Minor code improvements in the documentation

Author: HereAround

experimental/FTheoryTools/docs/src/g4.md

@@ -25,12 +25,9 @@ subject to **quantization**, **transversality**, and several additional consiste
 
 1. **Quantization (cf. [Witten 1997](@cite Wit97)):** ``G_4 + \frac{1}{2} c_2(\widehat{Y}_4) \in H^4(\widehat{Y}_4, \mathbb{Z})``
 
-2. **Transversality:** The flux must be orthogonal to both the base and the fiber in a precise cohomological
-sense.
+2. **Transversality:** The flux must be orthogonal to both the base and the fiber in a precise cohomological sense.
 
-3. Additional constraints, such as **D3-tadpole cancellation**, **self-duality**, and **primitivity**, are also
-typically imposed. One often requires that the ``G_4``-flux preserves the non-abelian gauge symmetry encoded in
-the fibration.
+3. Additional constraints, such as **D3-tadpole cancellation**, **self-duality**, and **primitivity**, are also typically imposed. One often requires that the ``G_4``-flux preserves the non-abelian gauge symmetry encoded in the fibration.
 
 These conditions ensure compatibility with fundamental physical principles like anomaly cancellation and D3-brane
 charge quantization.

experimental/FTheoryTools/docs/src/hypersurface.md

@@ -28,15 +28,8 @@ following ingredients:
 
 - A **base space** ``B``, over which the elliptic fibration is defined.
 - A **fiber ambient space** ``F``, in which the elliptic fiber appears as a hypersurface.
-- Two divisor classes ``D_1`` and ``D_2`` in ``\text{Cl}(B)``, and a choice of two
-homogeneous coordinates of the fiber ambient space ``F``. These two coordinates transform
-over the base ``B`` as sections of the line bundles associated to ``D_1`` and ``D_2``,
-respectively. All remaining homogeneous coordinates of ``F`` transform as sections of the
-trivial line bundle over ``B``.
-- A hypersurface equation defining the total space of the elliptic fibration as a section
-of the anti-canonical bundle ``\overline{K}_A`` of the full ambient space ``A``, which
-combines both the fiber ambient space ``F`` and the base space ``B``. This ensures that
-the hypersurface equation is Calabi–Yau.
+- Two divisor classes ``D_1`` and ``D_2`` in ``\text{Cl}(B)``, and a choice of two homogeneous coordinates of the fiber ambient space ``F``. These two coordinates transform over the base ``B`` as sections of the line bundles associated to ``D_1`` and ``D_2``, respectively. All remaining homogeneous coordinates of ``F`` transform as sections of the trivial line bundle over ``B``.
+- A hypersurface equation defining the total space of the elliptic fibration as a section of the anti-canonical bundle ``\overline{K}_A`` of the full ambient space ``A``, which combines both the fiber ambient space ``F`` and the base space ``B``. This ensures that the hypersurface equation is Calabi–Yau.
 
 It is worth noting that any elliptic fibration, for which the fiber ambient space is toric,
 can be cast into this form [KM-POPR15](@cite). Consequently, this approach allows for a

experimental/FTheoryTools/docs/src/tate.md

@@ -226,13 +226,8 @@ In F-theory, the standard approach to handling singular geometries is to replace
 **crepant resolutions**. This process preserves the Calabi–Yau condition and ensures the correct encoding of physical
 data. However, several important caveats apply:
 
-- Not all singularities admit crepant resolutions, rather some singularities are obstructed from being resolved without
-violating the Calabi–Yau condition. No algorithm is known to the authors that determines whether a given singularity
-admits a crepant resolution.
-- Likewise, no general algorithm is known for computing a crepant resolution of a given singular geometry. In practice,
-one applies all known resolution techniques, guided by mathematical structure and physical expectations. A particularly
-prominent strategy is a sequence of **blowups**. We discuss the available blowup functionality in
-[Functionality for all F-theory models](@ref).
+- Not all singularities admit crepant resolutions, rather some singularities are obstructed from being resolved without violating the Calabi–Yau condition. No algorithm is known to the authors that determines whether a given singularity admits a crepant resolution.
+- Likewise, no general algorithm is known for computing a crepant resolution of a given singular geometry. In practice, one applies all known resolution techniques, guided by mathematical structure and physical expectations. A particularly prominent strategy is a sequence of **blowups**. We discuss the available blowup functionality in [Functionality for all F-theory models](@ref).
 
