Toric varieties

This module provides support for (normal) toric varieties, corresponding to rational polyhedral fans. See also fano_variety for a more restrictive class of (weak) Fano toric varieties.

An excellent reference on toric varieties is the book “Toric Varieties” by David A. Cox, John B. Little, and Hal Schenck [CLS].

The interface to this module is provided through functions AffineToricVariety() and ToricVariety(), although you may also be interested in normalize_names().

Note

We do NOT build “general toric varieties” from affine toric varieties. Instead, we are using the quotient representation of toric varieties with the homogeneous coordinate ring (a.k.a. Cox’s ring or the total coordinate ring). This description works best for simplicial fans of the full dimension.

REFERENCES:

[CLS](1, 2) David A. Cox, John B. Little, Hal Schenck, “Toric Varieties”, Graduate Studies in Mathematics, Amer. Math. Soc., Providence, RI, 2011

AUTHORS:

  • Andrey Novoseltsev (2010-05-17): initial version.
  • Volker Braun (2010-07-24): Cohomology and characteristic classes added.

EXAMPLES:

We start with constructing the affine plane as an affine toric variety. First, we need to have a corresponding cone:

sage: quadrant = Cone([(1,0), (0,1)])

If you don’t care about variable names and the base field, that’s all we need for now:

sage: A2 = AffineToricVariety(quadrant)
sage: A2
2-d affine toric variety
sage: origin = A2(0,0)
sage: origin
[0 : 0]

Only affine toric varieties have points whose (homogeneous) coordinates are all zero.

sage: parent(origin)
Set of rational points of 2-d affine toric variety

As you can see, by default toric varieties live over the field of rational numbers:

sage: A2.base_ring()
Rational Field

While usually toric varieties are considered over the field of complex numbers, for computational purposes it is more convenient to work with fields that have exact representation on computers. You can also always do

sage: C2 = AffineToricVariety(quadrant, base_field=CC)
sage: C2.base_ring()
Complex Field with 53 bits of precision
sage: C2(1,2+i)
[1.00000000000000 : 2.00000000000000 + 1.00000000000000*I]

or even

sage: F = CC["a, b"].fraction_field()
sage: F.inject_variables()
Defining a, b
sage: A2 = AffineToricVariety(quadrant, base_field=F)
sage: A2(a,b)
[a : b]

OK, if you need to work only with affine spaces, AffineSpace() may be a better way to construct them. Our next example is the product of two projective lines realized as the toric variety associated to the face fan of the “diamond”:

sage: diamond = lattice_polytope.cross_polytope(2)
sage: diamond.vertices_pc()
M( 1,  0),
M( 0,  1),
M(-1,  0),
M( 0, -1)
in 2-d lattice M
sage: fan = FaceFan(diamond)
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1
2-d toric variety covered by 4 affine patches
sage: P1xP1.fan().rays()
M( 1,  0),
M( 0,  1),
M(-1,  0),
M( 0, -1)
in 2-d lattice M
sage: P1xP1.gens()
(z0, z1, z2, z3)

We got four coordinates - two for each of the projective lines, but their names are perhaps not very well chosen. Let’s make \((x,y)\) to be coordinates on the first line and \((s,t)\) on the second one:

sage: P1xP1 = ToricVariety(fan, coordinate_names="x s y t")
sage: P1xP1.gens()
(x, s, y, t)

Now, if we want to define subschemes of this variety, the defining polynomials must be homogeneous in each of these pairs:

sage: P1xP1.inject_variables()
Defining x, s, y, t
sage: P1xP1.subscheme(x)
Closed subscheme of 2-d toric variety
covered by 4 affine patches defined by:
  x
sage: P1xP1.subscheme(x^2 + y^2)
Closed subscheme of 2-d toric variety
covered by 4 affine patches defined by:
  x^2 + y^2
sage: P1xP1.subscheme(x^2 + s^2)
Traceback (most recent call last):
...
ValueError: x^2 + s^2 is not homogeneous
on 2-d toric variety covered by 4 affine patches!
sage: P1xP1.subscheme([x^2*s^2 + x*y*t^2 +y^2*t^2, s^3 + t^3])
Closed subscheme of 2-d toric variety
covered by 4 affine patches defined by:
  x^2*s^2 + x*y*t^2 + y^2*t^2,
  s^3 + t^3

While we don’t build toric varieties from affine toric varieties, we still can access the “building pieces”:

sage: patch = P1xP1.affine_patch(2)
sage: patch
2-d affine toric variety
sage: patch.fan().rays()
M(1, 0),
M(0, 1)
in 2-d lattice M
sage: patch.embedding_morphism()
Scheme morphism:
  From: 2-d affine toric variety
  To:   2-d toric variety covered by 4 affine patches
  Defn: Defined on coordinates by sending [x : s] to
        [x : s : 1 : 1]

The patch above was specifically chosen to coincide with our representation of the affine plane before, but you can get the other three patches as well. (While any cone of a fan will correspond to an affine toric variety, the main interest is usually in the generating fans as “the biggest” affine subvarieties, and these are precisely the patches that you can get from affine_patch().)

All two-dimensional toric varieties are “quite nice” because any two-dimensional cone is generated by exactly two rays. From the point of view of the corresponding toric varieties, this means that they have at worst quotient singularities:

sage: P1xP1.is_orbifold()
True
sage: P1xP1.is_smooth()
True
sage: TV = ToricVariety(NormalFan(diamond))
sage: TV.fan().rays()
N(-1,  1),
N( 1,  1),
N(-1, -1),
N( 1, -1)
in 2-d lattice N
sage: TV.is_orbifold()
True
sage: TV.is_smooth()
False

In higher dimensions worse things can happen:

sage: TV3 = ToricVariety(NormalFan(lattice_polytope.cross_polytope(3)))
sage: TV3.fan().rays()
N(-1, -1,  1),
N( 1, -1,  1),
N(-1,  1,  1),
N( 1,  1,  1),
N(-1, -1, -1),
N( 1, -1, -1),
N(-1,  1, -1),
N( 1,  1, -1)
in 3-d lattice N
sage: TV3.is_orbifold()
False

Fortunately, we can perform a (partial) resolution:

sage: TV3_res = TV3.resolve_to_orbifold()
sage: TV3_res.is_orbifold()
True
sage: TV3_res.fan().ngenerating_cones()
12
sage: TV3.fan().ngenerating_cones()
6

In this example we had to double the number of affine patches. The result is still singular:

sage: TV3_res.is_smooth()
False

You can resolve it further using resolve() method, but (at least for now) you will have to specify which rays should be inserted into the fan. See also CPRFanoToricVariety(), which can construct some other “nice partial resolutions.”

The intersection theory on toric varieties is very well understood, and there are explicit algorithms to compute many quantities of interest. The most important tools are the cohomology ring and the Chow group. For \(d\)-dimensional compact toric varieties with at most orbifold singularities, the rational cohomology ring \(H^*(X,\QQ)\) and the rational Chow ring \(A^*(X,\QQ) = A_{d-*}(X)\otimes \QQ\) are isomorphic except for a doubling in degree. More precisely, the Chow group has the same rank

\[A_{d-k}(X) \otimes \QQ \simeq H^{2k}(X,\QQ)\]

and the intersection in of Chow cycles matches the cup product in cohomology.

