Global Tate Models

Global Tate models provide a powerful framework to systematically engineer elliptic fibrations with prescribed fiber singularities. They are widely used in F-theory model building because they make it easier to realize certain gauge symmetries and matter spectra geometrically. Their structure is governed by Tate’s algorithm, which relates the vanishing orders of specific polynomial coefficients to fiber singularities—classified via the refined Tate table.

What Is a Global Tate Model?

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

$a_1 \in H^0(B, \overline{K}_{B})$,
$a_2 \in H^0(B, \overline{K}_{B}^{\otimes 2})$,
$a_3 \in H^0(B, \overline{K}_{B}^{\otimes 3})$,
$a_4 \in H^0(B, \overline{K}_{B}^{\otimes 4})$,
$a_6 \in H^0(B, \overline{K}_{B}^{\otimes 6})$.

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 $\mathcal{O}_B$.

In this 5-fold ambient space, a global Tate model is the hypersurface defined by the vanishing of the Tate polynomial:

\[P_T = x^3 - y^2 - x y z a_1 + x^2 z^2 a_2 - y z^3 a_3 + x z^4 a_4 + z^6 a_6\,.\]

In a sufficiently small open neighborhood $U \subset B$ of a point $p \in B$, the polynomial locally takes the form:

\[y^2 + a_1(q) x y z + a_3(q) y z^3 = x^3 + a_2(q) x^2 z^2 + a_4(q) x z^4 + a_6(q) z^6\,,\]

where the $a_i(q)$ are the values of the Tate sections at a local coordinate $q \in U$. This formulation is known as a Tate model. One may define the elliptic fibration globally by specifying a single Tate polynomial as above. Such constructions, known as global Tate models, are strictly less general than Weierstrass Models but have proven especially useful for model building in F-theory: Engineering the desired fiber singularities—crucial for F-theory model building—is often simpler for global Tate models than for Weierstrass Models.

Much like the Kodaira classification, in a global Tate model the singularities of the elliptic fiber are characterized by the vanishing orders of the Tate sections $a_i$, although no additional monodromy needs to be taken into account (with a few notable exceptions). The complete classification is summarized in the following table—commonly referred to as the Tate table—which is taken from [Wei10]:

sing. type	$\mathrm{ord}(\Delta)$	singularity	group $G$	$a_1$	$a_2$	$a_3$	$a_4$	$a_6$
$I_0$	$0$			$0$	$0$	$0$	$0$	$0$
$I_1$	$1$			$0$	$0$	$1$	$1$	$1$
$I_2$	$2$	$A_1$	$SU(2)$	$0$	$0$	$1$	$1$	$2$
$I_{2k}^{ns}$	$2k$	$C_k$	$Sp(k)$	$0$	$0$	$k$	$k$	$2k$
$I_{2k}^s$	$2k$	$A_{2k-1}$	$SU(2k)$	$0$	$1$	$k$	$k$	$2k$
$I_{2k+1}^{ns}$	$2k+1$		$Sp(k)$	$0$	$0$	$k+1$	$k+1$	$2k+1$
$I_{2k+1}^{s}$	$2k+1$	$A_{2k}$	$SU(2k+1)$	$0$	$1$	$k$	$k+1$	$2k+1$
$II$	$2$			$1$	$1$	$1$	$1$	$1$
$III$	$3$	$A_1$	$SU(2)$	$1$	$1$	$1$	$1$	$2$
$IV^{ns}$	$4$		$Sp(1)$	$1$	$1$	$1$	$2$	$2$
$IV^s$	$4$	$A_2$	$SU(3)$	$1$	$1$	$1$	$2$	$3$
$I_0^{*ns}$	$6$	$G_2$	$G_2$	$1$	$1$	$2$	$2$	$3$
$I_0^{*ss}$	$6$	$B_3$	$SO(7)$	$1$	$1$	$2$	$2$	$4$
$I_0^{*s}$	$6$	$D_4$	$SO(8)$	$1$	$1$	$2$	$2$	$4$
$I_1^{*ns}$	$7$	$B_4$	$SO(9)$	$1$	$1$	$2$	$3$	$4$
$I_1^{*s}$	$7$	$D_5$	$SO(10)$	$1$	$1$	$2$	$3$	$5$
$I_2^{*ns}$	$8$	$B_5$	$SO(11)$	$1$	$1$	$3$	$3$	$5$
$I_2^{*s}$	$8$	$D_6$	$SO(12)$	$1$	$1$	$3$	$3$	$5$
$I_{2k-3}^{*ns}$	$2k+3$	$B_{2k}$	$SO(4k+1)$	$1$	$1$	$k$	$k+1$	$2k$
$I_{2k-3}^{*s}$	$2k+3$	$D_{2k+1}$	$SO(4k+2)$	$1$	$1$	$k$	$k+1$	$2k+1$
$I_{2k-2}^{*ns}$	$2k+4$	$B_{2k+1}$	$SO(4k+3)$	$1$	$1$	$k+1$	$k+1$	$2k+1$
$I_{2k-2}^{*s}$	$2k+4$	$D_{2k+2}$	$SO(4k+4)$	$1$	$1$	$k+1$	$k+1$	$2k+1$
$IV^{*ns}$	$8$	$F_4$	$F_4$	$1$	$2$	$2$	$3$	$4$
$IV^{*s}$	$8$	$E_6$	$E_6$	$1$	$2$	$2$	$3$	$5$
$III^*$	$9$	$E_7$	$E_7$	$1$	$2$	$3$	$3$	$5$
$II^*$	$10$	$E_8$	$E_8$	$1$	$2$	$3$	$4$	$5$
non-min.	$12$			$1$	$2$	$3$	$4$	$6$

