# Weierstrass models

A Weierstrass model describes a particular form of an elliptic fibration. We focus on an elliptic fibration over a complete base $B$. Consider the weighted projective space $\mathbb{P}^{2,3,1}$ with coordinates $x, y, z$. In addition, consider

- $f \in H^0( B, \overline{K}_{B}^{\otimes 4} )$,
- $g \in H^0( B, \overline{K}_{B}^{\otimes 6} )$,

Then form a $\mathbb{P}^{2,3,1}$-bundle over $B$ such that

- $x$ transforms as a section of $2 \overline{K}_{B}$,
- $y$ transforms as a section of $3 \overline{K}_{B}$,
- $z$ transforms as a section of $0 \overline{K}_{B} = \mathcal{O}_{B}$.

In this 5-fold ambient space, a Weierstrass model is the hypersurface defined by the vanishing of the Weierstrass polynomial $P_W = x^3 - y^2 + f x z^4 + g z^6$.

Crucially, for non-trivial F-theory settings, the elliptic fibration in question must be singular. In fact, by construction, one usually engineers certain singularities. This can be read-off from the Weierstrass table, which we have reproduced from Timo Weigand (2018) with small corrections:

type | $\mathrm{ord}(f)$ | $\mathrm{ord}(g)$ | $\mathrm{ord}(\Delta)$ | sing. | monodromy cover | algebra $\mathfrak{g}$ | comp. |
---|---|---|---|---|---|---|---|

$I_0$ | $\geq 0$ | $\geq 0$ | 0 | ||||

$I_1$ | $0$ | $0$ | $1$ | ||||

$II$ | $\geq 1$ | $1$ | $2$ | ||||

$III$ | $1$ | $\geq 2$ | $3$ | $A_1$ | $\mathfrak{su}(2)$ | ||

$IV^{ns}$ | $\geq 2$ | $2$ | $4$ | $A_2$ | $\left. \psi^2 - \frac{g}{w^2} \right|_{w = 0}$ | $\mathfrak{sp}(1)$ | $1$ |

$IV^{s}$ | $\geq 2$ | $2$ | $4$ | $A_2$ | $\left. \psi^2 - \frac{g}{w^2} \right|_{w = 0}$ | $\mathfrak{su}(3)$ | $2$ |

$I_m^{ns}$ | $0$ | $0$ | $m$ | $A_{m - 1}$ | $\left. \psi^2 + \frac{9g}{2f} \right|_{w = 0}$ | $\mathfrak{sp}(\lfloor \frac{m}{2} \rfloor])$ | $1$ |

$I_m^{s}$ | $0$ | $0$ | $m$ | $A_{m - 1}$ | $\left. \psi^2 + \frac{9g}{2f} \right|_{w = 0}$ | $\mathfrak{su}(m)$ | $2$ |

$I_0^{*ns}$ | $\geq 2$ | $\geq 3$ | $6$ | $D_4$ | $\left. \psi^3 + \psi \cdot \frac{f}{w^2} + \frac{g}{w^3} \right|_{w = 0}$ | $\mathfrak{g}_2$ | $1$ |

$I_0^{*ss}$ | $\geq 2$ | $\geq 3$ | $6$ | $D_4$ | $\left. \psi^3 + \psi \cdot \frac{f}{w^2} + \frac{g}{w^3} \right|_{w = 0}$ | $\mathfrak{so}(7)$ | $2$ |

$I_0^{*s}$ | $\geq 2$ | $\geq 3$ | $6$ | $D_4$ | $\left. \psi^3 + \psi \cdot \frac{f}{w^2} + \frac{g}{w^3} \right|_{w = 0}$ | $\mathfrak{so}(8)$ | $3$ |

$I_{2n-5}^{*ns}$ ($n \geq 3$) | $2$ | $3$ | $2n+1$ | $D_{2n-1}$ | $\left. \psi^2 + \frac{1}{4} \left( \frac{\Delta}{w^{2n+1}} \right) \left( \frac{2wf}{9g} \right)^3 \right|_{w = 0}$ | $\mathfrak{so}(4n-3)$ | $1$ |

$I_{2n-5}^{*s}$ ($n \geq 3$) | $2$ | $3$ | $2n+1$ | $D_{2n-1}$ | $\left. \psi^2 + \frac{1}{4} \left( \frac{\Delta}{w^{2n+1}} \right) \left( \frac{2wf}{9g} \right)^3 \right|_{w = 0}$ | $\mathfrak{so}(4n-2)$ | $2$ |

