Weierstrass Models
Weierstrass models are central to many constructions in F-theory. Such a model describes a particular form of an elliptic fibration. Our primary focus is on Weierstrass models built over toric base spaces. These models receive the most comprehensive support. Weierstrass models over more general bases, such as arbitrary schemes or unspecified base spaces, are under development.
What Is a Weierstrass Model?
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 [Wei18] 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. |
Constructing Weierstrass Models
Weierstrass models are most robustly supported over toric base spaces. While there are plans to extend support to toric schemes, covered schemes, and unspecified base families, these directions remain experimental and are reserved for future development. This section briefly discusses constructing Weierstrass models over arbitrary bases, but primarily focuses on working with a concrete toric base.
Unspecified Base Spaces
As mentioned, a Weierstrass model is defined as a hypersurface within an ambient space. Thus, constructing the ambient space is the first step. When working with a family of base spaces, this ambient space can be treated symbolically—flexible enough to represent an entire family while still allowing algebraic manipulations. In particular, it is often useful to construct a Weierstrass model without explicitly specifying the base space. This is especially common when engineering singularities through specific factorizations of the Weierstrass sections f and g, as seen in the Weierstrass table. This method is prevalent in F-theory model building, where singularities are engineered independently of the full base geometry.
To facilitate such workflows, users can define the Weierstrass sections f and g as elements in a multivariate polynomial ring. The variables in this ring are interpreted as sections of line bundles on the auxiliary base space.
The mathematical meaning and implementation details of symbolic base families are subtle and under active discussion. We do not elaborate on them further here. Let us only mention that users can create such models with the following constructor, though support for these models may be limited:
weierstrass_model — Methodweierstrass_model(auxiliary_base_ring::MPolyRing, auxiliary_base_grading::Matrix{Int64}, d::Int, weierstrass_f::MPolyRingElem, weierstrass_g::MPolyRingElem)Constructs a Weierstrass model over an unspecified base space.
This method is intended for workflows where the base space is not concretely fixed, such as in singularity engineering or symbolic studies. The variables in auxiliary_base_ring are interpreted as sections of line bundles on the (unspecified) base space. The corresponding line bundles are encoded by the auxiliary_base_grading matrix.
Grading convention:
- The first row of the grading corresponds to twists by the anticanonical bundle $\overline{K}_B$.
- Subsequent rows correspond to additional user-defined gradings.
To support typical F-theory constructions, a variable Kbar representing a section of $\overline{K}_B$ is automatically included unless already provided.
Note: This interface is more symbolic and less robust than the constructors for concrete toric bases.
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 baseThis function is part of the experimental code in Oscar. Please read here for more details.
To check whether a Weierstrass model has been constructed over a concrete base space or over a symbolic family, use is_base_space_fully_specified. This returns true if the base is fully specified, and false otherwise.
Concrete Toric Base Spaces
Full support exists for constructing Weierstrass models over concrete, complete toric base spaces. Completeness is a technical assumption: it ensures that the set of global sections of a line bundle forms a finite-dimensional vector space, enabling OSCAR to handle these sets efficiently. In the future, this restriction may be relaxed. For now, completeness checks—though sometimes slow—are performed in many methods involving Weierstrass models. To skip them and improve performance, use the optional keyword:
completeness_check = falseWe proceed under the assumption that the base space is a fixed, complete toric variety.
In this setting, constructing a suitable ambient space becomes significantly more efficient. First, we focus on toric ambient spaces. Second, rather than enumerating large numbers of ambient spaces from polytope triangulations—a resource-intensive process—we adopt a performance-optimized approach. Starting from the rays and maximal cones of the toric base, we extend them to construct a compatible polyhedral fan defining the toric ambient space. This avoids triangulation entirely and yields a single suitable ambient toric variety. The resulting space may not be smooth and may differ from those commonly used in the F-theory literature, but it is guaranteed to be compatible with the elliptic fibration structure.