In this case, you should work with the cohomology ring description because it is much faster. For example, here is a weighted projective space with a curve of \(\ZZ_3\)-orbifold singularities:

sage: P4_11133 = toric_varieties.P4_11133()
sage: P4_11133.is_smooth(), P4_11133.is_orbifold()
(False, True)
sage: cone = P4_11133.fan(3)[8]
sage: cone.is_smooth(), cone.is_simplicial()
(False, True)
sage: HH = P4_11133.cohomology_ring();  HH
Rational cohomology ring of a 4-d CPR-Fano toric variety covered by 5 affine patches
sage: P4_11133.cohomology_basis()
(([1],), ([z4],), ([z4^2],), ([z4^3],), ([z4^4],))

Every cone defines a torus orbit closure, and hence a (co)homology class:

sage: HH.gens()
([3*z4], [3*z4], [z4], [z4], [z4])
sage: map(HH, P4_11133.fan(1))
[[3*z4], [3*z4], [z4], [z4], [z4]]
sage: map(HH, P4_11133.fan(4) )
[[9*z4^4], [9*z4^4], [9*z4^4], [9*z4^4], [9*z4^4]]
sage: HH(cone)
[3*z4^3]

We can compute intersection numbers by integrating top-dimensional cohomology classes:

sage: D = P4_11133.divisor(0)
sage: HH(D)
[3*z4]
sage: P4_11133.integrate( HH(D)^4 )
9
sage: P4_11133.integrate( HH(D) * HH(cone) )
1

Although computationally less efficient, we can do the same computations with the rational Chow group:

sage: AA = P4_11133.Chow_group(QQ)
sage: map(AA, P4_11133.fan(1))  # long time (5s on sage.math, 2012)
[( 0 | 0 | 0 | 3 | 0 ), ( 0 | 0 | 0 | 3 | 0 ), ( 0 | 0 | 0 | 1 | 0 ), ( 0 | 0 | 0 | 1 | 0 ), ( 0 | 0 | 0 | 1 | 0 )]
sage: map(AA, P4_11133.fan(4))  # long time (5s on sage.math, 2012)
[( 1 | 0 | 0 | 0 | 0 ), ( 1 | 0 | 0 | 0 | 0 ), ( 1 | 0 | 0 | 0 | 0 ), ( 1 | 0 | 0 | 0 | 0 ), ( 1 | 0 | 0 | 0 | 0 )]
sage: AA(cone).intersection_with_divisor(D)  # long time (4s on sage.math, 2013)
( 1 | 0 | 0 | 0 | 0 )
sage: AA(cone).intersection_with_divisor(D).count_points()  # long time
1

The real advantage of the Chow group is that

  • it works just as well over \(\ZZ\), so torsion information is also easily available, and
  • its combinatorial description also works over worse-than-orbifold singularities. By contrast, the cohomology groups can become very complicated to compute in this case, and one usually only has a spectral sequence but no toric algorithm.

Below you will find detailed descriptions of available functions. If you are familiar with toric geometry, you will likely see that many important objects and operations are unavailable. However, this module is under active development and hopefully will improve in future releases of Sage. If there are some particular features that you would like to see implemented ASAP, please consider reporting them to the Sage Development Team or even implementing them on your own as a patch for inclusion!

sage.schemes.toric.variety.AffineToricVariety(cone, *args, **kwds)

Construct an affine toric variety.

INPUT:

This cone will be used to construct a rational polyhedral fan, which will be passed to ToricVariety() with the rest of positional and keyword arguments.

OUTPUT:

Note

The generating rays of the fan of this variety are guaranteed to be listed in the same order as the rays of the original cone.

EXAMPLES:

We will create the affine plane as an affine toric variety:

sage: quadrant = Cone([(1,0), (0,1)])
sage: A2 = AffineToricVariety(quadrant)
sage: origin = A2(0,0)
sage: origin
[0 : 0]
sage: parent(origin)
Set of rational points of 2-d affine toric variety

Only affine toric varieties have points whose (homogeneous) coordinates are all zero.

class sage.schemes.toric.variety.CohomologyClass(cohomology_ring, representative)

Bases: sage.rings.quotient_ring_element.QuotientRingElement

An element of the CohomologyRing.

Warning

You should not create instances of this class manually. The generators of the cohomology ring as well as the cohomology classes associated to cones of the fan can be obtained from ToricVariety_field.cohomology_ring().

EXAMPLES:

sage: P2 = toric_varieties.P2()
sage: P2.cohomology_ring().gen(0)
[z]
sage: HH = P2.cohomology_ring()
sage: HH.gen(0)
[z]
sage: cone = P2.fan(1)[0];  HH(cone)
[z]
deg()

The degree of the cohomology class.

OUTPUT:

An integer \(d\) such that the cohomology class is in degree \(2d\). If the cohomology class is of mixed degree, the highest degree is returned.

EXAMPLES:

sage: P2 = toric_varieties.P2()
sage: P2.cohomology_ring().gen(0).deg()
1
sage: P2.cohomology_ring().zero().deg()
-1
exp()

Exponentiate self.

Note

The exponential \(\exp(x)\) of a rational number \(x\) is usually not rational. Therefore, the cohomology class must not have a constant (degree zero) part. The coefficients in the Taylor series of \(\exp\) are rational, so any cohomology class without constant term can be exponentiated.

OUTPUT

The cohomology class \(\exp(\) self \()\) if the constant part vanishes, otherwise a ValueError is raised.

EXAMPLES:

sage: P2 = toric_varieties.P2()
sage: H_class = P2.cohomology_ring().gen(0)
sage: H_class
[z]
sage: H_class.exp()
[1/2*z^2 + z + 1]
part_of_degree(d)

Project the (mixed-degree) cohomology class to the given degree.

\[\mathop{pr}\nolimits_d:~ H^\bullet(X_\Delta,\QQ) \to H^{2d}(X_\Delta,\QQ)\]

INPUT:

  • An integer d

OUTPUT:

  • The degree-2d part of the cohomology class.

EXAMPLES:

sage: P1xP1 = toric_varieties.P1xP1()
sage: t = P1xP1.cohomology_ring().gen(0)
sage: y = P1xP1.cohomology_ring().gen(2)
sage: 3*t+4*t^2*y+y+t*y+t+1
[t*y + 4*t + y + 1]
sage: (3*t+4*t^2*y+y+t*y+t+1).part_of_degree(1)
[4*t + y]
class sage.schemes.toric.variety.CohomologyRing(variety)

Bases: sage.rings.quotient_ring.QuotientRing_generic, sage.structure.unique_representation.UniqueRepresentation

The (even) cohomology ring of a toric variety.

Irregardles of the variety’s base ring, we always work with the variety over \(\CC\) and its topology.

The cohomology is always the singular cohomology with \(\QQ\)-coefficients. Note, however, that the cohomology of smooth toric varieties is torsion-free, so there is no loss of information in that case.

Currently, the toric variety must not be “too singular”. See ToricVariety_field.cohomology_ring() for a detailed description of which toric varieties are admissible. For such varieties the odd-dimensional cohomology groups vanish.

Warning

You should not create instances of this class manually. Use ToricVariety_field.cohomology_ring() to generate the cohomology ring.

INPUT:

  • variety – a toric variety. Currently, the toric variety must be at least an orbifold. See ToricVariety_field.cohomology_ring() for a detailed description of which toric varieties are admissible.

EXAMPLES:

sage: P2 = toric_varieties.P2()
sage: P2.cohomology_ring()
Rational cohomology ring of a 2-d CPR-Fano toric variety covered by 3 affine patches

This is equivalent to:

sage: from sage.schemes.toric.variety import CohomologyRing
sage: CohomologyRing(P2)
Rational cohomology ring of a 2-d CPR-Fano toric variety covered by 3 affine patches
gen(i)

Return the generators of the cohomology ring.

INPUT:

  • i – integer.

OUTPUT:

The i-th generator of the cohomology ring. If we denote the toric variety by X, then this generator is associated to the ray X.fan().ray(i), which spans the one-cone X.fan(1)[i]

EXAMPLES:

sage: P2 = toric_varieties.P2()
sage: P2.cohomology_ring().gen(2)
[z]
gens()

Return the generators of the cohomology ring.

OUTPUT:

A tuple of generators, one for each toric divisor of the toric variety X. The order is the same as the ordering of the rays of the fan X.fan().rays(), which is also the same as the ordering of the one-cones in X.fan(1)

EXAMPLES:

sage: P2 = toric_varieties.P2()
sage: P2.cohomology_ring().gens()
([z], [z], [z])
sage.schemes.toric.variety.ToricVariety(fan, coordinate_names=None, names=None, coordinate_indices=None, base_ring=Rational Field, base_field=None)

Construct a toric variety.

INPUT:

  • fanrational polyhedral fan;
  • coordinate_names – names of variables for the coordinate ring, see normalize_names() for acceptable formats. If not given, indexed variable names will be created automatically;
  • names – an alias of coordinate_names for internal use. You may specify either names or coordinate_names, but not both;
  • coordinate_indices – list of integers, indices for indexed variables. If not given, the index of each variable will coincide with the index of the corresponding ray of the fan;
  • base_ring – base ring of the toric variety (default: \(\QQ\)). Must be a field.
  • base_field – alias for base_ring. Takes precedence if both are specified.

OUTPUT:

EXAMPLES:

We will create the product of two projective lines:

sage: fan = FaceFan(lattice_polytope.cross_polytope(2))
sage: fan.rays()
M( 1,  0),
M( 0,  1),
M(-1,  0),
M( 0, -1)
in 2-d lattice M
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1.gens()
(z0, z1, z2, z3)

Let’s create some points:

sage: P1xP1(1,1,1,1)
[1 : 1 : 1 : 1]
sage: P1xP1(0,1,1,1)
[0 : 1 : 1 : 1]
sage: P1xP1(0,1,0,1)
Traceback (most recent call last):
...
TypeError: coordinates (0, 1, 0, 1)
are in the exceptional set!

