Toric Blowdown Morphisms (Experimental)

It is a common goal in algebraic geometry to resolve singularities. Certainly, (sub)varieties of toric varieties are no exception and we provide a growing set of functionality for such tasks.

In general, resolutions need not be toric. Indeed, some of the functionality below requires fully-fledge schemes machinery, which – as of this writing (October 2023) – is still in Oscar's experimental state. For this reason, the methods below should be considered experimental.

Constructors

Blowups of toric varieties are obtained from star subdivisions of polyhedral fans. In the most general form, a star subdivision is defined by a new primitive element in the fan. Below, we refer to this new primitive element as new_ray. In addition to this new_ray, our design of toric blowdown morphisms requires an underlying toric morphism. With an eye towards covered schemes as possible return value, any toric blowdown morphism must also know (to compute) its blowup center in the form of an ideal sheaf. The following constructor allows to set this ideal sheaf center upon construction:

toric_blowdown_morphism(bl::ToricMorphism, new_ray::AbstractVector{<:IntegerUnion}, center::AbsIdealSheaf)

The "working-horse" constructor however is the following:

toric_blowdown_morphism(Y::NormalToricVariety, new_ray::AbstractVector{<:IntegerUnion}, coordinate_name::String)

This constructor will, among others, construct the underlying toric morphism. In addition, we can then specify a name for the coordinate in the Cox ring that is assigned to new_ray.

Blowdown morphisms from blowing up toric varieties

The following methods blow up toric varieties. The center of the blowup can be provided in different formats. We discuss the methods in ascending generality.

For our most specialized blow-up method, we focus on the n-th cone in the fan of the variety v in question. This cone need not be maximal! The ensuing star subdivision will subdivide this cone about its "diagonal" (the sum of all its rays). The result of this will always be a toric variety:

blow_up — Method

blow_up(v::NormalToricVariety, n::Int; coordinate_name::String = "e")

Blow up the toric variety by subdividing the n-th cone in the list of all cones of the fan of v. This cone need not be maximal. This function returns the corresponding blowdown morphism.

By default, we pick "e" as the name of the homogeneous coordinate for the exceptional divisor. As third optional argument one can supply a custom variable name.

Examples

julia> P3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> blow_down_morphism = blow_up(P3, 5)
Toric blowdown morphism

julia> bP3 = domain(blow_down_morphism)
Normal toric variety

julia> cox_ring(bP3)
Multivariate polynomial ring in 5 variables over QQ graded by
  x1 -> [1 0]
  x2 -> [0 1]
  x3 -> [0 1]
  x4 -> [1 0]
  e -> [1 -1]

source

More generally, we can provide a primitive element in the fan of the variety in question. The resulting star subdivision leads to a polyhedral fan, or put differently, the blow-up along this center is always toric:

blow_up — Method

blow_up(v::NormalToricVariety, new_ray::AbstractVector{<:IntegerUnion}; coordinate_name::String = "e")

Blow up the toric variety by subdividing the fan of the variety with the provided new ray. Note that this ray must be a primitive element in the lattice Z^d, with d the dimension of the fan. This function returns the corresponding blowdown morphism.

By default, we pick "e" as the name of the homogeneous coordinate for the exceptional divisor. As third optional argument one can supply a custom variable name.

Examples

julia> P3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> blow_down_morphism = blow_up(P3, [0, 1, 1])
Toric blowdown morphism

julia> bP3 = domain(blow_down_morphism)
Normal toric variety

julia> cox_ring(bP3)
Multivariate polynomial ring in 5 variables over QQ graded by
  x1 -> [1 0]
  x2 -> [0 1]
  x3 -> [0 1]
  x4 -> [1 0]
  e -> [1 -1]

julia> typeof(center(blow_down_morphism))
Oscar.ToricIdealSheafFromCoxRingIdeal{NormalToricVariety, AbsAffineScheme, Ideal, Map}

julia> Oscar.ideal_in_cox_ring(center(blow_down_morphism))
Ideal generated by
  x2
  x3

Notice that in the above example, the blowup center is not just an ideal sheaf. Rather, it is an ideal sheaf that knows its datum, in the form of an ideal, in the Cox ring. Sadly, we cannot always (at least not yet) compute such a datum. The following example demonstrates such a case.