$I_{2n-4}^{*ns}$ ($n \geq 3$) | $2$ | $3$ | $2n+2$ | $D_{2n}$ | $\left. \psi^2 + \left( \frac{\Delta}{w^{2n+2}} \right) \left( \frac{2wf}{9g} \right)^2 \right|_{w = 0}$ | $\mathfrak{so}(4n-1)$ | $1$ |

$I_{2n-4}^{*s}$ ($n \geq 3$) | $2$ | $3$ | $2n+2$ | $D_{2n}$ | $\left. \psi^2 + \left( \frac{\Delta}{w^{2n+2}} \right) \left( \frac{2wf}{9g} \right)^2 \right|_{w = 0}$ | $\mathfrak{so}(4n)$ | $2$ |

$IV^{*ns}$ | $\geq 3$ | $4$ | $8$ | $E_6$ | $\left. \psi^2 - \frac{g}{w^4} \right|_{w = 0}$ | $\mathfrak{f}_4$ | $1$ |

$IV^{*s}$ | $\geq 3$ | $4$ | $8$ | $E_6$ | $\left. \psi^2 - \frac{g}{w^4} \right|_{w = 0}$ | $\mathfrak{e}_6$ | $2$ |

$III^*$ | $3$ | $\geq 5$ | $9$ | $E_7$ | $\mathfrak{e}_7$ | ||

$II^*$ | $\geq 4$ | $5$ | $10$ | $E_8$ | $\mathfrak{e}_8$ | ||

non-min. | $\geq 4$ | $\geq 6$ | $\geq 12$ | non-can. |

## Constructors

We aim to provide support for Weierstrass models over the following bases:

- a toric variety,
- a toric scheme,
- a (covered) scheme.

Often, one also wishes to obtain information about a Weierstrass model without explicitly specifying the base space. Also for this application, we provide support. Finally, we provide support for some standard constructions.

Before we detail these constructors, we must comment on the constructors over toric base spaces. Namely, in order to construct a Weierstrass model as a hypersurface in an ambient space, we first wish to construct the ambient space in question. For a toric base, one way to achieve this is to first focus on the Cox ring of the toric ambient space. This ring must be graded such that the Weierstrass polynomial is homogeneous and cuts out a Calabi-Yau hypersurface. Given this grading, one can perform a triangulation task. Typically, this combinatorial task is very demanding, consumes a lot of computational power and takes a long time to complete. Even more, it will yield a large, often huge, number of candidate ambient spaces of which the typical user will only pick one. For instance, a common and often appropriate choice is a toric ambient space which contains the toric base space in a manifest way.

To circumvent this very demanding computation, our toric constructors operate in the opposite direction. That is, they begin by extracting the rays and maximal cones of the chosen toric base space. Subsequently, those rays and cones are extended to form one of the many toric ambient spaces. This proves hugely superior in performance than going through the triangulation task of enumerating all possible toric ambient spaces. One downside of this strategy is that the so-constructed ambient space need not be smooth.

### A toric variety as base space

We require that the provided toric base space is complete. This is a technical limitation as of now. The functionality of OSCAR only allows us to compute a section basis (or a finite subset thereof) for complete toric varieties. In the future, this could be extended.

However, completeness is an expensive check. Therefore, we provide an optional argument which one can use to disable this check if desired. To this end, one passes the optional argument `completeness_check = false`

as last argument to the constructor. The following examples demonstrate this:

`weierstrass_model`

— Method`weierstrass_model(base::NormalToricVariety; completeness_check::Bool = true)`

This method constructs a Weierstrass model over a given toric base 3-fold. The Weierstrass sections $f$ and $g$ are taken with (pseudo)random coefficients.

**Examples**

```
julia> w = weierstrass_model(sample_toric_variety(); completeness_check = false)
Weierstrass model over a concrete base
```

`weierstrass_model`

— Method`weierstrass_model(base::NormalToricVariety, f::MPolyRingElem, g::MPolyRingElem; completeness_check::Bool = true)`

This method operates analogously to `weierstrass_model(base::NormalToricVarietyType)`

. The only difference is that the Weierstrass sections $f$ and $g$ can be specified with non-generic values.

**Examples**

```
julia> base = sample_toric_variety()
Normal toric variety
julia> f = sum([rand(Int) * b for b in basis_of_global_sections(anticanonical_bundle(base)^4)]);
julia> g = sum([rand(Int) * b for b in basis_of_global_sections(anticanonical_bundle(base)^6)]);
julia> w = weierstrass_model(base, f, g; completeness_check = false)
Weierstrass model over a concrete base
```

### A (covered) scheme as base space

This functionality does not yet exist.