We cannot set to zero both coordinates of the same projective line!

Let’s change the names of the variables. We have to re-create our toric variety:

sage: P1xP1 = ToricVariety(fan, "x s y t")
sage: P1xP1.gens()
(x, s, y, t)

Now \((x, y)\) correspond to one line and \((s, t)\) to the other one.

sage: P1xP1.inject_variables()
Defining x, s, y, t
sage: P1xP1.subscheme(x*s-y*t)
Closed subscheme of 2-d toric variety
covered by 4 affine patches defined by:
  x*s - y*t

Here is a shorthand for defining the toric variety and homogeneous coordinates in one go:

sage: P1xP1.<a,b,c,d> = ToricVariety(fan)
sage: (a^2+b^2) * (c+d)
a^2*c + b^2*c + a^2*d + b^2*d
class sage.schemes.toric.variety.ToricVariety_field(fan, coordinate_names, coordinate_indices, base_field)

Bases: sage.misc.cachefunc.ClearCacheOnPickle, sage.schemes.generic.ambient_space.AmbientSpace

Construct a toric variety associated to a rational polyhedral fan.

Warning

This class does not perform any checks of correctness of input. Use ToricVariety() and AffineToricVariety() to construct toric varieties.

INPUT:

  • fanrational polyhedral fan;
  • coordinate_names – names of variables, see normalize_names() for acceptable formats. If None, indexed variable names will be created automatically;
  • coordinate_indices – list of integers, indices for indexed variables. If None, the index of each variable will coincide with the index of the corresponding ray of the fan;
  • base_field – base field of the toric variety.

OUTPUT:

TESTS:

sage: fan = FaceFan(lattice_polytope.cross_polytope(2))
sage: P1xP1 = ToricVariety(fan)
Aut_dimension()

Return the dimension of the automorphism group

There are three kinds of symmetries of toric varieties:

  • Toric automorphisms (rescaling of homogeneous coordinates)
  • Demazure roots. These are translations \(x_i \to x_i + \epsilon x^m\) of a homogeneous coordinate \(x_i\) by a monomial \(x^m\) of the same homogeneous degree.
  • Symmetries of the fan. These yield discrete subgroups.

OUTPUT:

An integer. The dimension of the automorphism group. Equals the dimension of the \(M\)-lattice plus the number of Demazure roots.

EXAMPLES:

sage: P2 = toric_varieties.P2()
sage: P2.Aut_dimension()
8

TESTS:

sage: toric_varieties.A1().Aut_dimension()
Traceback (most recent call last):
...
NotImplementedError: Aut_dimension() is only implemented for complete toric varieties.
Chern_character(deg=None)

Return the Chern character (of the tangent bundle) of the toric variety.

INPUT:

  • deg – integer (optional). The degree of the Chern character.

OUTPUT:

  • If the degree is specified, the degree-deg part of the Chern character.
  • If no degree is specified, the total Chern character.

REFERENCES:

EXAMPLES:

sage: dP6 = toric_varieties.dP6()
sage: dP6.Chern_character()
[3*w^2 + y + 2*v + 2*z + w + 2]
sage: dP6.ch()
[3*w^2 + y + 2*v + 2*z + w + 2]
sage: dP6.ch(1) == dP6.c(1)
True
Chern_class(deg=None)

Return Chern classes of the (tangent bundle of the) toric variety.

INPUT:

  • deg – integer (optional). The degree of the Chern class.

OUTPUT:

  • If the degree is specified, the deg-th Chern class.
  • If no degree is specified, the total Chern class.

REFERENCES:

EXAMPLES:

sage: X = toric_varieties.dP6()
sage: X.Chern_class()
[-6*w^2 + y + 2*v + 2*z + w + 1]
sage: X.c()
[-6*w^2 + y + 2*v + 2*z + w + 1]
sage: X.c(1)
[y + 2*v + 2*z + w]
sage: X.c(2)
[-6*w^2]
sage: X.integrate( X.c(2) )
6
sage: X.integrate( X.c(2) ) == X.Euler_number()
True
Chow_group(base_ring=Integer Ring)

Return the toric Chow group.

INPUT:

  • base_ring – either ZZ (default) or QQ. The coefficient ring of the Chow group.

OUTPUT:

A sage.schemes.toric.chow_group.ChowGroup_class

EXAMPLES:

sage: A = toric_varieties.P2().Chow_group(); A
Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches
sage: A.gens()
(( 1 | 0 | 0 ), ( 0 | 1 | 0 ), ( 0 | 0 | 1 ))
Demazure_roots()

Return the Demazure roots.

OUTPUT:

The roots as points of the \(M\)-lattice.

REFERENCES:

[Demazure]M. Demazure Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. Ecole Norm. Sup. 1970, 3, 507–588.
[Bazhov]Ivan Bazhov: On orbits of the automorphism group on a complete toric variety. Arxiv 1110.4275, doi:10.1007/s13366-011-0084-0.

EXAMPLE:

sage: P2 = toric_varieties.P2()
sage: P2.Demazure_roots()
(M(-1, 0), M(-1, 1), M(0, -1), M(0, 1), M(1, -1), M(1, 0))

Here are the remaining three examples listed in [Bazhov], Example 2.1 and 2.3:

sage: s = 3
sage: cones = [(0,1),(1,2),(2,3),(3,0)]
sage: Hs = ToricVariety(Fan(rays=[(1,0),(0,-1),(-1,s),(0,1)], cones=cones))
sage: Hs.Demazure_roots()
(M(-1, 0), M(1, 0), M(0, 1), M(1, 1), M(2, 1), M(3, 1))

sage: P11s = ToricVariety(Fan(rays=[(1,0),(0,-1),(-1,s)], cones=[(0,1),(1,2),(2,0)]))
sage: P11s.Demazure_roots()
(M(-1, 0), M(1, 0), M(0, 1), M(1, 1), M(2, 1), M(3, 1))
sage: P11s.Demazure_roots() == Hs.Demazure_roots()
True

sage: Bs = ToricVariety(Fan(rays=[(s,1),(s,-1),(-s,-1),(-s,1)], cones=cones))
sage: Bs.Demazure_roots()
()

TESTS:

sage: toric_varieties.A1().Demazure_roots()
Traceback (most recent call last):
...
NotImplementedError: Demazure_roots() is only implemented for complete toric varieties.
Euler_number()

Return the topological Euler number of the toric variety.

Sometimes, this is also called the Euler characteristic. chi() is a synonym for Euler_number().

REFERENCES:

EXAMPLES:

sage: P1xP1 = toric_varieties.P1xP1()
sage: P1xP1.Euler_number()
4
sage: P1xP1.chi()
4
K()

Returns the canonical divisor of the toric variety.

EXAMPLES:

Lets test that the del Pezzo surface \(dP_6\) has degree 6, as its name implies:

sage: dP6 = toric_varieties.dP6()
sage: HH = dP6.cohomology_ring()
sage: dP6.K()
-V(x) - V(u) - V(y) - V(v) - V(z) - V(w)
sage: dP6.integrate( HH(dP6.K())^2 )
6
Kaehler_cone()

Return the closure of the Kähler cone of self.

OUTPUT:

Note

This cone sits in the rational divisor class group of self and the choice of coordinates agrees with rational_class_group().

EXAMPLES:

sage: P1xP1 = toric_varieties.P1xP1()
sage: Kc = P1xP1.Kaehler_cone()
sage: Kc
2-d cone in 2-d lattice
sage: Kc.rays()
Divisor class [0, 1],
Divisor class [1, 0]
in Basis lattice of The toric rational divisor class group
of a 2-d CPR-Fano toric variety covered by 4 affine patches
sage: [ divisor_class.lift() for divisor_class in Kc.rays() ]
[V(x), V(s)]
sage: Kc.lattice()
Basis lattice of The toric rational divisor class group of a
2-d CPR-Fano toric variety covered by 4 affine patches
Mori_cone()

Returns the Mori cone of self.