Users can construct Weierstrass models over such concrete toric bases with the following constructors:
weierstrass_model — Methodweierstrass_model(base::NormalToricVariety; completeness_check::Bool = true)Constructs a Weierstrass model over a given toric base space. The Weierstrass sections $f$ and $g$ are automatically generated with (pseudo)random coefficients.
julia> w = weierstrass_model(projective_space(NormalToricVariety, 2); completeness_check = false)
Weierstrass model over a concrete baseThis function is part of the experimental code in Oscar. Please read here for more details.
weierstrass_model — Methodweierstrass_model(base::NormalToricVariety, f::MPolyRingElem, g::MPolyRingElem; completeness_check::Bool = true)Constructs a Weierstrass model over a given toric base space $X$. The Weierstrass sections $f$ and $g$ are explicitly specified by the user as polynomials in the Cox ring of $X$.
julia> chosen_base = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> f = generic_section(anticanonical_bundle(chosen_base)^4);
julia> g = generic_section(anticanonical_bundle(chosen_base)^6);
julia> w = weierstrass_model(chosen_base, f, g; completeness_check = false)
Weierstrass model over a concrete baseThis function is part of the experimental code in Oscar. Please read here for more details.
For convenience—ideal for quick experiments and educational use—we also support constructors for Weierstrass models over commonly used base spaces:
weierstrass_model_over_projective_space — Methodweierstrass_model_over_projective_space(d::Int)Constructs a Weierstrass model over the $d$-dimensional projective space, represented as a toric variety. The Weierstrass sections $f$ and $g$ are automatically generated with pseudorandom coefficients.
julia> weierstrass_model_over_projective_space(3)
Weierstrass model over a concrete baseThis function is part of the experimental code in Oscar. Please read here for more details.
weierstrass_model_over_hirzebruch_surface — Methodweierstrass_model_over_hirzebruch_surface(r::Int)Constructs a Weierstrass model over the Hirzebruch surface $F_r$, represented as a toric variety. The Weierstrass sections $f$ and $g$ are automatically generated with pseudorandom coefficients.
julia> weierstrass_model_over_hirzebruch_surface(1)
Weierstrass model over a concrete baseThis function is part of the experimental code in Oscar. Please read here for more details.
weierstrass_model_over_del_pezzo_surface — Methodweierstrass_model_over_del_pezzo_surface(b::Int)Constructs a Weierstrass model over the del Pezzo surface $\text{dP}_b$, represented as a toric variety. The Weierstrass sections $f$ and $g$ are automatically generated with pseudorandom coefficients.
julia> weierstrass_model_over_del_pezzo_surface(3)
Weierstrass model over a concrete baseThis function is part of the experimental code in Oscar. Please read here for more details.
Famous Weierstrass Models
Several Weierstrass models have gained popularity in the F-theory community. These models are often associated with specific publications and may be informally referred to by author names or recognizable keywords. For these established constructions, we provide support through the specialized literature_model interface, which is discussed on the page Literature Models.
Attributes of Weierstrass Models
Weierstrass models are one way to represent an elliptic fibration as a hypersurface in an ambient space. While different representations may vary in implementation details, they share a common structure in broad strokes. As such, many attributes and properties are shared across model representations. These shared components—such as base_space, ambient_space, and fiber_ambient_space—are documented on the page Functionality for all F-theory models.
Below, we list the attributes that are specific to Weierstrass models and do not generally apply to other representations (such as Global Tate Models or Hypersurface Models):
weierstrass_section_f — Methodweierstrass_section_f(w::WeierstrassModel)Returns the Weierstrass section $f$ of the Weierstrass model.
julia> w = weierstrass_model_over_projective_space(3)
Weierstrass model over a concrete base
julia> weierstrass_section_f(w);This function is part of the experimental code in Oscar. Please read here for more details.
weierstrass_section_g — Methodweierstrass_section_g(w::WeierstrassModel)Returns the Weierstrass section $g$ of the Weierstrass model.
julia> w = weierstrass_model_over_projective_space(3)
Weierstrass model over a concrete base
julia> weierstrass_section_g(w);This function is part of the experimental code in Oscar. Please read here for more details.
weierstrass_polynomial — Methodweierstrass_polynomial(w::WeierstrassModel)Returns the Weierstrass polynomial of the model.
