Tate's Algorithm for F-theory GUTs with two U(1)s

We present a systematic study of elliptic fibrations for F-theory realizations of gauge theories with two U(1) factors. In particular, we determine a new class of SU(5) x U(1)^2 fibrations, which can be used to engineer Grand Unified Theories, with multiple, differently charged, 10 matter representations. To determine these models we apply Tate's algorithm to elliptic fibrations with two U(1) symmetries, which are realized in terms of a cubic in P^2. In the process, we find fibers which are not characterized solely in terms of vanishing orders, but with some additional specialization, which plays a key role in the construction of these novel SU(5) models with multiple 10 matter. We also determine a table of Tate-like forms for Kodaira fibers with two U(1)s.


Introduction
F-theory [1][2][3] on elliptically fibered Calabi-Yau manifolds has proven to be a successful framework to realize supersymmetric non-abelian gauge theories, in particular Grand Unified Theories (GUTs) [4][5][6]. Although GUTs are an appealing framework for supersymmetric model building 3 , it is well known that they can suffer from fast proton decay, which, however, can be obviated by having additional discrete or continuous symmetries. In this paper we consider F-theory compactifications that give rise to GUTs with two additional U (1)s, which can potentially be used to suppress certain proton decay operators 4 . In F-theory abelian gauge factors have their genesis in geometric properties of the compactification manifold, namely in the existence of additional rational sections of the elliptic fibration. We carry out a systematic procedure to constrain which such fibrations can give rise to gauge groups G × U (1) 2 .
It is natural to ask whether there is a systematic way to explore the full range of possible low-energy theories with two additional abelian gauge factors which have an F-theory realization. One approach to address this question is to apply Tate's algorithm [44][45][46] to elliptic fibrations with two additional rational sections. This is the approach that we take in this paper and indeed we show that there is a large class of new elliptic fibrations with phenomenologically interesting properties not seen from the top constructions. While Tate's algorithm is a comprehensive method to obtain the form of any elliptic fibration with two rational sections there is a caveat that it is sometimes difficult in practice to proceed with the algorithm without making simplifications at the cost of generality.
The starting point for the application of Tate's algorithm in this context is the realization of the elliptic fiber as a cubic in P 2 [34,35,38,39]. Tate's algorithm involves the study of the discriminant of this cubic equation, which captures the information about the singularities 3 See [7][8][9] for some nice reviews of GUT model building in F-theory. 4 Discrete symmetries have been studied in local and global F-theory model building in, e.g. [10][11][12][13][14].
of the fiber. The singular fibers of an elliptic surface were classified by Kodaira [47,48] and Néron [49], and they belong to an ADE-type classification; Tate's algorithm is a systematic procedure to determine the type of singular fiber. The ADE type of the singular fiber determines the non-abelian part of the gauge symmetry. Tate's algorithm was applied to the Weierstrass form for an elliptic fibration where there are generically no U (1)s in [45,46], and in [32] to the quartic equation in P (1,1,2) which realizes a single U (1) [28]. The application of the algorithm to the cubic in P 2 will constrain the form of the fibrations which realize a G × U (1) 2 symmetry, for some non-abelian gauge group G, which are phenomenologically interesting for model building.
As a result of Tate's algorithm we find a collection of elliptic fibrations which realize the gauge symmetry SU (5) × U (1) 2 where the non-abelian symmetry is the minimal simple Lie group containing the Standard Model gauge group. The fibrations found encompass all of the SU (5) models with two U (1)s in the literature which we are aware of, and includes previously unknown models which, in many cases, have exciting phenomenological features, such as having multiple, differently charged, 10 matter curves. We also determine fibrations that lead to E 6 and SO(10) gauge groups with two U (1)s.
Our results are not restricted to F-theoretic GUT model building, and we hope that they are also useful in other areas of F-theory, for example in direct constructions of the Standard Model [50,51], in the determination of the network of resolutions of elliptic fibrations [52][53][54][55][56], or in the recent relationship drawn between elliptic fibrations with U (1)s and genus one fibrations with multisections [57][58][59].
In section 2 we present a summary where we highlight the fibrations found in the application of Tate's algorithm to the cubic equation, up to fibers realizing SU (5). We also present a table of a particularly nice kind of realizations for Kodaira fibers I n and I * n . In section 3 we recap the embedding of the elliptic fibration as a cubic hypersurface in a P 2 fibration and give details of the resolution and intersection procedures. Section 4 contains Tate's algorithm proper, up to the I 5 , or SU (5), singular fibers. In section 5 the U (1) charges of the various 10 and 5 matter curves that appear in the models from the SU (5) singular fibers are determined.
In section 6 Tate's algorithm is continued from where it was left off in section 4 and we obtain fibrations that have a non-abelian component corresponding to an exceptional Lie algebra.