OUTPUT:

Note

  • The Mori cone is dual to the Kähler cone.
  • We think of the Mori cone as living inside the row span of the Gale transform matrix (computed by self.fan().Gale_transform()).
  • The points in the Mori cone are the effective curves in the variety.
  • The i-th entry in each Mori vector is the intersection number of the curve corresponding to the generator of the i-th ray of the fan with the corresponding divisor class. The very last entry is associated to the orgin of the fan lattice.
  • The Mori vectors are also known as the gauged linear sigma model charge vectors.

EXAMPLES:

sage: P4_11169 = toric_varieties.P4_11169_resolved()
sage: P4_11169.Mori_cone()
2-d cone in 7-d lattice
sage: P4_11169.Mori_cone().rays()
(3, 2, 0, 0, 0,  1, -6),
(0, 0, 1, 1, 1, -3,  0)
in Ambient free module of rank 7
over the principal ideal domain Integer Ring
Spec(cone=None, names=None)

Return the spectrum associated to the dual cone.

Let \(\sigma \in N_\RR\) be a cone and \(\sigma^\vee \cap M\) the associated semigroup of lattice points in the dual cone. Then

\[S = \CC[\sigma^\vee \cap M]\]

is a \(\CC\)-algebra. It is spanned over \(\CC\) by the points of \(\sigma \cap N\), addition is formal linear combination of lattice points, and multiplication of lattice points is the semigroup law (that is, addition of lattice points). The \(\CC\)-algebra \(S\) then defines a scheme \(\mathop{Spec}(S)\).

For example, if \(\sigma=\{(x,y)|x\geq 0,y\geq 0\}\) is the first quadrant then \(S\) is the polynomial ring in two variables. The associated scheme is \(\mathop{Spec}(S) = \CC^2\).

The same construction works over any base field, this introduction only used \(\CC\) for simplicity.

INPUT:

  • cone – a Cone. Can be omitted for an affine toric variety, in which case the (unique) generating cone is used.
  • names – (optional). Names of variables for the semigroup ring, see normalize_names() for acceptable formats. If not given, indexed variable names will be created automatically.

Output:

The spectrum of the semigroup ring \(\CC[\sigma^\vee \cap M]\).

EXAMPLES:

sage: quadrant = Cone([(1,0),(0,1)])
sage: AffineToricVariety(quadrant).Spec()
Spectrum of Multivariate Polynomial Ring in z0, z1 over Rational Field

A more interesting example:

sage: A2Z2 = Cone([(0,1),(2,1)])
sage: AffineToricVariety(A2Z2).Spec(names='u,v,t')
Spectrum of Quotient of Multivariate Polynomial Ring
in u, v, t over Rational Field by the ideal (-u*v + t^2)
Stanley_Reisner_ideal()

Return the Stanley-Reisner ideal.

OUTPUT:

  • The Stanley-Reisner ideal in the polynomial ring over \(\QQ\) generated by the homogeneous coordinates.

EXAMPLES:

sage: fan = Fan([[0,1,3],[3,4],[2,0],[1,2,4]], [(-3, -2, 1), (0, 0, 1), (3, -2, 1), (-1, -1, 1), (1, -1, 1)])
sage: X = ToricVariety(fan, coordinate_names='A B C D E', base_field=GF(5))
sage: SR = X.Stanley_Reisner_ideal(); SR
Ideal (A*E, C*D, A*B*C, B*D*E) of Multivariate Polynomial Ring in A, B, C, D, E over Rational Field
Td(deg=None)

Return the Todd class (of the tangent bundle) of the toric variety.

INPUT:

  • deg – integer (optional). The desired degree part.

OUTPUT:

  • If the degree is specified, the degree-deg part of the Todd class.
  • If no degree is specified, the total Todd class.

REFERENCES:

EXAMPLES:

sage: dP6 = toric_varieties.dP6()
sage: dP6.Todd_class()
[-w^2 + 1/2*y + v + z + 1/2*w + 1]
sage: dP6.Td()
[-w^2 + 1/2*y + v + z + 1/2*w + 1]
sage: dP6.integrate( dP6.Td() )
1
Todd_class(deg=None)

Return the Todd class (of the tangent bundle) of the toric variety.

INPUT:

  • deg – integer (optional). The desired degree part.

OUTPUT:

  • If the degree is specified, the degree-deg part of the Todd class.
  • If no degree is specified, the total Todd class.

REFERENCES:

EXAMPLES:

sage: dP6 = toric_varieties.dP6()
sage: dP6.Todd_class()
[-w^2 + 1/2*y + v + z + 1/2*w + 1]
sage: dP6.Td()
[-w^2 + 1/2*y + v + z + 1/2*w + 1]
sage: dP6.integrate( dP6.Td() )
1
affine_algebraic_patch(cone=None, names=None)

Return the patch corresponding to cone as an affine algebraic subvariety.

INPUT:

  • cone – a Cone \(\sigma\) of the fan. It can be omitted for an affine toric variety, in which case the single generating cone is used.

OUTPUT:

A affine algebraic subscheme corresponding to the patch \(\mathop{Spec}(\sigma^\vee \cap M)\) associated to the cone \(\sigma\).

See also affine_patch(), which expresses the patches as subvarieties of affine toric varieties instead.

EXAMPLES:

sage: cone = Cone([(0,1),(2,1)])
sage: A2Z2 = AffineToricVariety(cone)
sage: A2Z2.affine_algebraic_patch()
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
  -z0*z1 + z2^2
sage: A2Z2.affine_algebraic_patch(Cone([(0,1)]), names='x, y, t')
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
  1
affine_patch(i)

Return the i-th affine patch of self.

INPUT:

  • i – integer, index of a generating cone of the fan of self.

OUTPUT:

  • affine toric variety corresponding to the i-th generating cone of the fan of self.

The result is cached, so the i-th patch is always the same object in memory.

See also affine_algebraic_patch(), which expresses the patches as subvarieties of affine space instead.

EXAMPLES:

sage: fan = FaceFan(lattice_polytope.cross_polytope(2))
sage: P1xP1 = ToricVariety(fan, "x s y t")
sage: patch0 = P1xP1.affine_patch(0)
sage: patch0
2-d affine toric variety
sage: patch0.embedding_morphism()
Scheme morphism:
  From: 2-d affine toric variety
  To:   2-d toric variety covered by 4 affine patches
  Defn: Defined on coordinates by sending [x : t] to
        [x : 1 : 1 : t]
sage: patch1 = P1xP1.affine_patch(1)
sage: patch1.embedding_morphism()
Scheme morphism:
  From: 2-d affine toric variety
  To:   2-d toric variety covered by 4 affine patches
  Defn: Defined on coordinates by sending [y : t] to
        [1 : 1 : y : t]
sage: patch1 is P1xP1.affine_patch(1)
True
c(deg=None)

Return Chern classes of the (tangent bundle of the) toric variety.

INPUT:

  • deg – integer (optional). The degree of the Chern class.

OUTPUT:

  • If the degree is specified, the deg-th Chern class.
  • If no degree is specified, the total Chern class.

REFERENCES:

EXAMPLES:

sage: X = toric_varieties.dP6()
sage: X.Chern_class()
[-6*w^2 + y + 2*v + 2*z + w + 1]
sage: X.c()
[-6*w^2 + y + 2*v + 2*z + w + 1]
sage: X.c(1)
[y + 2*v + 2*z + w]
sage: X.c(2)
[-6*w^2]
sage: X.integrate( X.c(2) )
6
sage: X.integrate( X.c(2) ) == X.Euler_number()
True
cartesian_product(other, coordinate_names=None, coordinate_indices=None)

Return the Cartesian product of self with other.

INPUT:

  • other – a toric variety;
  • coordinate_names – names of variables for the coordinate ring, see normalize_names() for acceptable formats. If not given, indexed variable names will be created automatically;
  • coordinate_indices – list of integers, indices for indexed variables. If not given, the index of each variable will coincide with the index of the corresponding ray of the fan.

OUTPUT:

– a toric variety.

EXAMPLES:

sage: P1 = ToricVariety(Fan([Cone([(1,)]), Cone([(-1,)])]))
sage: P1xP1 = P1.cartesian_product(P1); P1xP1
2-d toric variety covered by 4 affine patches
sage: P1xP1.fan().rays()
N+N(-1,  0),
N+N( 1,  0),
N+N( 0, -1),
N+N( 0,  1)
in 2-d lattice N+N
ch(deg=None)

Return the Chern character (of the tangent bundle) of the toric variety.

INPUT:

  • deg – integer (optional). The degree of the Chern character.

OUTPUT:

  • If the degree is specified, the degree-deg part of the Chern character.
  • If no degree is specified, the total Chern character.