### Base space not specified

A Weierstrass model can also be constructed over a base space that is not fully specified. Rather, it assumes that a base space exists such that the Weierstrass sections $f$ and $g$ are well-defined, so that the Weierstrass model in question is well-defined.

For many practical applications, one wishes to assume a further specialize the Weierstrass sections $f$ and $g$. This has the advantage that one can engineer singularity loci or even the singularity type over a specific locus. To some extend, this is the backbone of many F-theory constructions. It is useful to consider a polynomial ring whose variables are the sections used in the desired factorization of the Weierstrass sections $f$ and $g$. In theory, one can consider the indeterminates of this polynomial ring as local coordinate of an auxiliary base space. Indeed, for our computer implementation the polynomial ring with these indeterminates serve as the coordinate ring of an auxiliary *toric* base space. Despite this auxiliary base space being toric, the predictions from such an analysis are not limited to the world of toric varieties.

For constructions along these lines, we support the following constructor:

`weierstrass_model`

— Method`weierstrass_model(auxiliary_base_ring::MPolyRing, auxiliary_base_grading::Matrix{Int64}, d::Int, weierstrass_f::MPolyRingElem, weierstrass_g::MPolyRingElem; toric_sample = true)`

This method constructs a Weierstrass model over a base space that is not fully specified.

Note that many studies in the literature use the class of the anticanonical bundle in their analysis. We anticipate this by adding this class as a variable of the auxiliary base space, unless the user already provides this grading. Our convention is that the first grading refers to Kbar and that the homogeneous variable corresponding to this class carries the name "Kbar".

The following example illustrates this approach.

**Examples**

```
julia> auxiliary_base_ring, (f, g, Kbar, v) = QQ["f", "g", "Kbar", "u"]
(Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[f, g, Kbar, u])
julia> auxiliary_base_grading = [4 6 1 0]
1×4 Matrix{Int64}:
4 6 1 0
julia> w = weierstrass_model(auxiliary_base_ring, auxiliary_base_grading, 3, f, g)
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base
julia> w2 = weierstrass_model(auxiliary_base_ring, auxiliary_base_grading, 3, f, g; toric_sample = false)
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base
```

### Standard constructions

We provide convenient constructions of Weierstrass models over famous base spaces. Currently, we support the following:

`weierstrass_model_over_projective_space`

— Method`weierstrass_model_over_projective_space(d::Int)`

This method constructs a Weierstrass model over the projective space.

**Examples**

```
julia> weierstrass_model_over_projective_space(3)
Weierstrass model over a concrete base
```

`weierstrass_model_over_hirzebruch_surface`

— Method`weierstrass_model_over_hirzebruch_surface(r::Int)`

This method constructs a Weierstrass model over a Hirzebruch surface.

**Examples**

```
julia> weierstrass_model_over_hirzebruch_surface(1)
Weierstrass model over a concrete base
```

`weierstrass_model_over_del_pezzo_surface`

— Method`weierstrass_model_over_del_pezzo_surface(b::Int)`

This method constructs a Weierstrass model over a del-Pezzo surface.

**Examples**

```
julia> weierstrass_model_over_del_pezzo_surface(3)
Weierstrass model over a concrete base
```

### Literature models

Certain Weierstrass 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. Currently, we provide support for the following:

`su5_weierstrass_model_over_arbitrary_3d_base`

— Method`su5_weierstrass_model_over_arbitrary_3d_base()`

Return the SU(5) Weierstrass model over an arbitrary 3-dimensional base space. For more details see e.g. Timo Weigand (2018) and references therein.

```
julia> tm = su5_weierstrass_model_over_arbitrary_3d_base()
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base
```

## Attributes

### Basic attributes

For all Weierstrass models – irrespective over whether the base is toric or not – we support the following attributes:

`weierstrass_section_f`

— Method`weierstrass_section_f(w::WeierstrassModel)`

Return the polynomial $f$ used for the construction of the Weierstrass model.

```
julia> w = su5_weierstrass_model_over_arbitrary_3d_base()
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base
julia> weierstrass_section_f(w);
```

