Implement ambient space models for G4-flux candidates

[HereAround]

See the doc string for more details. Hopefully, this is sufficiently clear.

I have taken the liberty to profile this new function a lot, since we want to showcase this very soon (or so I believe). For the doctest example g4_amb_list = ambient_space_models_of_g4_fluxes(qsm_model, check = false); I found:

• My first working implementation: About 1.5 * 10^6 allocs.
• Current implementation: Only about 24,000 allocs.

The main improvement came from a more efficient identification of remaining_vars_list, but there were quite a number of other tricks I employed. Overall, I am very happy with these improvements.

@apturner @emikelsons You may find it entertaining that for "the" big model, I expect that ambient_space_models_of_g4_fluxes completes in no more than 10 minutes on most personal computers. On mine, even only about 5 to 6 minutes. (The first run will of course always be somewhat slower...)

[HereAround]

A few more numbers for "the" big model @apturner @emikelsons :

• b_4(X_Sigma) = 815, X_Sigma the toric ambient space of the resolved model. So we have H^(2,2) \cong Q^815, and this would be our "naive" basis for G4-flux candidates.
• The newly introduced method in this PR identifies 629 basis elements of H^(2,2) which restrict - most likely - non-trivially to Y^4.

For details on what I mean by "most likely", see the changes in g4.md.

[emikelsons]

Calling the new method on the SU(5)xU(1) restricted Tate model leads to an error, the following code reproduces the error:
julia> B3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> Kbar = anticanonical_divisor_class(B3)
Divisor class on a normal toric variety

julia> t = literature_model(arxiv_id = "1109.3454",
equation = "3.1", base_space = B3, defining_classes = Dict("w"=>Kbar));
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!

julia> ambient_space_models_of_g4_fluxes(t, check = false);

[emikelsons]

I think the problem lies on line 137 of the basis_of_h22 code, where an error occurs, if remaining_relations are empty

[HereAround]

Calling the new method on the SU(5)xU(1) restricted Tate model leads to an error, the following code reproduces the error: julia> B3 = projective_space(NormalToricVariety, 3) Normal toric variety

julia> Kbar = anticanonical_divisor_class(B3) Divisor class on a normal toric variety

julia> t = literature_model(arxiv_id = "1109.3454", equation = "3.1", base_space = B3, defining_classes = Dict("w"=>Kbar)); Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!

julia> ambient_space_models_of_g4_fluxes(t, check = false);

Good catch. Thank you!

I think the problem lies on line 137 of the basis_of_h22 code, where an error occurs, if remaining_relations are empty

Indeed, I did not catch the case for which there are no remaining relations. 9e586c0 should fix this.

[emikelsons]

Looks good!

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 84.47%. Comparing base (2814275) to head (ed02e4e).
Report is 284 commits behind head on master.

Additional details and impacted files

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  Coverage   84.47%   84.47%           
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  Files         641      641           
  Lines       85394    85427   +33     
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+ Hits        72133    72168   +35     
+ Misses      13261    13259    -2     

Files with missing lines	Coverage Δ
...al/FTheoryTools/src/G4Fluxes/special_attributes.jl	97.08% <100.00%> (+4.23%)	⬆️