Toric Blowups (Experimental)

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 morphism.

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

Examples

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

julia> f = blow_up(P3, 5)
Toric blowup morphism

julia> bP3 = domain(f)
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]

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

Blow up the toric variety by subdividing the fan of the variety with the provided new ray. This function returns the corresponding morphism.

Note that this ray must be a primitive element in the lattice Z^d, with d the dimension of the fan. In particular, it is currently impossible to blow up along a ray which corresponds to a non-Q-Cartier divisor.

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

Examples

In the example below center_unnormalized(f) has type ToricIdealSheafFromCoxRingIdeal and we can access the corresponding ideal in the Cox ring using Oscar.ideal_in_cox_ring.

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

julia> f = blow_up(P3, [0, 2, 3])
Toric blowup morphism

julia> bP3 = domain(f)
Normal toric variety

julia> cox_ring(bP3)
Multivariate polynomial ring in 5 variables over QQ graded by
  x1 -> [1 0]
  x2 -> [1 2]
  x3 -> [1 3]
  x4 -> [1 0]
  e -> [0 -1]

julia> typeof(center_unnormalized(f))
Oscar.ToricIdealSheafFromCoxRingIdeal{NormalToricVariety, AbsAffineScheme, Ideal, Map}

julia> Oscar.ideal_in_cox_ring(center_unnormalized(f))
Ideal generated by
  x2^2
  x3^3

In the below example, center_unnormalized(f) has type IdealSheaf and we cannot access the corresponding ideal in the Cox ring.

Examples

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

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

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

julia> f = blow_up(v, [0, 1])
Toric blowup morphism

julia> center_unnormalized(f)
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_unnormalized(f))
IdealSheaf{NormalToricVariety, AbsAffineScheme, Ideal, Map}

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 prime 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

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

Blow up the toric variety along 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> f = 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)

underlying_morphism(bl::ToricBlowupMorphism)

Return the underlying toric morphism of a toric blowup. 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> f = blow_up(P3, [0, 1, 1])
Toric blowup morphism

julia> Oscar.underlying_morphism(f)
Toric morphism

index_of_new_ray(bl::ToricBlowupMorphism)

Return the index of the new ray used in the construction of the toric blowup.

Examples

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

julia> f = blow_up(P3, [0, 1, 1])
Toric blowup morphism

julia> index_of_new_ray(f)
5

center_data(bl::ToricBlowupMorphism)

Returns the ideal, ideal sheaf or ray that was used to construct the morphism.

Examples

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

julia> f = blow_up(P3, [0, 2, 3])
Toric blowup morphism

julia> center_data(f)
3-element Vector{Int64}:
 0
 2
 3

center_unnormalized(bl::ToricBlowupMorphism)

Returns an ideal sheaf I such that the normalization of the blowup along I gives the morphism bl.

Examples

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

julia> f = blow_up(P3, [0, 2, 3])
Toric blowup morphism

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

exceptional_prime_divisor(bl::ToricBlowupMorphism)

Return the exceptional prime Weil divisor (as a toric divisor) of the ray used to construct the toric blowup. Note that this divisor need not be Cartier and this divisor need not coincide with the locus where the morphism is not an isomorphism.

Examples

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

julia> f = blow_up(P3, [0, 2, 3])
Toric blowup morphism

julia> E = exceptional_prime_divisor(f)
Torus-invariant, prime divisor on a normal toric variety

julia> is_cartier(E)
false

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

Computes the total transform of an ideal sheaf along a blowup.

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 blowup 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)