Overview and Summary
For the reader's convenience, the key results are summarized in this section. For those interested simply in the new SU (5) models we refer to section 5.
An elliptic fibration with two additional rational sections, which gives rise to a gauge theory with two additional U (1)s, can be realized as a hypersurface in a P 2 fibration, as in [34,35,38,39], given by the equation s 1 w 3 + s 2 w 2 x + s 3 wx 2 + s 5 w 2 y + s 6 wxy + s 7 x 2 y + s 8 wy 2 + s 9 xy 2 = 0 , Often the pertinent information from the equation (2.1) is just the vanishing orders of the s i in z, which we will refer to through n i = ord z (s i ) . (2.4) A shorthand for the equation will be the tuple of positive integers (n 1 , n 2 , n 3 , n 5 , n 6 , n 7 , n 8 , n 9 ) representing the vanishing orders. It will not always be possible to express a fibration just through a set of vanishing orders, but there will also be non-trivial relations among the coefficients of the equation. We will refer to fibrations of this form as non-canonical models.
This will be the result of solving in full generality the polynomials which appear in the discriminant as a necessary condition for enhancing the singular fiber. In particular the fact that the coefficients of our fibration belong to a unique factorization domain [46,60] will be used. Schematically we will refer to these fibrations via the shorthand notation I nc i : (n 1 , n 2 , n 3 , n 5 , n 6 , n 7 , n 8 , n 9 ) [s 1,n 1 , s 2,n 2 , s 3,n 3 , s 5,n 5 , s 6,n 6 , s 7,n 7 , s 8,n 8 , s 9,n 9 ] , (2. 5) where the term in square brackets denotes any specialization of the leading non-vanishing coefficients in the expansion of the s i , and the I represents the Kodaira fiber type. Often, for ease of reading, a dash will be inserted to indicate that a particular coefficient is unspecialized. The exponent of the index nc will signal how many non-canonical enhancements of the discriminant were used in order to obtain the singular fiber, that is, how many times solving a polynomial in the discriminant did not require just setting some of the expansion coefficients to zero, but also some additional cancellation.
There is a last piece of notation that needs to be explained before the results can be presented. Since the elliptic fibration has three sections, it will be seen in section 4, where the algorithm is studied in detail, that the discriminant will reflect the fact that the sections can intersect the components of the resolved fiber in multiple different ways. Thus, a number of (non-)canonical forms for each Kodaira  We refer to section 6 for more details about the notation for representing singular fibers corresponding to other types of Kodaira singular fibers.
All of the fibers found and determined are presented in the following summary tables, where the fibers are grouped first by the Kodaira type and then by the degree of canonicality: • In table 1 we list the singular fibers up to vanishing order ord z (∆) = 3. These include fibers of type I 1 ,I 2 ,I 3 , II, and III.
• In table 2 we list the singular fibers at vanishing order ord z (∆) = 4. These include both type I 4 and type IV Kodaira fibers.
• In table 3

Setup
In this section the general setup for the discussion of singular elliptic fibrations with a rank two Mordell-Weil group is provided. First it is explained in more detail that such a fibration can be embedded into a P 2 fibration via a cubic hypersurface equation. This is done in section 3.1. In section 3.2 the symmetries of this cubic equation are detailed and it is demonstrated how they lead to a redundancy of singular fiber types. Some constraints are chosen, listed at the head of section 3.2, to eliminate this redundancy. All the properties of the construction used in the resolution and study of the singular fibers found are documented in section 3.3.

