Hypersurface models

Introduction

A hypersurface model describes a particular form of an elliptic fibration. We make the following assumptions:

The generic fiber is a hypersurface in a 2-dimensional toric fiber ambient space $F$.
The first two (homogeneous) coordinates in the coordinate ring of $F$ transform in the

divisor classes $D_1$ and $D_2$ over the base $B$ of the elliptic fibration. The remaining coordinates of $F$ are assumed to transform in the trivial bundle over $B$. See [KM-POPR15] for more details.

Our tools are most powerful if the base space $B$ is a toric variety. Then, based on the above information, it is possible to compute a toric ambient space $A$ for the elliptic fibration. The elliptic fibration is then a hypersurface in this toric space $A$. Furthermore, since we assume that this fibration is Calabi-Yau, it is clear that the hypersurface equation is a (potentially very special) section of the $\overline{K}_A$. This hypersurface equation completes the information required about a hypersurface model.

Certainly, one often wishes to extend beyond this setting:

Oftentimes, a special hypersurface equation is chosen. In the toric setting above, this will

be a polynomial in the Cox ring of the toric ambient space. Since said ambient space, and therefore also its Cox ring, are computed in the cause of the above construction, we do not support one constructor, which immediately accepts a special hypersurface equation. Rather, in this case, the user must first use one of the general constructors described in the next section. Subsequently, the tune function, described in the methods section below, can be employed.

Bases other than toric spaces matter. Often, the F-theory literature will not even assume

one particular base space but rather an entire family of base spaces. We extend our machinery to these cases, but such more general settings are typically significantly more limited than the toric setting. This limitation originates from the nature of the matter – the more general the geometry, the less is known.

Constructors

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

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

[Often, one also wishes to obtain information about a hypersurface model without explicitly specifying the base space. For this, please use our tune function, described in the methods section below.]

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 hypersurface model, we first have to construct the ambient space in question. For a toric base, one way to achieve this is by means of triangulations. However, this is a rather time consuming and computationally challenging task, which leads to a huge number of ambient spaces. Even more, typically one wishes to only pick one of thees many ambient spaces. 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 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.

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:

hypersurface_model — Method

hypersurface_model(base::NormalToricVariety; completeness_check::Bool = true)

Construct a hypersurface model. This constructor takes $\mathbb{P}^{2,3,1}$ as fiber ambient space with coordinates $[x:y:z]$ and ensures that $x$ transforms as $2 \overline{K}_{B_3}$ and $y$ as $3 \overline{K}_{B_3}$.

Examples

julia> base = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> hypersurface_model(base; completeness_check = false)
Hypersurface model over a concrete base