# Literature constructions

Certain models have been studied in the physics literature over and over again. Thereby, these constructions became famous and some were given special names. We aim to provide support for such standard constructions. An example of such a model is the following:

`su5_tate_model_over_arbitrary_3d_base`

— Method`su5_tate_model_over_arbitrary_3d_base()`

Return the SU(5) Tate model over an arbitrary 3-dimensional base space. For more details see e.g. [Wei18] and references therein.

```
julia> tm = su5_tate_model_over_arbitrary_3d_base()
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base
julia> v = ambient_space(tm)
A family of spaces of dimension d = 5
```

This function is part of the experimental code in Oscar. Please read here for more details.

More generally, we support literature constructions.

`literature_model`

— Method`literature_model(; doi::String="", arxiv_id::String="", version::String="", equation::String="", model_parameters::Dict{String,<:Any} = Dict{String,Any}(), base_space::FTheorySpace = affine_space(NormalToricVariety, 0), model_sections::Dict{String, <:Any} = Dict{String,Any}(), defining_classes::Dict{String, <:Any} = Dict{String,Any}(), completeness_check::Bool = true)`

Many models have been created in the F-theory literature. A significant number of them have even been given specific names, for instance the "U(1)-restricted SU(5)-GUT model". This method has access to a database, from which it can look up such literature models.

Currently, you can provide any combination of the following optional arguments to the method `literature_model`

:

`doi`

: A string representing the DOI of the publication that

introduced the model in question.

`equation`

: A string representing the number of the equation that introduced

the model in question. For papers, that were posted on the arXiv, we can instead of the `doi`

also provide the following:

`arxiv_id`

: A string that represents the arXiv identifier of the paper that

introduced the model in question.

`version`

: A string representing the version of the arXiv upload.

The method `literature_model`

attempts to find a model in our database for which the provided data matches the information in our record. If no such model can be found, or multiple models exist with information matching the provided information, then an error is raised.

Some literature models require additional parameters to specified to single out a model from a family of models. Such models can be provided using the optional argument `model_parameters`

, which should be a dictionary such as `Dict("k" => 5)`

.

```
julia> t = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> v = ambient_space(t)
A family of spaces of dimension d = 5
julia> coordinate_ring(v)
Multivariate polynomial ring in 8 variables w, a1, a21, a32, ..., z
over rational field
```

It is also possible to construct a literature model over a particular base. Currently, this feature is only supported for toric base spaces.

```
julia> B3 = projective_space(NormalToricVariety, 3)
Normal toric variety
julia> w = torusinvariant_prime_divisors(B3)[1]
Torus-invariant, prime divisor on a normal toric variety
julia> t2 = literature_model(arxiv_id = "1109.3454", equation = "3.1", base_space = B3, defining_classes = Dict("w" => w), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!
Global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> length(singular_loci(t2))
2
```

Of course, this is also possible for Weierstrass models.

```
julia> B2 = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> b = torusinvariant_prime_divisors(B2)[1]
Torus-invariant, prime divisor on a normal toric variety
julia> w = literature_model(arxiv_id = "1208.2695", equation = "B.19", base_space = B2, defining_classes = Dict("b" => b), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!
Weierstrass model over a concrete base -- U(1) Weierstrass model based on arXiv paper 1208.2695 Eq. (B.19)
julia> length(singular_loci(w))
1
```

For convenience, we also support a simplified constructor. Instead of the meta data of the article, this constructor accepts an integer, which specifies the position of this model in our database.

```
julia> B2 = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> b = torusinvariant_prime_divisors(B2)[1]
Torus-invariant, prime divisor on a normal toric variety
julia> w = literature_model(3, base_space = B2, defining_classes = Dict("b" => b), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!
Weierstrass model over a concrete base -- U(1) Weierstrass model based on arXiv paper 1208.2695 Eq. (B.19)
julia> length(singular_loci(w))
1
```