Constructing Global Tate Models

Global Tate 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 outlines constructing global Tate models over arbitrary bases, but primarily focuses on working with a concrete toric base.

Unspecified Base Spaces

As with Weierstrass Models, constructing a global Tate model begins with building a suitable ambient space. When the base is not fully specified, the ambient space can be treated symbolically. This symbolic flexibility is particularly useful when engineering singular fibers via specified vanishing orders or factorizations of the Tate sections $a_i$. This method is foundational to F-theory model building.

To support such workflows, the Tate sections $a_i$ are defined as indeterminates of a multivariate polynomial ring. The indeterminates are interpreted as sections of line bundles on an auxiliary base space. This allows engineering desired vanishing profiles and exploring families of models over a space of base geometries. A typical use case involves factoring the Tate sections $a_i$ to control the singularity type over a specific divisor. For instance, to realize an $I_5$ fiber over the divisor $\{w = 0\}$ in the unspecified base space, one might set:

$a_1 = a_{10} \times w^0$,
$a_2 = a_{21} \times w^1$,
$a_3 = a_{32} \times w^2$,
$a_4 = a_{43} \times w^3$,
$a_6 = a_{65} \times w^5$.

This formalism underlies many symbolic constructions in F-theory model building. The mathematical interpretation of symbolic bases is subtle and under active development. Users can construct such models with the following constructor:

global_tate_model — Method

global_tate_model(auxiliary_base_ring::MPolyRing, auxiliary_base_grading::Matrix{Int64}, d::Int, ais::Vector{T}) where {T<:MPolyRingElem}

This method constructs a global Tate 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 code exemplifies this approach.

Examples

julia> auxiliary_base_ring, (a10, a21, a32, a43, a65, w) = QQ[:a10, :a21, :a32, :a43, :a65, :w];

julia> auxiliary_base_grading = [1 2 3 4 6 0; 0 -1 -2 -3 -5 1]
2×6 Matrix{Int64}:
 1   2   3   4   6  0
 0  -1  -2  -3  -5  1

julia> a1 = a10;

julia> a2 = a21 * w;

julia> a3 = a32 * w^2;

julia> a4 = a43 * w^3;

julia> a6 = a65 * w^5;

julia> ais = [a1, a2, a3, a4, a6];

julia> t = global_tate_model(auxiliary_base_ring, auxiliary_base_grading, 3, ais)
Assuming that the first row of the given grading is the grading under Kbar

Global Tate model over a not fully specified base

Experimental

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