 After applying a resolution strategy, one obtains a **partially resolved** model. For the reasons stated above, OSCAR
 does not currently verify whether the model has been fully resolved—i.e., whether all resolvable singularities have been

experimental/FTheoryTools/docs/src/weierstrass.md

@@ -206,13 +206,8 @@ In F-theory, the standard approach to handling singular geometries is to replace
 via **crepant resolutions**. This process preserves the Calabi–Yau condition and ensures the correct encoding
 of physical data. However, several important caveats apply:
 
-- Not all singularities admit crepant resolutions, rather some singularities are obstructed from being resolved
-without violating the Calabi–Yau condition. No algorithm is known to the authors that determines whether a
-given singularity admits a crepant resolution.
-- Likewise, no general algorithm is known for computing a crepant resolution of a given singular geometry.
-In practice, one applies all known resolution techniques, guided by mathematical structure and physical
-expectations. A particularly prominent strategy is a sequence of **blowups**. We discuss the available blowup
-functionality in [Functionality for all F-theory models](@ref).
+- Not all singularities admit crepant resolutions, rather some singularities are obstructed from being resolved without violating the Calabi–Yau condition. No algorithm is known to the authors that determines whether a given singularity admits a crepant resolution.
+- Likewise, no general algorithm is known for computing a crepant resolution of a given singular geometry. In practice, one applies all known resolution techniques, guided by mathematical structure and physical expectations. A particularly prominent strategy is a sequence of **blowups**. We discuss the available blowup functionality in [Functionality for all F-theory models](@ref).
 
 After applying a resolution strategy, one obtains a **partially resolved** model. For the reasons stated above,
 OSCAR does not currently verify whether the model has been fully resolved—i.e., whether all resolvable

experimental/FTheoryTools/src/FamilyOfG4Fluxes/constructors.jl

@@ -10,7 +10,7 @@ identify ambient space candidates of G4-fluxes. In terms of these
 candidates, we can define a family of G4-fluxes as:
 - ``\mathbb{Z}``-linear combinations, provided by a matrix ``\text{mat}_{\text{int}}``,
 - ``\mathbb{Q}``-linear combinations, provided by a matrix ``\text{mat}_{\text{rat}}``,
-- a shift ``---``resembling the appearance of ``\frac{1}{2} \cdot c_2`` in the flux quantization condition``---`` provided by a vector ``\text{offset}``.
+- a shift—resembling the appearance of ``\frac{1}{2} \cdot c_2`` in the flux quantization condition—provided by a vector ``\text{offset}``.
 
 For convenience we also allow to only provide ``\text{mat}_{\text{int}}``or ``\text{mat}_{\text{rat}}``. In this case, the shift is taken to be zero.
 

experimental/FTheoryTools/src/G4Fluxes/auxiliary.jl

@@ -400,7 +400,7 @@ and ``b``-th toric variables defines a cohomology class on the ambient toric var
 ``H^{2,2}(X_\Sigma, \mathbb{Q})`` to ``\widehat{Y}_4``.
 
 This symbolic representation can be used to reconstruct cohomology class generators 
-via Cox ring monomials. For the actual cohomology class generators, see ``gens_of_h22_hypersurface``.
+via Cox ring monomials. For the actual cohomology class generators, see `gens_of_h22_hypersurface`.
 
 Use `check = false` to skip completeness and simplicity verification.
 
@@ -502,10 +502,10 @@ to linear combinations of generators of ``S \subseteq H^{2,2}(\widehat{Y}_4, \ma
 where ``\widehat{Y}_4`` is the (smooth) hypersurface associated to the F-theory model and
 ``S`` the restriction of ``H^{2,2}(X_\Sigma, \mathbb{Q})`` to ``\widehat{Y}_4``.
 
-The generating set of ``S`` is the one returned by ``gens_of_h22_hypersurface``, 
+The generating set of ``S`` is the one returned by `gens_of_h22_hypersurface`,
 and this converter enables expressing any ambient class in terms of these restricted generators.
 
-For the analogous map in the ambient toric variety, see ``converter_dict_h22_ambient``.
+For the analogous map in the ambient toric variety, see `converter_dict_h22_ambient`.
 
 Use `check = false` to skip completeness and simplicity verification.
 
@@ -548,10 +548,10 @@ end
 Given an F-theory model `m` defined as a hypersurface in a simplicial and
 complete toric space ``X_\Sigma``, this method computes a basis of
 ``H^{2,2}(X_\Sigma, \mathbb{Q})`` (using the method `basis_of_h22`) and then
-filters out ``---``based on fairly elementary, sufficent but not necessary checks``---``
-basis elements whose restriction to the hypersurface in question is non-trivial. The
-list of these basis elements ---``cast into G4-flux ambient space candidates``---
-is then returned by this method.
+filters out—based on fairly elementary, sufficent but not necessary checks—basis
+elements whose restriction to the hypersurface in question is non-trivial. The
+list of these basis elements—cast into ``G_4``-flux ambient space candidates—is then
+returned by this method.
 
 Use `check = false` to skip completeness and simplicity verification.
 

experimental/FTheoryTools/src/TateModels/methods.jl

@@ -5,7 +5,6 @@ Determine the fiber of a (singular) global Tate model over a particular base loc
 
 !!! warning
     This method may run for very long time and is currently not tested as part of the regular OSCAR CI due to its excessive run times.
-```
 """
 function analyze_fibers(model::GlobalTateModel, centers::Vector{<:Vector{<:Integer}})