Similarly, also hypersurface models are supported:

```
julia> h = literature_model(arxiv_id = "1208.2695", equation = "B.5")
Assuming that the first row of the given grading is the grading under Kbar
Hypersurface model over a not fully specified base
julia> explicit_model_sections(h)
Dict{String, MPolyRingElem} with 5 entries:
"c2" => c2
"c1" => c1
"c3" => c3
"b" => b
"c0" => c0
julia> B2 = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> b = torusinvariant_prime_divisors(B2)[1]
Torus-invariant, prime divisor on a normal toric variety
julia> h2 = literature_model(arxiv_id = "1208.2695", equation = "B.5", base_space = B2, defining_classes = Dict("b" => b))
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!
Hypersurface model over a concrete base
julia> hypersurface_equation_parametrization(h2)
b*w*v^2 - c0*u^4 - c1*u^3*v - c2*u^2*v^2 - c3*u*v^3 + w^2
```

A yet more special instance of literature model are the F-theory QSMs. Those consist of 708 families of geometries, each of which is obtained from triangulations of polytopes in the 3-dimensional Kreuzer-Skarke list. In particular, for each of these 708 families, a lot of information is known. Still, those geometries are also somewhat involved. Let us demonstrate this on the F-theory QSM based on the 4th polytope in the 3-dimensional Kreuzer-Skarke list. We can create and study this model as follows (TODO: currently under development):

```
julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4));
julia> model = qsm_model.hs_model
Hypersurface model over a concrete base
julia> cox_ring(base_space(model))
Multivariate polynomial ring in 29 variables over QQ graded by
x1 -> [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
x2 -> [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
x3 -> [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
x4 -> [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
x5 -> [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
x6 -> [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
x7 -> [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
x8 -> [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
x9 -> [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
x10 -> [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
x11 -> [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
x12 -> [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
x13 -> [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0]
x14 -> [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0]
x15 -> [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]
x16 -> [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0]
x17 -> [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0]
x18 -> [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0]
x19 -> [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0]
x20 -> [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0]
x21 -> [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0]
x22 -> [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0]
x23 -> [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0]
x24 -> [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0]
x25 -> [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0]
x26 -> [1 4 0 1 2 3 1 2 3 4 0 1 1 1 1 1 1 1 1 4 0 2 3 4 4 5]
x27 -> [3 2 2 0 -1 -1 1 0 0 1 1 2 0 1 -1 0 -2 -1 -3 -2 -2 1 1 -1 0 0]
x28 -> [-2 -4 -1 -1 -1 -2 -1 -2 -3 -4 -1 -2 -2 -2 -1 -1 0 0 1 -2 1 -2 -3 -3 -3 -4]
x29 -> [0 -1 0 -1 -1 -1 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 -1 0 -1 -1 -1 -1 -1]
julia> qsm_model.Kbar3
6
julia> qsm_model.triang_quick
true
julia> qsm_model.estimated_number_of_triangulations
212533333333
julia> qsm_model.dual_graph
Undirected graph with 21 nodes and the following edges:
(5, 1)(6, 5)(7, 6)(8, 7)(9, 4)(9, 8)(10, 1)(11, 4)(12, 3)(12, 10)(13, 3)(13, 11)(14, 1)(15, 4)(16, 3)(17, 3)(18, 2)(18, 14)(19, 2)(19, 15)(20, 2)(20, 16)(21, 2)(21, 17)
julia> qsm_model.components_of_simplified_dual_graph
4-element Vector{String}:
"C0"
"C1"
"C2"
"C3"
julia> qsm_model.simplified_dual_graph
Undirected graph with 4 nodes and the following edges:
(2, 1)(3, 1)(3, 2)(4, 1)(4, 2)(4, 3)
```

This function is part of the experimental code in Oscar. Please read here for more details.

## Attributes

For literature models, we provide the following attributes referencing meta data:

`arxiv_id`

— Method`arxiv_id(m::AbstractFTheoryModel)`

Return the `arxiv_id`

of the preprint that introduced the given model. If no `arxiv_id`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> arxiv_id(m)
"1109.3454"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`arxiv_doi`

— Method`arxiv_doi(m::AbstractFTheoryModel)`

Return the `arxiv_doi`

of the preprint that introduced the given model. If no `arxiv_doi`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> arxiv_doi(m)
"10.48550/arXiv.1109.3454"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`arxiv_link`