REFERENCES:

EXAMPLES:

sage: dP6 = toric_varieties.dP6()
sage: dP6.Chern_character()
[3*w^2 + y + 2*v + 2*z + w + 2]
sage: dP6.ch()
[3*w^2 + y + 2*v + 2*z + w + 2]
sage: dP6.ch(1) == dP6.c(1)
True
change_ring(F)

Return a toric variety over F and otherwise the same as self.

INPUT:

  • F – field.

OUTPUT:

Note

There is no need to have any relation between F and the base field of self. If you do want to have such a relation, use base_extend() instead.

EXAMPLES:

sage: P1xA1 = toric_varieties.P1xA1()
sage: P1xA1.base_ring()
Rational Field
sage: P1xA1_RR = P1xA1.change_ring(RR)
sage: P1xA1_RR.base_ring()
Real Field with 53 bits of precision
sage: P1xA1_QQ = P1xA1_RR.change_ring(QQ)
sage: P1xA1_QQ.base_ring()
Rational Field
sage: P1xA1_RR.base_extend(QQ)
Traceback (most recent call last):
...
ValueError: no natural map from the base ring
(=Real Field with 53 bits of precision)
to R (=Rational Field)!
sage: R = PolynomialRing(QQ, 2, 'a')
sage: P1xA1.change_ring(R)
Traceback (most recent call last):
...
TypeError: need a field to construct a toric variety!
 Got Multivariate Polynomial Ring in a0, a1 over Rational Field
chi()

Return the topological Euler number of the toric variety.

Sometimes, this is also called the Euler characteristic. chi() is a synonym for Euler_number().

REFERENCES:

EXAMPLES:

sage: P1xP1 = toric_varieties.P1xP1()
sage: P1xP1.Euler_number()
4
sage: P1xP1.chi()
4
cohomology_basis(d=None)

Return a basis for the cohomology of the toric variety.

INPUT:

  • d (optional) – integer.

OUTPUT:

  • Without the optional argument, a list whose d-th entry is a basis for \(H^{2d}(X,\QQ)\)
  • If the argument is an integer d, returns basis for \(H^{2d}(X,\QQ)\)

EXAMPLES:

sage: X = toric_varieties.dP8()
sage: X.cohomology_basis()
(([1],), ([z], [y]), ([y*z],))
sage: X.cohomology_basis(1)
([z], [y])
sage: X.cohomology_basis(dimension(X))[0] == X.volume_class()
True
cohomology_ring()

Return the cohomology ring of the toric variety.

OUTPUT:

  • If the toric variety is is over \(\CC\) and has at most finite orbifold singularities: \(H^\bullet(X,\QQ)\) as a polynomial quotient ring.
  • Other cases are not handled yet.

Note

  • Toric varieties over any field of characteristic 0 are treated as if they were varieties over \(\CC\).
  • The integral cohomology of smooth toric varieties is torsion-free, so in this case there is no loss of information when going to rational coefficients.
  • self.cohomology_ring().gen(i) is the divisor class corresponding to the i-th ray of the fan.

EXAMPLES:

sage: X = toric_varieties.dP6()
sage: X.cohomology_ring()
Rational cohomology ring of a 2-d CPR-Fano toric variety covered by 6 affine patches
sage: X.cohomology_ring().defining_ideal()
Ideal (-u - y + z + w, x - y - v + w, x*y, x*v, x*z, u*v, u*z, u*w, y*z, y*w, v*w) of Multivariate Polynomial Ring in x, u, y, v, z, w over Rational Field
sage: X.cohomology_ring().defining_ideal().ring()
Multivariate Polynomial Ring in x, u, y, v, z, w over Rational Field
sage: X.variable_names()
('x', 'u', 'y', 'v', 'z', 'w')
sage: X.cohomology_ring().gens()
([y + v - w], [-y + z + w], [y], [v], [z], [w])

TESTS:

The cohomology ring is a circular reference that is potentially troublesome on unpickling, see trac ticket #15050 and trac ticket #15149

sage: variety = toric_varieties.P(1)
sage: a = [variety.cohomology_ring(), variety.cohomology_basis(), variety.volume_class()]
sage: b = [variety.Todd_class(), variety.Chern_class(), variety.Chern_character(), variety.Kaehler_cone(), variety.Mori_cone()]
sage: loads(dumps(variety)) == variety
True
coordinate_ring()

Return the coordinate ring of self.

For toric varieties this is the homogeneous coordinate ring (a.k.a. Cox’s ring and total ring).

OUTPUT:

  • polynomial ring.

EXAMPLES:

sage: P1xP1 = toric_varieties.P1xP1()
sage: P1xP1.coordinate_ring()
Multivariate Polynomial Ring in s, t, x, y
over Rational Field

TESTS:

sage: R = toric_varieties.A1().coordinate_ring();  R
Multivariate Polynomial Ring in z over Rational Field
sage: type(R)
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular'>
count_points()

Return the number of points of self.

This is an alias for point_set().cardinality(), see cardinality() for details.

EXAMPLES:

sage: o = lattice_polytope.cross_polytope(3)
sage: V = ToricVariety(FaceFan(o))
sage: V2 = V.change_ring(GF(2))
sage: V2.point_set().cardinality()
27
sage: V2.count_points()
27
dimension_singularities()

Return the dimension of the singular set.

OUTPUT:

Integer. The dimension of the singular set of the toric variety. Often the singular set is a reducible subvariety, and this method will return the dimension of the largest-dimensional component.

Returns -1 if the toric variety is smooth.

EXAMPLES:

sage: toric_varieties.P4_11169().dimension_singularities()
1
sage: toric_varieties.Conifold().dimension_singularities()
0
sage: toric_varieties.P2().dimension_singularities()
-1
divisor(arg, base_ring=None, check=True, reduce=True)

Return a divisor.

INPUT:

The arguments are the same as in sage.schemes.toric.divisor.ToricDivisor(), with the exception of defining a divisor with a single integer: this method considers it to be the index of a ray of the fan() of self.

OUTPUT:

EXAMPLES:

sage: dP6 = toric_varieties.dP6()
sage: dP6.coordinate_ring()
Multivariate Polynomial Ring in x, u, y, v, z, w
over Rational Field
sage: dP6.divisor(range(6))
V(u) + 2*V(y) + 3*V(v) + 4*V(z) + 5*V(w)
sage: dP6.inject_variables()
Defining x, u, y, v, z, w
sage: dP6.divisor(x*u^3)
V(x) + 3*V(u)

You can also construct divisors based on ray indices:

sage: dP6.divisor(0)
V(x)
sage: for i in range(0,dP6.fan().nrays()):
...       print dP6.divisor(i),
...       print ': generated by ray',
...       dP6.fan().ray(i)
V(x) : generated by ray N(0, 1)
V(u) : generated by ray N(-1, 0)
V(y) : generated by ray N(-1, -1)
V(v) : generated by ray N(0, -1)
V(z) : generated by ray N(1, 0)
V(w) : generated by ray N(1, 1)

TESTS:

We check that the issue trac ticket #12812 is resolved:

sage: sum(dP6.divisor(i) for i in range(3))
V(x) + V(u) + V(y)
divisor_group(base_ring=Integer Ring)

Return the group of Weil divisors.

INPUT:

  • base_ring – the coefficient ring, usually ZZ (default) or QQ.

OUTPUT:

The (free abelian) group of Cartier divisors, that is, formal linear combinations of polynomial equations over the coefficient ring base_ring.

These need not be toric (=defined by monomials), but allow general polynomials. The output will be an instance of sage.schemes.generic.divisor_group.DivisorGroup_generic.

Warning

You almost certainly want the group of toric divisors, see toric_divisor_group(). The toric divisor group is generated by the rays of the fan. The general divisor group has no toric functionality implemented.

EXAMPLES:

sage: dP6 = toric_varieties.dP6()
sage: Div = dP6.divisor_group(); Div
Group of ZZ-Divisors on 2-d CPR-Fano toric variety
covered by 6 affine patches
sage: Div(x)
V(x)
embedding_morphism()

Return the default embedding morphism of self.

Such a morphism is always defined for an affine patch of a toric variety (which is also a toric varieties itself).

OUTPUT:

  • scheme morphism if the default embedding morphism was defined for self, otherwise a ValueError exception is raised.

