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 [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.

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

Experimental

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