Embedding
By the algebro-geometric construction in [28,34,35,38,39], an elliptic fibration with rank two Mordell-Weil group can be embedded into a P 2 fibration by the hypersurface equation s 1 w 3 + s 2 w 2 x + s 3 wx 2 + s 5 w 2 y + s 6 wxy + s 7 x 2 y + s 8 wy 2 + s 9 xy 2 = 0 , as seen in the previous section. Some explanation of this construction is given in appendix D.
Here [w : x : y] are the projective coordinates of the fibration and the s i are elements of the base coordinate ring, R. It can be seen that this has three marked points, where w, x, and y take values in the fraction field, K, associated to R. Specifically the three marked points are which we label as Σ 0 , Σ 2 , and Σ 1 respectively.
We will work in an open neighbourhood in the base, around the singular locus, which has coordinate z such that the singular locus will occur at the origin of this open neighbourhood. In such a local patch we can specify the s i as expansions in z, We also introduce the simplifying notation
These lop relations and the Z 2 symmetry generate a family of equivalences by applying them repeatedly and in different orders. To choose an appropriate element of each equivalence class the procedure shall be as follows: • Use the Z 2 symmetry to fix n 9 ≥ n 7 .
• Apply lop one to reduce n 7 to 0.
• Apply lop two to reduce the least valued of n 8 and n 9 − n 7 to zero.
• Apply the Z 2 symmetry.
In this way one can often choose a representative of a particular lop-equivalence class where n 7 = n 9 = 0. In the application of Tate's algorithm enhancements which move a form out of this lop-equivalence class will not be considered. In this way the redundancies inherent in the cubic equation (2.1) shall be removed. The remainder of this subsection shall be devoted to showing that these relations hold.

Resolutions, Intersections, and the Shioda Map
To determine the Kodaira type, including the distribution of the marked points, of the codimension one singularity in the fibration specified by (2.1) one often explicitly constructs the resolved geometry via a sequence of algebraic resolutions. In the context of elliptic fibrations such resolutions have been constructed in [26,52,55,56,[61][62][63][64][65][66]. In this section we set up the framework to discuss the resolved geometries and the intersection computations, for example of U (1) charges of matter curves, that are carried out as part of the analysis of the singular fibers found. In particular details are given about the embedding of the fibration as a hypersurface in an ambient fivefold, the details of how the intersection numbers between curves and fibral divisors are computed, and on the construction of the U (1) charge generators.
Consider the ambient fivefold X 5 = P 2 (O ⊕ O(α) ⊕ O(β)) which is the projectivization of line bundles over a base space B 3 . The elliptically fibered Calabi-Yau fourfold will be realized as the hypersurface in this X 5 cut out by the cubic equation (2.1). The terms in the homogeneous polynomial are then sections of the following line bundles Here c 1 is a shorthand notation for π * c 1 (B 3 ). In practice, the first step in any explicit determination of a singular fiber is to blow up the P 2 fibration to a dP 2 fibration by the substitution w → l 1 l 2 w, x → l 1 x, and y → l 2 y and taking the proper transform, as was also the procedure in [34,35,38,39].
The geometry is then specified by the equation s 1 l 2 1 l 2 2 w 3 +s 2 l 2 1 l 2 w 2 x+s 3 l 2 1 wx 2 +s 5 l 1 l 2 2 w 2 y +s 6 l 1 l 2 wxy +s 7 l 1 x 2 y +s 8 l 2 2 wy 2 +s 9 l 2 xy 2 = 0 , (3.22) in dP 2 . After these blow ups the fiber coordinates in this equation are sections of the line As can be seen from the blow ups which mapped P 2 to dP 2 the marked point [0 : 0 : 1] has been mapped to the exceptional divisor l 1 , similarly for [0 : 1 : 0] and l 2 . As such the marked points Σ 0 , Σ 1 , and Σ 2 have been related to the divisors l 1 , w, and l 2 respectively.
As the marked points form sections they are restricted to intersect, in codimension one, only a single multiplicity one component of the singular fiber [67]. The dP 2 intersection ring is not freely generated due to the projective relations which hold in dP 2 . These relations are, using standard projective coordinate notation, These correspond to the relations in the intersection ring The strategy, as it was in [61,65], will be to choose a basis of the intersection ring and repeatedly apply these relations, including any that come from exceptional divisor classes introduced in the resolution. In this way the intersection numbers between curves and fibral divisors can be computed. In this paper the resolutions and intersections were carried out using the Mathematica package Smooth [68].
Given an elliptic fibration with multiple rational sections there remains the construction of the generators of the U (1) symmetries, that is the generators of the Mordell-Weil group.
The Mordell-Weil group is a finitely generated abelian group [69] Z ⊕ · · · ⊕ Z ⊕ G , (3.26) where G is some finite torsion group 6 . There is a map, known as the Shioda map, which constructs from rational sections the generators of the Mordell-Weil group. This map is discussed in detail in [28,72,73].
The Shioda map associates to each rational section, σ i , a divisor s(σ i ) such that where F j are the exceptional curves and B is the dual to the class of the base B 3 . Reduction on the F j gives rise to gauge bosons which should be uncharged under the abelian gauge symmetry. This is ensured by the conditions (3.27).
The charge of a particular matter curve C with respect to the U (1) generator associated to the rational section σ i is given by the intersection number s(σ i ) · C. The constraints (3.27) determine the U (1) charges from s(σ i ) up to an overall scale. We shall always consider the zero-section to be the rational section associated with the introduction of the l 1 in the blow up to dP 2 .
As was alluded to in section 3.1 it is not always the case that a fibration that arises from the algorithm can be specified purely in terms of the vanishing orders of the coefficients.
Sometimes it is necessary to also include some specialization of the coefficients in the zexpansion of the coefficients of the equation. Consider a discriminant of the form