EXAMPLES:

sage: fan = FaceFan(lattice_polytope.cross_polytope(2))
sage: P1xP1 = ToricVariety(fan, "x s y t")
sage: P1xP1.embedding_morphism()
Traceback (most recent call last):
...
ValueError: no default embedding was
defined for this toric variety!
sage: patch = P1xP1.affine_patch(0)
sage: patch
2-d affine toric variety
sage: patch.embedding_morphism()
Scheme morphism:
  From: 2-d affine toric variety
  To:   2-d toric variety covered by 4 affine patches
  Defn: Defined on coordinates by sending [x : t] to
        [x : 1 : 1 : t]
fan(dim=None, codim=None)

Return the underlying fan of self or its cones.

INPUT:

  • dim – dimension of the requested cones;
  • codim – codimension of the requested cones.

OUTPUT:

EXAMPLES:

sage: fan = FaceFan(lattice_polytope.cross_polytope(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1.fan()
Rational polyhedral fan in 2-d lattice M
sage: P1xP1.fan() is fan
True
sage: P1xP1.fan(1)[0]
1-d cone of Rational polyhedral fan in 2-d lattice M
inject_coefficients(scope=None, verbose=True)

Inject generators of the base field of self into scope.

This function is useful if the base field is the field of rational functions.

INPUT:

  • scope – namespace (default: global, not just the scope from which this function was called);
  • verbose – if True (default), names of injected generators will be printed.

OUTPUT:

  • none.

EXAMPLES:

sage: fan = FaceFan(lattice_polytope.cross_polytope(2))
sage: F = QQ["a, b"].fraction_field()
sage: P1xP1 = ToricVariety(fan, base_field=F)
sage: P1xP1.inject_coefficients()
Defining a, b

We check that we can use names a and b, Trac #10498 is fixed:

sage: a + b
a + b
sage: a + b in P1xP1.coordinate_ring()
True
integrate(cohomology_class)

Integrate a cohomology class over the toric variety.

INPUT:

  • cohomology_class – A cohomology class given as a polynomial in self.cohomology_ring()

OUTPUT:

The integral of the cohomology class over the variety. The volume normalization is given by volume_class(), that is, self.integrate(self.volume_class()) is always one (if the volume class exists).

EXAMPLES:

sage: dP6 = toric_varieties.dP6()
sage: HH = dP6.cohomology_ring()
sage: D = [ HH(c) for c in dP6.fan(dim=1) ]
sage: matrix([ [ D[i]*D[j] for i in range(0,6) ] for j in range(0,6) ])
[ [w^2] [-w^2]    [0]    [0]    [0] [-w^2]]
[[-w^2]  [w^2] [-w^2]    [0]    [0]    [0]]
[   [0] [-w^2]  [w^2] [-w^2]    [0]    [0]]
[   [0]    [0] [-w^2]  [w^2] [-w^2]    [0]]
[   [0]    [0]    [0] [-w^2]  [w^2] [-w^2]]
[[-w^2]    [0]    [0]    [0] [-w^2]  [w^2]]
sage: matrix([ [ dP6.integrate(D[i]*D[j]) for i in range(0,6) ] for j in range(0,6) ])
[-1  1  0  0  0  1]
[ 1 -1  1  0  0  0]
[ 0  1 -1  1  0  0]
[ 0  0  1 -1  1  0]
[ 0  0  0  1 -1  1]
[ 1  0  0  0  1 -1]

If the toric variety is an orbifold, the intersection numbers are usually fractional:

sage: P2_123 = toric_varieties.P2_123()
sage: HH = P2_123.cohomology_ring()
sage: D = [ HH(c) for c in P2_123.fan(dim=1) ]
sage: matrix([ [ P2_123.integrate(D[i]*D[j]) for i in range(0,3) ] for j in range(0,3) ])
[2/3   1 1/3]
[  1 3/2 1/2]
[1/3 1/2 1/6]
sage: A = P2_123.Chow_group(QQ)
sage: matrix([ [ A(P2_123.divisor(i))
...              .intersection_with_divisor(P2_123.divisor(j))
...              .count_points() for i in range(0,3) ] for j in range(0,3) ])
[2/3   1 1/3]
[  1 3/2 1/2]
[1/3 1/2 1/6]
is_affine()

Check if self is an affine toric variety.

An affine toric variety is a toric variety whose fan is the face lattice of a single cone. See also AffineToricVariety().

OUTPUT:

Boolean.

EXAMPLES:

sage: toric_varieties.A2().is_affine()
True
sage: toric_varieties.P1xA1().is_affine()
False
is_complete()

Check if self is complete.

OUTPUT:

  • True if self is complete and False otherwise.

EXAMPLES:

sage: P1xP1 = toric_varieties.P1xP1()
sage: P1xP1.is_complete()
True
sage: P1xP1.affine_patch(0).is_complete()
False
is_homogeneous(polynomial)

Check if polynomial is homogeneous.

The coordinate ring of a toric variety is multigraded by relations between generating rays of the underlying fan.

INPUT:

  • polynomial – polynomial in the coordinate ring of self or its quotient.

OUTPUT:

  • True if polynomial is homogeneous and False otherwise.

EXAMPLES:

We will use the product of two projective lines with coordinates \((x, y)\) for one and \((s, t)\) for the other:

sage: P1xP1.<x,y,s,t> = toric_varieties.P1xP1()
sage: P1xP1.is_homogeneous(x - y)
True
sage: P1xP1.is_homogeneous(x*s + y*t)
True
sage: P1xP1.is_homogeneous(x - t)
False
sage: P1xP1.is_homogeneous(1)
True

Note that by homogeneous, we mean well-defined with respect to the homogeneous rescalings of self. So a polynomial that you would usually not call homogeneous can be homogeneous if there are no homogeneous rescalings, for example:

sage: A1.<z> = toric_varieties.A1()
sage: A1.is_homogeneous(z^3+z^7)
True

Finally, the degree group is really the Chow group \(A_{d-1}(X)\) and can contain torsion. For example, take \(\CC^2/\ZZ_2\). Here, the Chow group is \(A_{d-1}(\CC^2/\ZZ_2) = \ZZ_2\) and distinguishes even-degree homogeneous polynomials from odd-degree homogeneous polynomials:

sage: A2_Z2.<x,y> = toric_varieties.A2_Z2()
sage: A2_Z2.is_homogeneous(x+y+x^3+y^5+x^3*y^4)
True
sage: A2_Z2.is_homogeneous(x^2+x*y+y^4+(x*y)^5+x^4*y^4)
True
sage: A2_Z2.is_homogeneous(x+y^2)
False
is_isomorphic(another)

Check if self is isomorphic to another.

INPUT:

OUTPUT:

  • True if self and another are isomorphic, False otherwise.

EXAMPLES:

sage: TV1 = toric_varieties.P1xA1()
sage: TV2 = toric_varieties.P1xP1()

Only the most trivial case is implemented so far:

sage: TV1.is_isomorphic(TV1)
True
sage: TV1.is_isomorphic(TV2)
Traceback (most recent call last):
...
NotImplementedError:
isomorphism check is not yet implemented!
is_orbifold()

Check if self has only quotient singularities.

A toric variety with at most orbifold singularities (in this sense) is often called a simplicial toric variety. In this package, we generally try to avoid this term since it mixes up differential geometry and cone terminology.

OUTPUT:

  • True if self has at most quotient singularities by finite groups, False otherwise.

EXAMPLES:

sage: fan1 = FaceFan(lattice_polytope.cross_polytope(2))
sage: P1xP1 = ToricVariety(fan1)
sage: P1xP1.is_orbifold()
True
sage: fan2 = NormalFan(lattice_polytope.cross_polytope(3))
sage: TV = ToricVariety(fan2)
sage: TV.is_orbifold()
False
is_smooth()

Check if self is smooth.

OUTPUT:

  • True if self is smooth and False otherwise.

EXAMPLES:

sage: fan1 = FaceFan(lattice_polytope.cross_polytope(2))
sage: P1xP1 = ToricVariety(fan1)
sage: P1xP1.is_smooth()
True
sage: fan2 = NormalFan(lattice_polytope.cross_polytope(2))
sage: TV = ToricVariety(fan2)
sage: TV.is_smooth()
False
linear_equivalence_ideal()

Return the ideal generated by linear relations

OUTPUT:

  • The ideal generated by the linear relations of the rays in the polynomial ring over \(\QQ\) generated by the homogeneous coordinates.

