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 [CLS2011].
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.
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()
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: list(map(HH, P4_11133.fan(1)))
[[3*z4], [3*z4], [z4], [z4], [z4]]
sage: list(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: list(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: list(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 toToricVariety()
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 aValueError
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. SeeToricVariety_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 byX
, then this generator is associated to the rayX.fan().ray(i)
, which spans the one-coneX.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 fanX.fan().rays()
, which is also the same as the ordering of the one-cones inX.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:
fan
–rational polyhedral fan
;coordinate_names
– names of variables for the coordinate ring, seenormalize_names()
for acceptable formats. If not given, indexed variable names will be created automatically;names
– an alias ofcoordinate_names
for internal use. You may specify eithernames
orcoordinate_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 forbase_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.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()
andAffineToricVariety()
to construct toric varieties.INPUT:
fan
–rational polyhedral fan
;coordinate_names
– names of variables, seenormalize_names()
for acceptable formats. IfNone
, indexed variable names will be created automatically;coordinate_indices
– list of integers, indices for indexed variables. IfNone
, 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:
- 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
- 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
– eitherZZ
(default) orQQ
. 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() (( 0 | 0 | 1 ), ( 0 | 1 | 0 ), ( 1 | 0 | 0 ))
- Demazure_roots()¶
Return the Demazure roots.
OUTPUT:
The roots as points of the \(M\)-lattice.
REFERENCES:
EXAMPLES:
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 [Baz2011], 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() ()
- Euler_number()¶
Return the topological Euler number of the toric variety.
Sometimes, this is also called the Euler characteristic.
chi()
is a synonym forEuler_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:
cone
.
Note
This cone sits in the rational divisor class group of
self
and the choice of coordinates agrees withrational_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(y), V(t)] 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:
cone
.
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 thei
-th ray of the fan with the corresponding divisor class. The very last entry is associated to the origin 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
– aCone
. 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, seenormalize_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
– aCone
\(\sigma\) 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 ofself
.INPUT:
i
– integer, index of a generating cone of the fan ofself
.
OUTPUT:
affine
toric variety
corresponding to thei
-th generating cone of the fan ofself
.
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 [y : t] to [1 : 1 : y : 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 [s : y] to [1 : s : y : 1] 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
withother
.INPUT:
other
– atoric variety
;coordinate_names
– names of variables for the coordinate ring, seenormalize_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 asself
.INPUT:
F
– field.
OUTPUT:
toric variety
overF
.
Note
There is no need to have any relation between
F
and the base field ofself
. If you do want to have such a relation, usebase_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 forEuler_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 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 thei
-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])
- 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
- count_points()¶
Return the number of points of
self
.This is an alias for
point_set().cardinality()
, seecardinality()
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 thefan()
ofself
.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(list(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(dP6.fan().nrays()): ....: print('{} : generated by ray {}'.format(dP6.divisor(i), ....: 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)
- divisor_group(base_ring=Integer Ring)¶
Return the group of Weil divisors.
INPUT:
base_ring
– the coefficient ring, usuallyZZ
(default) orQQ
.
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 forself
, otherwise aValueError
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 [y : t] to [1 : 1 : y : 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:
rational polyhedral fan
if no parameters were given,tuple
ofcones
otherwise.
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
intoscope
.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
– ifTrue
(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
andb
, trac ticket #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 inself.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
ifself
is complete andFalse
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 ofself
or its quotient.
OUTPUT:
True
ifpolynomial
is homogeneous andFalse
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 toanother
.INPUT:
another
-toric variety
.
OUTPUT:
True
ifself
andanother
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
ifself
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
ifself
is smooth andFalse
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 [CLS2011] for more details.
INPUT:
cone
– acone
of the fan.
OUTPUT:
a torus orbit closure associated to
cone
as atoric variety
.
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]
- plot(**options)¶
Plot
self
, i.e. the corresponding fan.INPUT:
any options for toric plots (see
toric_plotter.options
), none are mandatory.
OUTPUT:
a plot.
Note
The difference between
X.plot()
andX.fan().plot()
is that in the first case default ray labels correspond to variables ofX
.EXAMPLES:
sage: X = toric_varieties.Cube_deformation(4) sage: X.plot() Graphics3d Object
- 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 ofself
and necessary indexed ones. Seenormalize_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 underlyingrational 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()
withmake_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
- sheaves¶
Return the factory object for sheaves on the toric variety.
See
sage.schemes.toric.sheaf.constructor.SheafLibrary
for details.EXAMPLES:
sage: dP6 = toric_varieties.dP6() sage: dP6.sheaves Sheaf constructor on 2-d CPR-Fano toric variety covered by 6 affine patches sage: dP6.sheaves.trivial_bundle() Rank 1 bundle on 2-d CPR-Fano toric variety covered by 6 affine patches.
- subscheme(polynomials)¶
Return the subscheme of
self
defined bypolynomials
.INPUT:
polynomials
– list of polynomials in the coordinate ring ofself
.
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, usuallyZZ
(default) orQQ
.
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 ofsage.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 Poincaré dual of a single point. If it does not exist, aValueError
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 [KS]:
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
- sage.schemes.toric.variety.is_CohomologyClass(x)¶
Check whether
x
is a cohomology class of a toric variety.INPUT:
x
– anything.
OUTPUT:
True
orFalse
depending on whetherx
is an instance ofCohomologyClass
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:
True
ifx
is atoric variety
andFalse
otherwise.
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 asprefix
(even ifprefix
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 withprefix
. If given, must be of lengthngens
;return_prefix
– ifTrue
, the last element of the returned list will contain the prefix determined fromnames
or given as the parameterprefix
. 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:
If necessary, split
names
into separate names.If the last name ends with “+”, put it into
prefix
.If
ngens
was given, add to the names obtained so far as many indexed names as necessary to get this number. If thek
-th name of the total list of names is indexed, it isprefix + str(indices[k])
. If there were already more names thanngens
, discard “extra” ones.Check if constructed names are valid. See
certify_names()
for details.If the option
return_prefix=True
was given, addprefix
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=list(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 oft0
. 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=list(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: variable name '123' does not start with a letter
A more subtle one:
sage: normalize_names("x1", 4, prefix="x") Traceback (most recent call last): ... ValueError: variable name 'x1' appears more than once