0, 0, 1), (1, 0, 0), (0, 1, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0), (1, 1, 1)]

sage: ni = toric_varieties.P1xP1(base_ring=F2).point_set()._naive_enumerator()
sage: list(ni.coordinate_iter())
[(0, 1, 0, 1), (1, 0, 0, 1), (1, 0, 1, 0),
 (0, 1, 1, 0), (0, 1, 1, 1), (1, 0, 1, 1),
 (1, 1, 0, 1), (1, 1, 1, 0), (1, 1, 1, 1)]

TESTS:

sage: V = ToricVariety(Fan([Cone([(1,1)])]), base_ring=GF(3))
sage: ni = V.point_set()._naive_enumerator()
sage: list(ni.coordinate_iter())
[(0, 1), (0, 2), (1, 1), (1, 2), (2, 1), (2, 2)]
orbit(point)

Return the orbit of homogeneous coordinates under rescalings.

OUTPUT:

The set of all homogeneous coordinates that are equivalent to point.

EXAMPLES:

sage: ne = toric_varieties.P2_123(base_ring=GF(7)).point_set()._naive_enumerator()
sage: sorted(ne.orbit([1, 0, 0]))
[(1, 0, 0), (2, 0, 0), (4, 0, 0)]
sage: sorted(ne.orbit([0, 1, 0]))
[(0, 1, 0), (0, 6, 0)]
sage: sorted(ne.orbit([0, 0, 1]))
[(0, 0, 1), (0, 0, 2), (0, 0, 3), (0, 0, 4), (0, 0, 5), (0, 0, 6)]
sage: sorted(ne.orbit([1, 1, 0]))
[(1, 1, 0), (1, 6, 0), (2, 1, 0), (2, 6, 0), (4, 1, 0), (4, 6, 0)]
rescalings()

Return the rescalings of homogeneous coordinates.

OUTPUT:

A tuple containing all points that are equivalent to \([1:1:\dots:1]\), the distinguished point of the big torus orbit.

EXAMPLES:

sage: ni = toric_varieties.P2_123(base_ring=GF(5)).point_set()._naive_enumerator()
sage: ni.rescalings()
((1, 1, 1), (1, 4, 4), (4, 2, 3), (4, 3, 2))

sage: ni = toric_varieties.dP8(base_ring=GF(3)).point_set()._naive_enumerator()
sage: ni.rescalings()
((1, 1, 1, 1), (1, 2, 2, 2), (2, 1, 2, 1), (2, 2, 1, 2))

sage: ni = toric_varieties.P1xP1(base_ring=GF(3)).point_set()._naive_enumerator()
sage: ni.rescalings()
((1, 1, 1, 1), (1, 1, 2, 2), (2, 2, 1, 1), (2, 2, 2, 2))
roots(n)

Return the n-th roots in the base field

EXAMPLES:

sage: ne = toric_varieties.P2(base_ring=GF(5)).point_set()._naive_enumerator()
sage: ne.roots(2)
(1, 4)
sage: ne.roots(3)
(1,)
sage: ne.roots(4)
(1, 2, 3, 4)
units()

Return the units in the base field.

EXAMPLES:

sage: ne = toric_varieties.P2(base_ring=GF(5)).point_set()._naive_enumerator()
sage: ne.units()
(1, 2, 3, 4)

Previous topic

Set of homomorphisms between two toric varieties.

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