EXAMPLES:

sage: fan = Fan([[0,1,3],[3,4],[2,0],[1,2,4]], [(-3, -2, 1), (0, 0, 1), (3, -2, 1), (-1, -1, 1), (1, -1, 1)])
sage: X = ToricVariety(fan, coordinate_names='A B C D E', base_field=GF(5))
sage: lin = X.linear_equivalence_ideal(); lin
Ideal (-3*A + 3*C - D + E, -2*A - 2*C - D - E, A + B + C + D + E) of Multivariate Polynomial Ring in A, B, C, D, E over Rational Field
orbit_closure(cone)

Return the orbit closure of cone.

The cones \(\sigma\) of a fan \(\Sigma\) are in one-to-one correspondence with the torus orbits \(O(\sigma)\) of the corresponding toric variety \(X_\Sigma\). Each orbit is isomorphic to a lower dimensional torus (of dimension equal to the codimension of \(\sigma\)). Just like the toric variety \(X_\Sigma\) itself, these orbits are (partially) compactified by lower-dimensional orbits. In particular, one can define the closure \(V(\sigma)\) of the torus orbit \(O(\sigma)\) in the ambient toric variety \(X_\Sigma\), which is again a toric variety.

See Proposition 3.2.7 of [CLS] for more details.

INPUT:

  • cone – a cone of the fan.

OUTPUT:

EXAMPLES:

sage: P1xP1 = toric_varieties.P1xP1()
sage: H = P1xP1.fan(1)[0]
sage: V = P1xP1.orbit_closure(H);  V
1-d toric variety covered by 2 affine patches
sage: V.embedding_morphism()
Scheme morphism:
  From: 1-d toric variety covered by 2 affine patches
  To:   2-d CPR-Fano toric variety covered by 4 affine patches
  Defn: Defined by embedding the torus closure associated to the 1-d
        cone of Rational polyhedral fan in 2-d lattice N.
sage: V.embedding_morphism().as_polynomial_map()
Scheme morphism:
  From: 1-d toric variety covered by 2 affine patches
  To:   2-d CPR-Fano toric variety covered by 4 affine patches
  Defn: Defined on coordinates by sending [z0 : z1] to
        [0 : 1 : z1 : z0]

TESTS:

sage: A2 = toric_varieties.A2()
sage: A2.orbit_closure(A2.fan(2)[0])
0-d affine toric variety
plot(**options)

Plot self, i.e. the corresponding fan.

INPUT:

OUTPUT:

  • a plot.

Note

The difference between X.plot() and X.fan().plot() is that in the first case default ray labels correspond to variables of X.

EXAMPLES:

sage: X = toric_varieties.Cube_deformation(4)
sage: X.plot()
rational_class_group()

Return the rational divisor class group of self.

Let \(X\) be a toric variety.

The Weil divisor class group \(\mathop{Cl}(X)\) is a finitely generated abelian group and can contain torsion. Its rank equals the number of rays in the fan of \(X\) minus the dimension of \(X\).

The rational divisor class group is \(\mathop{Cl}(X) \otimes_\ZZ \QQ\) and never includes torsion. If \(X\) is smooth, this equals the Picard group of \(X\), whose elements are the isomorphism classes of line bundles on \(X\). The group law (which we write as addition) is the tensor product of the line bundles. The Picard group of a toric variety is always torsion-free.

OUTPUT:

Note

  • Coordinates correspond to the rows of self.fan().gale_transform().
  • Kaehler_cone() yields a cone in this group.

EXAMPLES:

sage: P1xA1 = toric_varieties.P1xA1()
sage: P1xA1.rational_class_group()
The toric rational divisor class group
of a 2-d toric variety covered by 2 affine patches
resolve(**kwds)

Construct a toric variety whose fan subdivides the fan of self.

The name of this function reflects the fact that usually such subdivisions are done for resolving singularities of the original variety.

INPUT:

This function accepts only keyword arguments, none of which are mandatory.

  • coordinate_names – names for coordinates of the new variety. If not given, will be constructed from the coordinate names of self and necessary indexed ones. See normalize_names() for the description of acceptable formats;
  • coordinate_indices – coordinate indices which should be used for indexed variables of the new variety;
  • all other arguments will be passed to subdivide() method of the underlying rational polyhedral fan, see its documentation for the available options.

OUTPUT:

EXAMPLES:

First we will “manually” resolve a simple orbifold singularity:

sage: cone = Cone([(1,1), (-1,1)])
sage: fan = Fan([cone])
sage: TV = ToricVariety(fan)
sage: TV.is_smooth()
False
sage: TV_res = TV.resolve(new_rays=[(0,1)])
sage: TV_res.is_smooth()
True
sage: TV_res.fan().rays()
N( 1, 1),
N(-1, 1),
N( 0, 1)
in 2-d lattice N
sage: [cone.ambient_ray_indices() for cone in TV_res.fan()]
[(0, 2), (1, 2)]

Now let’s “automatically” partially resolve a more complicated fan:

sage: fan = NormalFan(lattice_polytope.cross_polytope(3))
sage: TV = ToricVariety(fan)
sage: TV.is_smooth()
False
sage: TV.is_orbifold()
False
sage: TV.fan().nrays()
8
sage: TV.fan().ngenerating_cones()
6
sage: TV_res = TV.resolve(make_simplicial=True)
sage: TV_res.is_smooth()
False
sage: TV_res.is_orbifold()
True
sage: TV_res.fan().nrays()
8
sage: TV_res.fan().ngenerating_cones()
12
sage: TV.gens()
(z0, z1, z2, z3, z4, z5, z6, z7)
sage: TV_res.gens()
(z0, z1, z2, z3, z4, z5, z6, z7)
sage: TV_res = TV.resolve(coordinate_names="x+",
...                       make_simplicial=True)
sage: TV_res.gens()
(x0, x1, x2, x3, x4, x5, x6, x7)
resolve_to_orbifold(**kwds)

Construct an orbifold whose fan subdivides the fan of self.

It is a synonym for resolve() with make_simplicial=True option.

INPUT:

  • this function accepts only keyword arguments. See resolve() for documentation.

OUTPUT:

EXAMPLES:

sage: fan = NormalFan(lattice_polytope.cross_polytope(3))
sage: TV = ToricVariety(fan)
sage: TV.is_orbifold()
False
sage: TV.fan().nrays()
8
sage: TV.fan().ngenerating_cones()
6
sage: TV_res = TV.resolve_to_orbifold()
sage: TV_res.is_orbifold()
True
sage: TV_res.fan().nrays()
8
sage: TV_res.fan().ngenerating_cones()
12
subscheme(polynomials)

Return the subscheme of self defined by polynomials.

INPUT:

  • polynomials – list of polynomials in the coordinate ring of self.

OUTPUT:

EXAMPLES:

We will construct a subscheme of the product of two projective lines with coordinates \((x, y)\) for one and \((s, t)\) for the other:

sage: P1xP1.<x,y,s,t> = toric_varieties.P1xP1()
sage: X = P1xP1.subscheme([x*s + y*t, x^3+y^3])
sage: X
Closed subscheme of 2-d CPR-Fano toric variety
covered by 4 affine patches defined by:
  x*s + y*t,
  x^3 + y^3
sage: X.defining_polynomials()
(x*s + y*t, x^3 + y^3)
sage: X.defining_ideal()
Ideal (x*s + y*t, x^3 + y^3)
of Multivariate Polynomial Ring in x, y, s, t
over Rational Field
sage: X.base_ring()
Rational Field
sage: X.base_scheme()
Spectrum of Rational Field
sage: X.structure_morphism()
Scheme morphism:
  From: Closed subscheme of 2-d CPR-Fano toric variety
        covered by 4 affine patches defined by:
  x*s + y*t,
  x^3 + y^3
  To:   Spectrum of Rational Field
  Defn: Structure map
toric_divisor_group(base_ring=Integer Ring)

Return the group of toric (T-Weil) divisors.

INPUT:

  • base_ring – the coefficient ring, usually ZZ (default) or QQ.

OUTPUT:

The free Abelian agroup of toric Weil divisors, that is, formal base_ring-linear combinations of codimension-one toric subvarieties. The output will be an instance of sage.schemes.toric.divisor.ToricDivisorGroup.

The \(i\)-th generator of the divisor group is the divisor where the \(i\)-th homogeneous coordinate vanishes, \(\{z_i=0\}\).

EXAMPLES:

sage: dP6 = toric_varieties.dP6()
sage: TDiv = dP6.toric_divisor_group(); TDiv
Group of toric ZZ-Weil divisors on 2-d CPR-Fano toric variety
covered by 6 affine patches
sage: TDiv == dP6.toric_divisor_group()
True
sage: TDiv.gens()
(V(x), V(u), V(y), V(v), V(z), V(w))
sage: dP6.coordinate_ring()
Multivariate Polynomial Ring in x, u, y, v, z, w over Rational Field
volume_class()

Return the cohomology class of the volume form on the toric variety.