— Method`arxiv_link(m::AbstractFTheoryModel)`

Return the `arxiv_link`

(formatted as string) to the arXiv version of the paper that introduced the given model. If no `arxiv_link`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> arxiv_link(m)
"https://arxiv.org/abs/1109.3454v2"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`arxiv_model_equation_number`

— Method`arxiv_model_equation_number(m::AbstractFTheoryModel)`

Return the `arxiv_model_equation_number`

in which the given model was introduced in the arXiv preprint in our record. If no `arxiv_model_equation_number`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> arxiv_model_equation_number(m)
"3.1"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`arxiv_model_page`

— Method`arxiv_model_page(m::AbstractFTheoryModel)`

Return the `arxiv_model_page`

on which the given model was introduced in the arXiv preprint in our record. If no `arxiv_model_page`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> arxiv_model_page(m)
"10"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`arxiv_model_section`

— Method`arxiv_model_section(m::AbstractFTheoryModel)`

Return the `arxiv_model_section`

in which the given model was introduced in the arXiv preprint in our record. If no `arxiv_model_section`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> arxiv_model_section(m)
"3"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`arxiv_version`

— Method`arxiv_version(m::AbstractFTheoryModel)`

Return the `arxiv_version`

of the arXiv preprint that introduced the given model. If no `arxiv_version`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> arxiv_version(m)
"2"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`associated_literature_models`

— Method`associated_literature_models(m::AbstractFTheoryModel)`

Return a list of the unique identifiers any `associated_literature_models`

of the given model. These are either other presentations (Weierstrass, Tate, ...) of the given model, or other version of the same model from a different paper in the literature. If no `associated_literature_models`

are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1507.05954", equation = "A.1")
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base -- U(1)xU(1) Weierstrass model based on arXiv paper 1507.05954 Eq. (A.1)
julia> associated_literature_models(m)
1-element Vector{String}:
"1507_05954-1"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`generating_sections`

— Method`generating_sections(m::AbstractFTheoryModel)`

Return a list of the known Mordell–Weil generating sections of the given model. If no generating sections are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> generating_sections(m)
1-element Vector{Vector{QQMPolyRingElem}}:
[0, 0, 1]
```

This function is part of the experimental code in Oscar. Please read here for more details.

`journal_doi`

— Method`journal_doi(m::AbstractFTheoryModel)`

Return the `journal_doi`

of the publication that introduced the given model. If no `journal_doi`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> journal_doi(m)
"10.1016/j.nuclphysb.2011.12.013"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`journal_link`

— Method`journal_link(m::AbstractFTheoryModel)`

Return the `journal_link`

(formatted as string) to the published version of the paper that introduced the given model. If no `journal_link`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> journal_link(m)
"https://www.sciencedirect.com/science/article/pii/S0550321311007115"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`journal_model_equation_number`

— Method`journal_model_equation_number(m::AbstractFTheoryModel)`

Return the `journal_model_equation_number`

in which the given model was introduced in the published paper in our record. If no `journal_model_equation_number`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> journal_model_equation_number(m)
"3.1"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`journal_model_page`

— Method`journal_model_page(m::AbstractFTheoryModel)`

Return the `journal_model_page`

on which the given model was introduced in the published paper in our record. If no `journal_model_page`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> journal_model_page(m)
"9"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`journal_model_section`

— Method`journal_model_section(m::AbstractFTheoryModel)`

Return the `journal_model_section`

in which the given model was introduced in the published paper in our record. If no `journal_model_section`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> journal_model_section(m)
"3"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`journal_name`

— Method`journal_name(m::AbstractFTheoryModel)`

Return the `journal_volume`

of the published paper in which the given model was introduced. If no `journal_volume`

are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> journal_volume(m)
"858"
```

`journal_pages`

— Method`journal_pages(m::AbstractFTheoryModel)`

Return the `journal_pages`

of the published paper in which the given model was introduced. If no `journal_pages`

are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> journal_pages(m)
"1–47"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`journal_report_numbers`

— Method`journal_report_numbers(m::AbstractFTheoryModel)`

Return the `journal_report_numbers`

of the published paper in which the given model was introduced. If no `journal_report_numbers`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1507.05954", equation = "A.1")
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base -- U(1)xU(1) Weierstrass model based on arXiv paper 1507.05954 Eq. (A.1)
julia> journal_report_numbers(m)
3-element Vector{String}:
"UPR-1274-T"
"CERN-PH-TH-2015-157"
"MIT-CTP-4678"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`journal_volume`