Examples

julia> rs = [1 1; -1 1]
2×2 Matrix{Int64}:
  1  1
 -1  1

julia> max_cones = IncidenceMatrix([[1, 2]])
1×2 IncidenceMatrix
[1, 2]

julia> v = normal_toric_variety(max_cones, rs)
Normal toric variety

julia> bu = blow_up(v, [0, 1])
Toric blowdown morphism

julia> center(bu)
Sheaf of ideals
  on normal, non-smooth toric variety
with restriction
  1: Ideal (x_3_1, x_2_1, x_1_1)

julia> typeof(center(bu))
IdealSheaf{NormalToricVariety, AbsAffineScheme, Ideal, Map}

source

Most generally, we encode the blowup center by a homogeneous ideal in the Cox ring. Such blowups center may easily lead to non-toric blowups, i.e. the return value of the following method could well be non-toric.

blow_up — Method

blow_up(v::NormalToricVariety, I::MPolyIdeal; coordinate_name::String = "e")

Blow up the toric variety by subdividing the cone in the list of all cones of the fan of v which corresponds to the provided ideal I. Note that this cone need not be maximal.

By default, we pick "e" as the name of the homogeneous coordinate for the exceptional divisor. As third optional argument one can supply a custom variable name.

Examples

julia> P3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> (x1,x2,x3,x4) = gens(cox_ring(P3))
4-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 x1
 x2
 x3
 x4

julia> I = ideal([x2,x3])
Ideal generated by
  x2
  x3

julia> bP3 = domain(blow_up(P3, I))
Normal toric variety

julia> cox_ring(bP3)
Multivariate polynomial ring in 5 variables over QQ graded by
  x1 -> [1 0]
  x2 -> [0 1]
  x3 -> [0 1]
  x4 -> [1 0]
  e -> [1 -1]

julia> I2 = ideal([x2 * x3])
Ideal generated by
  x2*x3

julia> b2P3 = blow_up(P3, I2);

julia> codomain(b2P3) == P3
true

source

Instead of providing the ideal, it is possible to turn a homogeneous ideal in the Cox ring into an ideal sheaf. Consequently, we also provide the support for the following method.

blow_up — Method

blow_up(m::NormalToricVariety, I::ToricIdealSheafFromCoxRingIdeal; coordinate_name::String = "e")

Blow up the toric variety along the center given by a toric ideal sheaf.

Examples

julia> P3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> x1, x2, x3, x4 = gens(cox_ring(P3))
4-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 x1
 x2
 x3
 x4

julia> II = ideal_sheaf(P3, ideal([x1*x2]))
Sheaf of ideals
  on normal toric variety
with restrictions
  1: Ideal (x_1_1*x_2_1)
  2: Ideal (x_2_2)
  3: Ideal (x_1_3)
  4: Ideal (x_1_4*x_2_4)

julia> blow_down_morphism = blow_up(P3, II)
Blowup
  of normal toric variety
  in sheaf of ideals with restrictions
    1b: Ideal (x_1_1*x_2_1)
    2b: Ideal (x_2_2)
    3b: Ideal (x_1_3)
    4b: Ideal (x_1_4*x_2_4)
with domain
  scheme over QQ covered with 4 patches
    1a: [x_1_1, x_2_1, x_3_1]   scheme(0)
    2a: [x_1_2, x_2_2, x_3_2]   scheme(0)
    3a: [x_1_3, x_2_3, x_3_3]   scheme(0)
    4a: [x_1_4, x_2_4, x_3_4]   scheme(0)
and exceptional divisor
  effective cartier divisor defined by
    sheaf of ideals with restrictions
      1a: Ideal (x_1_1*x_2_1)
      2a: Ideal (x_2_2)
      3a: Ideal (x_1_3)
      4a: Ideal (x_1_4*x_2_4)

source

Attributes

underlying_morphism — Method

underlying_morphism(bl::ToricBlowdownMorphism)

Return the underlying toric morphism of a toric blowdown morphism. Access to other attributes such as domain, codomain, covering_morphism are executed via underlying_morphism.