Note that we are using cohomology with compact supports. If the variety is non-compact this is dual to homology without any support condition. In particular, for non-compact varieties the volume form \(\mathrm{dVol}=\wedge_i(dx_i \wedge dy_i)\) does not define a (non-zero) cohomology class.

OUTPUT:

A CohomologyClass. If it exists, it is the class of the (properly normalized) volume form, that is, it is the Poincare dual of a single point. If it does not exist, a ValueError is raised.

EXAMPLES:

sage: P2 = toric_varieties.P2()
sage: P2.volume_class()
[z^2]

sage: A2_Z2 = toric_varieties.A2_Z2()
sage: A2_Z2.volume_class()
Traceback (most recent call last):
...
ValueError: Volume class does not exist.

If none of the maximal cones is smooth things get more tricky. In this case no torus-fixed point is smooth. If we want to count an ordinary point as \(1\), then a \(G\)-orbifold point needs to count as \(\frac{1}{|G|}\). For example, take \(\mathbb{P}^1\times\mathbb{P}^1\) with inhomogeneous coordinates \((t,y)\). Take the quotient by the action \((t,y)\mapsto (-t,-y)\). The \(\ZZ_2\)-invariant Weil divisors \(\{t=0\}\) and \(\{y=0\}\) intersect in a \(\ZZ_2\)-fixed point, so they ought to have intersection number \(\frac{1}{2}\). This means that the cohomology class \([t] \cap [y]\) should be \(\frac{1}{2}\) times the volume class. Note that this is different from the volume normalization chosen in [Schubert]:

sage: P1xP1_Z2 = toric_varieties.P1xP1_Z2()
sage: Dt = P1xP1_Z2.divisor(1);  Dt
V(t)
sage: Dy = P1xP1_Z2.divisor(3);  Dy
V(y)
sage: P1xP1_Z2.volume_class()
[2*t*y]

sage: HH = P1xP1_Z2.cohomology_ring()
sage: HH(Dt) * HH(Dy) == 1/2 * P1xP1_Z2.volume_class()
True

The fractional coefficients are also necessary to match the normalization in the rational Chow group for simplicial toric varieties:

sage: A = P1xP1_Z2.Chow_group(QQ)
sage: A(Dt).intersection_with_divisor(Dy).count_points()
1/2

REFERENCES:

[Schubert]Sheldon Katz and Stein Arild Stromme, A Maple package for intersection theory and enumerative geometry.
sage.schemes.toric.variety.certify_names(names)

Make sure that names are valid in Python.

INPUT:

  • names – list of strings.

OUTPUT:

  • none, but a ValueError exception is raised if names are invalid.

Each name must satisfy the following requirements:

  • Be non-empty.
  • Contain only (Latin) letters, digits, and underscores (“_”).
  • Start with a letter.

In addition, all names must be distinct.

EXAMPLES:

sage: from sage.schemes.toric.variety import certify_names
sage: certify_names([])
sage: certify_names(["a", "x0", "x_45"])
sage: certify_names(["", "x0", "x_45"])
Traceback (most recent call last):
...
ValueError: name must be nonempty!
sage: certify_names(["a", "0", "x_45"])
Traceback (most recent call last):
...
ValueError: name must start with a letter! Got 0
sage: certify_names(["a", "x0", "@_45"])
Traceback (most recent call last):
...
ValueError: name must be alphanumeric! Got @_45
sage: certify_names(["a", "x0", "x0"])
Traceback (most recent call last):
...
ValueError: names must be distinct! Got: ['a', 'x0', 'x0']
sage.schemes.toric.variety.is_CohomologyClass(x)

Check whether x is a cohomology class of a toric variety.

INPUT:

  • x – anything.

OUTPUT:

True or False depending on whether x is an instance of CohomologyClass

EXAMPLES:

sage: P2 = toric_varieties.P2()
sage: HH = P2.cohomology_ring()
sage: from sage.schemes.toric.variety import is_CohomologyClass
sage: is_CohomologyClass( HH.one() )
True
sage: is_CohomologyClass( HH(P2.fan(1)[0]) )
True
sage: is_CohomologyClass('z')
False
sage.schemes.toric.variety.is_ToricVariety(x)

Check if x is a toric variety.

INPUT:

  • x – anything.

OUTPUT:

Note

While projective spaces are toric varieties mathematically, they are not toric varieties in Sage due to efficiency considerations, so this function will return False.

EXAMPLES:

sage: from sage.schemes.toric.variety import is_ToricVariety
sage: is_ToricVariety(1)
False
sage: fan = FaceFan(lattice_polytope.cross_polytope(2))
sage: P = ToricVariety(fan)
sage: P
2-d toric variety covered by 4 affine patches
sage: is_ToricVariety(P)
True
sage: is_ToricVariety(ProjectiveSpace(2))
False
sage.schemes.toric.variety.normalize_names(names=None, ngens=None, prefix=None, indices=None, return_prefix=False)

Return a list of names in the standard form.

INPUT:

All input parameters are optional.

  • names – names given either as a single string (with individual names separated by commas or spaces) or a list of strings with each string specifying a name. If the last name ends with the plus sign, “+”, this name will be used as prefix (even if prefix was given explicitly);
  • ngens – number of names to be returned;
  • prefix – prefix for the indexed names given as a string;
  • indices – list of integers (default: range(ngens)) used as indices for names with prefix. If given, must be of length ngens;
  • return_prefix – if True, the last element of the returned list will contain the prefix determined from names or given as the parameter prefix. This is useful if you may need more names in the future.

OUTPUT:

  • list of names given as strings.

These names are constructed in the following way:

  1. If necessary, split names into separate names.
  2. If the last name ends with “+”, put it into prefix.
  3. If ngens was given, add to the names obtained so far as many indexed names as necessary to get this number. If the k-th name of the total list of names is indexed, it is prefix + str(indices[k]). If there were already more names than ngens, discard “extra” ones.
  4. Check if constructed names are valid. See certify_names() for details.
  5. If the option return_prefix=True was given, add prefix to the end of the list.

EXAMPLES:

As promised, all parameters are optional:

sage: from sage.schemes.toric.variety import normalize_names
sage: normalize_names()
[]

One of the most common uses is probably this one:

sage: normalize_names("x+", 4)
['x0', 'x1', 'x2', 'x3']

Now suppose that you want to enumerate your variables starting with one instead of zero:

sage: normalize_names("x+", 4, indices=range(1,5))
['x1', 'x2', 'x3', 'x4']

You may actually have an arbitrary enumeration scheme:

sage: normalize_names("x+", 4, indices=[1, 10, 100, 1000])
['x1', 'x10', 'x100', 'x1000']

Now let’s add some “explicit” names:

sage: normalize_names("x y z t+", 4)
['x', 'y', 'z', 't3']

Note that the “automatic” name is t3 instead of t0. This may seem weird, but the reason for this behaviour is that the fourth name in this list will be the same no matter how many explicit names were given:

sage: normalize_names("x y t+", 4)
['x', 'y', 't2', 't3']

This is especially useful if you get names from a user but want to specify all default names:

sage: normalize_names("x, y", 4, prefix="t")
['x', 'y', 't2', 't3']

In this format, the user can easily override your choice for automatic names:

sage: normalize_names("x y s+", 4, prefix="t")
['x', 'y', 's2', 's3']

Let’s now use all parameters at once:

sage: normalize_names("x, y, s+", 4, prefix="t",
...       indices=range(1,5), return_prefix=True)
['x', 'y', 's3', 's4', 's']

Note that you still need to give indices for all names, even if some of the first ones will be “wasted” because of the explicit names. The reason is the same as before - this ensures consistency of automatically generated names, no matter how many explicit names were given.

The prefix is discarded if ngens was not given:

sage: normalize_names("alpha, beta, gamma, zeta+")
['alpha', 'beta', 'gamma']

Finally, let’s take a look at some possible mistakes:

sage: normalize_names("123")
Traceback (most recent call last):
...
ValueError: name must start with a letter! Got 123

A more subtle one:

sage: normalize_names("x1", 4, prefix="x")
Traceback (most recent call last):
...
ValueError: names must be distinct! Got: ['x1', 'x1', 'x2', 'x3']

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