— Method`journal_volume(m::AbstractFTheoryModel)`

Return the `journal_volume`

of the published paper in which the given model was introduced. If no `journal_volume`

are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> journal_volume(m)
"858"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`journal_year`

— Method`journal_year(m::AbstractFTheoryModel)`

Return the `journal_year`

of the published paper in which the given model was introduced. If no `journal_year`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> journal_year(m)
"2012"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`literature_identifier`

— Method`literature_identifier(m::AbstractFTheoryModel)`

Return the `literature_identifier`

of the given mode, which is a unique string that distinguishes the model from all others in the literature model database. If no `literature_identifier`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> literature_identifier(m)
"1109_3454"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`model_parameters`

— Method`model_parameters(m::AbstractFTheoryModel)`

Return the `model_parameters`

of the given model. If no `model_parameters`

are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1212.2949", equation = "3.2", model_parameters = Dict("k" => 5))
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(2k+1) Tate model with parameter values (k = 5) based on arXiv paper 1212.2949 Eq. (3.2)
julia> model_parameters(m)
Dict{String, Int64} with 1 entry:
"k" => 5
```

This function is part of the experimental code in Oscar. Please read here for more details.

`paper_authors`

— Method`paper_authors(m::AbstractFTheoryModel)`

Return the `paper_authors`

of the paper that introduced the given model. If no `paper_authors`

are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> paper_authors(m)
3-element Vector{String}:
"Sven Krause"
"Christoph Mayrhofer"
"Timo Weigand"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`paper_buzzwords`

— Method`paper_buzzwords(m::AbstractFTheoryModel)`

Return the `paper_buzzwords`

of the paper that introduced the given model. If no `paper_buzzwords`

are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> paper_buzzwords(m)
4-element Vector{String}:
"GUT model"
"Tate"
"U(1)"
"SU(5)"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`paper_description`

— Method`paper_description(m::AbstractFTheoryModel)`

Return the `paper_description`

of the paper that introduced the given model. If no `paper_description`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> paper_description(m)
"SU(5)xU(1) restricted Tate model"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`paper_title`

— Method`paper_title(m::AbstractFTheoryModel)`

Return the `paper_title`

of the arXiv preprint that introduced the given model. If no `paper_title`

is known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> paper_title(m)
"\$G_4\$ flux, chiral matter and singularity resolution in F-theory compactifications"
```

This function is part of the experimental code in Oscar. Please read here for more details.

`related_literature_models`

— Method`related_literature_models(m::AbstractFTheoryModel)`

Return a list of the unique identifiers of any `related_literature_models`

of the given model. These are models that are introduced in the same paper as the given model, but that are distinct from the given model. If no `related_literature_models`

are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1212.2949", equation = "3.2", model_parameters = Dict("k" => 5))
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(2k+1) Tate model with parameter values (k = 5) based on arXiv paper 1212.2949 Eq. (3.2)
julia> related_literature_models(m)
6-element Vector{String}:
"1212_2949-2"
"1212_2949-3"
"1212_2949-4"
"1212_2949-5"
"1212_2949-6"
"1212_2949-7"
```

This function is part of the experimental code in Oscar. Please read here for more details.

Such meta data can be modified with setters. For instance, there is a function `set_description(m::AbstractFTheoryModel, description::String)`

, which takes the model in question as the first argument and the desired description - provided as string - as the second argument. Such a setter function exists for all of the above. If appropriate, we also offer a method that adds a new value. For instance, we have a function `add_paper_buzzword(m::AbstractFTheoryModel, addition::String)`

.

In addition, the following attributes are available to access advanced model information:

`resolutions`

— Method`resolutions(m::AbstractFTheoryModel)`

Return the list of all known resolutions for the given model. If no resolutions are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> resolutions(m)
1-element Vector{Vector{Vector}}:
[[["x", "y", "w"], ["y", "e1"], ["x", "e4"], ["y", "e2"], ["x", "y"]], ["e1", "e4", "e2", "e3", "s"]]
```

This function is part of the experimental code in Oscar. Please read here for more details.