Alias: hypersurface_equation(w::WeierstrassModel).
julia> w = weierstrass_model_over_projective_space(3)
Weierstrass model over a concrete base
julia> weierstrass_polynomial(w) == hypersurface_equation(w)
trueThis function is part of the experimental code in Oscar. Please read here for more details.
hypersurface_equation — Methodhypersurface_equation(w::WeierstrassModel)Alias for weierstrass_polynomial(w::WeierstrassModel).
This function is part of the experimental code in Oscar. Please read here for more details.
weierstrass_ideal_sheaf — Methodweierstrass_ideal_sheaf(w::WeierstrassModel)Returns the Weierstrass ideal sheaf of the Weierstrass model.
This method is relevant when the Weierstrass model cannot be represented by a single global polynomial—e.g., after non-toric blowups. In such cases, the model is defined locally by an ideal sheaf on each affine patch rather than by a global hypersurface equation.
julia> w = weierstrass_model_over_projective_space(2)
Weierstrass model over a concrete base
julia> weierstrass_ideal_sheaf(w)
Sheaf of ideals
on normal toric variety
with restrictions
1: Ideal with 1 generator
2: Ideal with 1 generator
3: Ideal with 1 generator
4: Ideal with 2 generators
5: Ideal with 2 generators
6: Ideal with 2 generators
7: Ideal with 5 generators
8: Ideal with 5 generators
9: Ideal with 5 generatorsThis function is part of the experimental code in Oscar. Please read here for more details.
calabi_yau_hypersurface — Methodcalabi_yau_hypersurface(w::WeierstrassModel)Returns the Calabi–Yau hypersurface that defines the Weierstrass model as a closed subvariety of its toric ambient space.
julia> w = weierstrass_model_over_projective_space(3)
Weierstrass model over a concrete base
julia> calabi_yau_hypersurface(w)
Closed subvariety of a normal toric varietyThis function is part of the experimental code in Oscar. Please read here for more details.
Singularities in Weierstrass Models
Let us emphasize again that in F-theory, singular elliptic fibrations are of central importance (cf. [Wei18] and references therein): singularities signal non-trivial physics.
A key quantity in this context is the discriminant locus of the fibration—i.e., the subset of the base space over which the elliptic fibers degenerate:
discriminant — Methoddiscriminant(w::WeierstrassModel)Returns the discriminant $\Delta = 4 f^3 + 27 g^2$ of the Weierstrass model.
julia> w = weierstrass_model_over_projective_space(3)
Weierstrass model over a concrete base
julia> discriminant(w);This function is part of the experimental code in Oscar. Please read here for more details.
Even more critical than the discriminant itself is its decomposition: the discriminant locus can split into multiple irreducible components. Over each such component, one can determine the singularity type of the fiber. The function singular_loci returns a list of these irreducible base loci, together with the corresponding singularity types:
singular_loci — Methodsingular_loci(w::WeierstrassModel)Returns 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.
Currently, the method either explicitly or implicitly assumes a toric variety as base space—either because the user provides one, or because an unspecified base is treated symbolically as such. This allows us to filter out trivial singular loci effectively.
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 [CLS11], $V(I)$ is trivial (i.e., empty) if and only if $B^l \subseteq I$ for some $l \geq 0$. This can be checked by testing whether the saturation $I : B^\infty$ equals the unit ideal.
By treating unspecified base spaces as symbolic toric varieties, we can extend this check to those cases as well.
Advanced technical details are available in BMT25.
The classification of singularities is based on a Monte Carlo algorithm, which involves random sampling. While extensively tested and highly reliable, the method’s probabilistic nature may lead to non-deterministic results in rare cases.
julia> w = weierstrass_model_over_projective_space(3)
Weierstrass model over a concrete base
julia> length(singular_loci(w))
1This function is part of the experimental code in Oscar. Please read here for more details.
We discuss singularities in greater depth—including how to deform models to achieve a desired singularity structure and how to resolve them—in Resolving F-Theory Models.