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methods.jl
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methods.jl
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@doc raw"""
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
```jldoctest
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)
```
"""
function total_transform(f::AbsSimpleBlowdownMorphism, II::AbsIdealSheaf)
return pullback(f, II)
end
function total_transform(f::AbsBlowdownMorphism, II::AbsIdealSheaf)
return pullback(f, II)
end