`resolution_generating_sections`

— Method`resolution_generating_sections(m::AbstractFTheoryModel)`

Return a list of lists of known Mordell–Weil generating sections for the given model after each known resolution. Each element of the outer list corresponds to a known resolution (in the same order), and each element of the list associated to a given resolution corresponds to a known generating section (in the same order). If no resolution generating sections are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> resolution_generating_sections(m)
1-element Vector{Vector{Vector{Vector{QQMPolyRingElem}}}}:
[[[0, 0, 1], [0, 0, 1], [0, 1], [0, 1], [0, 1], [a32, -a43]]]
```

This function is part of the experimental code in Oscar. Please read here for more details.

`resolution_zero_sections`

— Method`resolution_zero_sections(m::AbstractFTheoryModel)`

Return a list of known Mordell–Weil zero sections for the given model after each known resolution. Each element of the list corresponds to a known resolution (in the same order). If no resolution zero sections are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> resolution_zero_sections(m)
1-element Vector{Vector{Vector{QQMPolyRingElem}}}:
[[1, 1, 0], [1, 1, w], [1, 1], [1, 1], [1, 1], [1, 1]]
```

This function is part of the experimental code in Oscar. Please read here for more details.

`weighted_resolutions`

— Method`weighted_resolutions(m::AbstractFTheoryModel)`

Return the list of all known weighted resolutions for the given model. If no weighted resolutions are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> weighted_resolutions(m)
1-element Vector{Vector{Vector}}:
[Vector{Vector{Any}}[[["x", "y", "w"], [1, 1, 1]], [["x", "y", "w"], [1, 2, 1]], [["x", "y", "w"], [2, 2, 1]], [["x", "y", "w"], [2, 3, 1]], [["x", "y"], [1, 1]]], ["e1", "e4", "e2", "e3", "s"]]
```

This function is part of the experimental code in Oscar. Please read here for more details.

`weighted_resolution_generating_sections`

— Method`weighted_resolution_generating_sections(m::AbstractFTheoryModel)`

Return a list of lists of known Mordell–Weil generating sections for the given model after each known weighted resolution. Each element of the outer list corresponds to a known weighted resolution (in the same order), and each element of the list associated to a given weighted resolution corresponds to a known generating section (in the same order). If no weighted resolution generating sections are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> weighted_resolution_generating_sections(m)
1-element Vector{Vector{Vector{Vector{QQMPolyRingElem}}}}:
[[[0, 0, 1], [0, 0, 1], [0, 0, 1], [0, 0, 1], [0, 0, 1], [a32, -a43]]]
```

This function is part of the experimental code in Oscar. Please read here for more details.