`weierstrass_section_g`

— Method`weierstrass_section_g(w::WeierstrassModel)`

Return the polynomial $g$ used for the construction of the Weierstrass model.

```
julia> w = su5_weierstrass_model_over_arbitrary_3d_base()
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base
julia> weierstrass_section_g(w);
```

`weierstrass_polynomial`

— Method`weierstrass_polynomial(w::WeierstrassModel)`

Return the Weierstrass polynomial of the Weierstrass model.

```
julia> w = su5_weierstrass_model_over_arbitrary_3d_base()
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base
julia> weierstrass_polynomial(w);
```

In case the Weierstrass model is constructed over a not fully specified base, recall that we construct an auxiliary (toric) base space as well as an auxiliary (toric) ambient space. The (auxiliary) base and ambient space can be accessed with the following functions:

`base_space`

— Method`base_space(w::WeierstrassModel)`

Return the base space of the Weierstrass model.

```
julia> w = su5_weierstrass_model_over_arbitrary_3d_base()
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base
julia> base_space(w)
Normal toric variety
```

`ambient_space`

— Method`ambient_space(w::WeierstrassModel)`

Return the base space of the Weierstrass model.

```
julia> w = su5_weierstrass_model_over_arbitrary_3d_base()
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base
julia> ambient_space(w)
Normal toric variety
```

`fiber_ambient_space`

— Method`fiber_ambient_space(w::WeierstrassModel)`

Return the fiber ambient space of the Weierstrass model.

```
julia> w = su5_weierstrass_model_over_arbitrary_3d_base()
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base
julia> fiber_ambient_space(w)
Normal, non-affine, simplicial, projective, 2-dimensional toric variety without torusfactor
```

The following method allows to tell if the base/ambient space is auxiliary or not:

`base_fully_specified`

— Method`base_fully_specified(w::WeierstrassModel)`

Return `true`

is the Weierstrass model has a concrete base space and `false`

otherwise.

```
julia> w = su5_weierstrass_model_over_arbitrary_3d_base()
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base
julia> base_fully_specified(w)
false
```

The user can decide to get an information whenever an auxiliary base space, auxiliary ambient space or auxiliary hypersurface have been computed. To this end, one invokes `set_verbosity_level(:WeierstrassModel, 1)`

. More background information is available here.

### Advanced attributes

The following attributes are currently only supported in a toric setting:

`calabi_yau_hypersurface`

— Method`calabi_yau_hypersurface(w::WeierstrassModel)`

Return the Calabi-Yau hypersurface in the toric ambient space which defines the Weierstrass model.

```
julia> w = su5_weierstrass_model_over_arbitrary_3d_base()
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base
julia> calabi_yau_hypersurface(w)
Closed subvariety of a normal toric variety
```

Note that for applications in F-theory, *singular* elliptic fibrations are key (cf. Timo Weigand (2018) and references therein). Consequently the discriminant locus as well as the singular loci of the fibration in question are of ample importance:

`discriminant`

— Method`discriminant(w::WeierstrassModel)`

Return the discriminant $\Delta = 4 f^3 + 27 g^2$.

```
julia> w = su5_weierstrass_model_over_arbitrary_3d_base()
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base
julia> discriminant(w);
```

`singular_loci`

— Method`singular_loci(w::WeierstrassModel)`

Return the singular loci of the Weierstrass model, along with the order of vanishing of $(f, g, \Delta)$ at each locus and the refined Tate fiber type.

For the time being, we either explicitly or implicitly focus on toric varieties as base spaces. Explicitly, in case the user provides such a variety as base space, and implicitly, in case we work over a non-fully specified base. This has the advantage that we can "filter out" trivial singular loci.

Specifically, recall that every closed subvariety of a simplicial toric variety is of the form $V(I)$, where $I$ is a homogeneous ideal of the Cox ring. Let $B$ be the irrelevant ideal of this toric variety. Then, by proposition 5.2.6. of David A. Cox, John B. Little, Henry K. Schenck (2011), $V(I)$ is trivial/empty iff $B^l \subseteq I$ for a suitable $l \geq 0$. This can be checked by checking if the saturation $I:B^\infty$ is the ideal generated by $1$.

By treating a not-fully specified base space implicitly as toric space, we can extend this result straightforwardly to this situation also. This is the reason for constructing this auxiliary base space.

```
julia> w = su5_weierstrass_model_over_arbitrary_3d_base()
Assuming that the first row of the given grading is the grading under Kbar
Weierstrass model over a not fully specified base
julia> length(singular_loci(w))
2
```

## Methods

### Towards resolution of singularities

To come.