Fix blowups along rays and singular cones

[paemurru]

Fix blowups along singular cones when the cones is given by an index. Also, fix blowup and star subdivision along a single existing ray in the case where the ray is given by an index among the list of all cones.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 84.37%. Comparing base (a60b0d8) to head (489204d).
Report is 149 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #4454      +/-   ##
==========================================
- Coverage   84.42%   84.37%   -0.05%     
==========================================
  Files         668      672       +4     
  Lines       88363    88717     +354     
==========================================
+ Hits        74600    74855     +255     
- Misses      13763    13862      +99     

Files with missing lines	Coverage Δ
...erimental/Schemes/src/ToricBlowups/constructors.jl	91.89% <100.00%> (+0.84%)	⬆️
src/PolyhedralGeometry/PolyhedralFan/properties.jl	98.48% <100.00%> (+0.05%)	⬆️
...alGeometry/PolyhedralFan/standard_constructions.jl	100.00% <ø> (ø)

... and 52 files with indirect coverage changes

[HereAround]

@paemurru Thank you for working on this. Overall, this looks very good. A few comments:

• As stated above, could you point me to the line(s) that fixes the original issue that you face? I am just curious and want to appreciate your work.
• My vague recollection is that @lkastner , @HechtiDerLachs and I did not touch on blowups of singular cones. Since you seem to know this scenario well (at least the title of your PR indicates so), could you add a literature reference to the docs, that justifies the computational approach? (Sorry if it is already there and I overlooked it.) Again, this is to fully appreciate your work and verify that the code computes the correct mathematical object.
• Modulo "does the code compute the correct mathematical object" (i.e. the previous point), the code changes on the toric varieties side look fine to me.
• Regarding the code changes on the polyhedral geometry side, maybe @benlorenz could take a look?

[paemurru]

Bugs

There were two bugs before:

(1) Blowing up along a singular cone gave an error. If the intention was to allow only smooth cones, then this should have been checked using an @req and should have been documented.

One fix would have been to add the @req and document it. I took a different approach and implemented blowups along arbitrary nonzero cones.

The code that gave an error before:

julia> ray_generators = [[2, -1], [0, 1]]
2-element Vector{Vector{Int64}}:
 [2, -1]
 [0, 1]
julia> max_cones = IncidenceMatrix([[1, 2]])
1×2 IncidenceMatrix
[1, 2]
julia> X = normal_toric_variety(max_cones, ray_generators)
Normal toric variety
julia> g = blow_up(X, 2)
ERROR: ArgumentError: Cannot subdivide cone 2 as it is generated by a single ray
Stacktrace:
 [1] macro expansion
   @ ~/.julia/packages/AbstractAlgebra/xy6Re/src/Assertions.jl:599 [inlined]
 [2] star_subdivision(Sigma::NormalToricVariety, n::Int64)
   @ Oscar ~/.julia/dev/Oscar/src/PolyhedralGeometry/PolyhedralFan/standard_constructions.jl:255
 [3] blow_up(v::NormalToricVariety, n::Int64; coordinate_name::Nothing)
   @ Oscar ~/.julia/dev/Oscar/experimental/Schemes/src/ToricBlowups/constructors.jl:240
 [4] blow_up(v::NormalToricVariety, n::Int64)
   @ Oscar ~/.julia/dev/Oscar/experimental/Schemes/src/ToricBlowups/constructors.jl:236
 [5] top-level scope
   @ REPL[84]:1

(2) Blowing up along a (single) ray gave an error. The fix was just removing an @req in PolyhedralGeometry, namely

@req length(tau) > 1 "Cannot subdivide cone $n as it is generated by a single ray"

Below is the error message that appeared before.

julia> ray_generators = [[2, -1], [0, 1]]
2-element Vector{Vector{Int64}}:
 [2, -1]
 [0, 1]
julia> max_cones = IncidenceMatrix([[1, 2]])
1×2 IncidenceMatrix
[1, 2]
julia> X = normal_toric_variety(max_cones, ray_generators)
Normal toric variety
julia> f = blow_up(X, 1)
ERROR: ArgumentError: Could not identify position of exceptional ray
Stacktrace:
 [1] macro expansion
   @ ~/.julia/packages/AbstractAlgebra/xy6Re/src/Assertions.jl:599 [inlined]
 [2] (::Oscar.var"#_toric_blowup_morphism#9400")(v::NormalToricVariety, new_variety::NormalToricVariety, coordinate_name::String, exceptional_ray::Vector{…}, center_data::Vector{…})
   @ Oscar ~/.julia/dev/Oscar/experimental/Schemes/src/ToricBlowups/types.jl:38
 [3] Oscar.ToricBlowupMorphism(v::NormalToricVariety, new_variety::NormalToricVariety, coordinate_name::String, exceptional_ray::Vector{…}, center_data::Vector{…}, center_unnormalized::Oscar.ToricIdealSheafFromCoxRingIdeal{…})
   @ Oscar ~/.julia/dev/Oscar/experimental/Schemes/src/ToricBlowups/types.jl:69
 [4] blow_up(v::NormalToricVariety, n::Int64; coordinate_name::Nothing)
   @ Oscar ~/.julia/dev/Oscar/experimental/Schemes/src/ToricBlowups/constructors.jl:243
 [5] blow_up(v::NormalToricVariety, n::Int64)
   @ Oscar ~/.julia/dev/Oscar/experimental/Schemes/src/ToricBlowups/constructors.jl:236
 [6] top-level scope
   @ REPL[83]:1
Some type information was truncated. Use `show(err)` to see complete types.

Documentation

My last commit added the relevant citations to the docstring of blow_up.

Correctness

By Example 11.1.4 in CLS11, the star subdivision along a smooth cone and along the barycenter of the smooth cone coincide.

[HereAround]

Thank you for elaborating @paemurru . Just looked at your PR again, understanding it much better now. Looks good to me.

I will wait until tomorrow for feedback regarding your changes in the polyhedral geometry (e.g. by @benlorenz). If there are no complaints by the Friday meeting tomorrow, I will merge this PR.

[benlorenz]

A few small improvements and one correction for the check.