`weighted_resolution_zero_sections`

— Method`weighted_resolution_zero_sections(m::AbstractFTheoryModel)`

Return a list of known Mordell–Weil zero sections for the given model after each known weighted resolution. Each element of the list corresponds to a known weighted resolution (in the same order). If no weighted resolution zero sections are known, an error is raised.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> weighted_resolution_zero_sections(m)
1-element Vector{Vector{Vector{QQMPolyRingElem}}}:
[[1, 1, 0], [1, 1, w], [1, 1, w], [1, 1, w], [1, 1, w], [1, 1]]
```

This function is part of the experimental code in Oscar. Please read here for more details.

One can check if a model has a particular set of information. This is achieved with the following methods:

`has_arxiv_id(m::AbstractFTheoryModel)`

,`has_arxiv_doi(m::AbstractFTheoryModel)`

,`has_arxiv_link(m::AbstractFTheoryModel)`

,`has_arxiv_model_equation_number(m::AbstractFTheoryModel)`

,`has_arxiv_model_page(m::AbstractFTheoryModel)`

,`has_arxiv_model_section(m::AbstractFTheoryModel)`

,`has_arxiv_version(m::AbstractFTheoryModel)`

,`has_associated_literature_models(m::AbstractFTheoryModel)`

,`has_generating_sections(m::AbstractFTheoryModel)`

,`has_journal_doi(m::AbstractFTheoryModel)`

,`has_journal_link(m::AbstractFTheoryModel)`

,`has_journal_model_equation_number(m::AbstractFTheoryModel)`

,`has_journal_model_page(m::AbstractFTheoryModel)`

,`has_journal_model_section(m::AbstractFTheoryModel)`

,`has_journal_pages(m::AbstractFTheoryModel)`

,`has_journal_report_numbers(m::AbstractFTheoryModel)`

,`has_journal_volume(m::AbstractFTheoryModel)`

,`has_journal_year(m::AbstractFTheoryModel)`

,`has_literature_identifier(m::AbstractFTheoryModel)`

,`has_model_description(m::AbstractFTheoryModel)`

,`has_model_parameters(m::AbstractFTheoryModel)`

,`has_paper_authors(m::AbstractFTheoryModel)`

,`has_paper_buzzwords(m::AbstractFTheoryModel)`

,`has_paper_description(m::AbstractFTheoryModel)`

,`has_paper_title(m::AbstractFTheoryModel)`

,`has_related_literature_models(m::AbstractFTheoryModel)`

,`has_resolutions(m::AbstractFTheoryModel)`

,`has_resolution_generating_sections(m::AbstractFTheoryModel)`

,`has_resolution_zero_sections(m::AbstractFTheoryModel)`

,`has_weighted_resolutions(m::AbstractFTheoryModel)`

,`has_weighted_resolution_generating_sections(m::AbstractFTheoryModel)`

,`has_weighted_resolution_zero_sections(m::AbstractFTheoryModel)`

,`has_zero_section(m::AbstractFTheoryModel)`

.

## Methods

### Resolution(s) of a singular model

A central task in F-theory is to resolve a singular model. For literature models, we have stored resolutions in our data base. Upon construction of a literature model, we load these known resolutions.

In addition to listing the known resolutions with `resolutions(m::AbstractFTheoryModel)`

, the user might want to add a resolution. This can be achieved with the following method:

`add_resolution`

— Method`add_resolution(m::AbstractFTheoryModel, centers::Vector{Vector{String}}, exceptionals::Vector{String})`

Add a known resolution for a model.

```
julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar
Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> add_resolution(m, [["x", "y"], ["y", "s", "w"], ["s", "e4"], ["s", "e3"], ["s", "e1"]], ["s", "w", "e3", "e1", "e2"])
julia> length(resolutions(m))
2
```

This function is part of the experimental code in Oscar. Please read here for more details.

Provided that a resolution for a model is known, we can (attempt to) resolve the model.

`resolve`

— Method`resolve(m::AbstractFTheoryModel, index::Int)`

Resolve a model with the index-th resolution that is known.

```
julia> B3 = projective_space(NormalToricVariety, 3)
Normal toric variety
julia> w = torusinvariant_prime_divisors(B3)[1]
Torus-invariant, prime divisor on a normal toric variety
julia> t = literature_model(arxiv_id = "1109.3454", equation = "3.1", base_space = B3, defining_classes = Dict("w" => w), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!
Global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> t2 = resolve(t, 1)
Partially resolved global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> cox_ring(ambient_space(t2))
Multivariate polynomial ring in 12 variables over QQ graded by
x1 -> [1 0 0 0 0 0 0]
x2 -> [0 1 0 0 0 0 0]
x3 -> [0 1 0 0 0 0 0]
x4 -> [0 1 0 0 0 0 0]
x -> [0 0 1 0 0 0 0]
y -> [0 0 0 1 0 0 0]
z -> [0 0 0 0 1 0 0]
e1 -> [0 0 0 0 0 1 0]
e4 -> [0 0 0 0 0 0 1]
e2 -> [-1 -3 -1 1 -1 -1 0]
e3 -> [0 4 1 -1 1 0 -1]
s -> [2 6 -1 0 2 1 1]
julia> w2 = 2 * torusinvariant_prime_divisors(B3)[1]
Torus-invariant, non-prime divisor on a normal toric variety
julia> t3 = literature_model(arxiv_id = "1109.3454", equation = "3.1", base_space = B3, defining_classes = Dict("w" => w2), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!
Global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
julia> t4 = resolve(t3, 1)
Partially resolved global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
```

This function is part of the experimental code in Oscar. Please read here for more details.