Examples

julia> P3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> blow_down_morphism = blow_up(P3, [0, 1, 1])
Toric blowdown morphism

julia> Oscar.underlying_morphism(blow_down_morphism)
Toric morphism

source

index_of_new_ray — Method

index_of_new_ray(bl::ToricBlowdownMorphism)

Return the index of the new ray used in the construction of the toric blowdown morphism.

Examples

julia> P3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> blow_down_morphism = blow_up(P3, [0, 1, 1])
Toric blowdown morphism

julia> index_of_new_ray(blow_down_morphism)
5

source

center — Method

center(bl::ToricBlowdownMorphism)

Return the center of the toric blowdown morphism as ideal sheaf.

Examples

julia> P3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> blow_down_morphism = blow_up(P3, [0, 1, 1])
Toric blowdown morphism

julia> center(blow_down_morphism)
Sheaf of ideals
  on normal, smooth toric variety
with restrictions
  1: Ideal (x_3_1, x_2_1)
  2: Ideal (x_3_2, x_2_2)
  3: Ideal (1)
  4: Ideal (1)

source

exceptional_divisor — Method

exceptional_divisor(bl::ToricBlowdownMorphism)

Return the exceptional divisor (as toric divisor) of the toric blowdown morphism.

Examples

julia> P3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> blow_down_morphism = blow_up(P3, [0, 1, 1])
Toric blowdown morphism

julia> exceptional_divisor(blow_down_morphism)
Torus-invariant, cartier, prime divisor on a normal toric variety

source

Based on underlying_morphism, also the following attributes of toric morphisms are supported for toric blowdown morphisms:

grid_morphism(bl::ToricBlowdownMorphism),
morphism_on_torusinvariant_weil_divisor_group(bl::ToricBlowdownMorphism),
morphism_on_torusinvariant_cartier_divisor_group(bl::ToricBlowdownMorphism),
morphism_on_class_group(bl::ToricBlowdownMorphism),
morphism_on_picard_group(bl::ToricBlowdownMorphism).

The total and strict transform of ideal sheaves along toric blowdown morphisms can be computed:

total_transform — Method

total_transform(f::AbsSimpleBlowdownMorphism, II::IdealSheaf)

Computes the total transform of an ideal sheaf along a blowdown morphism.

In particular, this applies in the toric setting. However, note that currently (October 2023), ideal sheaves are only supported on smooth toric varieties.

Examples

julia> P2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> bl = blow_up(P2, [1, 1])
Toric blowdown morphism

julia> S = cox_ring(P2);

julia> x, y, z = gens(S);

julia> I = ideal_sheaf(P2, ideal([x*y]))
Sheaf of ideals
  on normal, smooth toric variety
with restrictions
  1: Ideal (x_1_1*x_2_1)
  2: Ideal (x_2_2)
  3: Ideal (x_1_3)

julia> total_transform(bl, I)
Sheaf of ideals
  on normal toric variety
with restrictions
  1: Ideal (x_1_1*x_2_1^2)
  2: Ideal (x_1_2^2*x_2_2)
  3: Ideal (x_2_3)
  4: Ideal (x_1_4)

source

Arithmetics

Toric blowdown morphisms can be added, subtracted and multiplied by rational numbers. The results of such operations will be toric morphisms, i.e. no longer attributed to the blowup of a certain locus. Arithmetics among toric blowdown morphisms and general toric morphisms is also supported, as well as equality for toric blowdown morphisms.