1). The solution to this particular polynomial is
where the pairs (σ 1 , σ 4 ) and (σ 2 , σ 3 ) are coprime. It is not generally possible to perform some shift of the coordinates in (2.1) to return this solution to an expression involving just vanishing orders. This is notably different from Tate's algorithm as carried out on the Weierstrass equation in [46]; there the equation includes monic terms unaccompanied by any coefficient, which often allows one to shift the variables to absorb these non-canonical like solutions into higher vanishing orders of the model.

Tate's Algorithm
In this section we will proceed through the algorithm [45,46], considering the discriminant of the elliptic fibration order by order in the expansion in terms of the base coordinate z. By enhancing the fiber of our elliptic fibration, we will see under which conditions on the sections s i the order of the discriminant will enhance and then study the resulting singular fibers.
This will be done systematically up to singular I 5 fibers for phenomenological reasons and in section 6 we will provide details for some of the exceptional singular fibers. In a step-by-step application of Tate's algorithm to the elliptic fibration (2.1) we find the various different types of Kodaira singular fibers decorated with the information of which sections intersect which components. The discriminant reflects the different ways in which the sections can intersect the multiplicity one fiber components (as explained in section 3.3), thus giving rise to an increased number of singular fibers over fibrations with fewer rational sections. The analysis will be carried out in parallel both for canonical models (determined only by the vanishing orders of the sections) and for non-canonical models (which require additional specialization arising from solving polynomials in the discriminant.)

Starting Points
In the following we will assume that the fibration develops a singularity along the locus z = 0 in the base. A singularity can be characterized by one of the following two criteria: • The leading order of the discriminant as a series expansion in z must vanish.
• The derivatives of D| z=0 in an affine patch must vanish along the z = 0 locus, where D is the equation for the fibration.
Since the leading order of the discriminant is a complicated and unenlightening expression, we will not present it here and instead study the derivatives of the equation of the fibration. This will turn out to be significantly simpler and we will see that the discriminant will enhance upon substitution of the conditions found by the derivative analysis. On the other hand, throughout our study of higher order singularities we will look only at the discriminant ignoring the derivative approach. Let us then study the equation for the elliptic fibration in the affine patch with coordinates (x, y), that is, where we can scale such that w = 1. Along the locus z = 0 we assume that the fiber becomes singular at the point (x 0 , y 0 ) and require the derivatives to vanish We can solve for s 2,0 and s 5,0 from the last two equations Upon substitution in the first equation we can solve for s 1,0 s 1,0 = s 3,0 x 2 0 + y 0 (s 8,0 y 0 + x 0 (s 6,0 + 2s 7,0 x 0 + 2s 9,0 y 0 )) . When s 1,0 , s 2,0 and s 5,0 satisfy the above requirements the discriminant indeed enhances to first order. We can bring the equation of the fibration in a canonical form, depending only on the vanishing orders of the coefficients, by performing the following coordinate shift We see that the singularity now sits at the origin of the affine patch and has generic coefficients in addition to {s 1,0 = s 2,0 = s 5,0 = 0}. This is an I 1 singular fiber, which is the only fiber at vanishing order ord z (∆) = 1 in Kodaira's classification. That this is indeed an I 1 fiber can also be seen by performing a linear approximation around the singular point and noting that we obtain two distinct tangent lines, which shows that this is indeed an ordinary double point. Since there is only one fiber component, all the three sections will intersect it, and we will denote the singular fiber This does not exhaust the possible ways to solve the three equations in (4.1). Indeed, we can look at the affine subspace y = 0 and see that we can find additional solutions. Note that we will not consider here the case x = 0 as this is related by the Z 2 symmetry discussed in section 3.2. The partial derivatives now read We see that if we require {s 1,0 = s 2,0 = s 3,0 = 0} the three equations are satisfied for the two solutions of the quadratic equation {s 5,0 + s 6,0 x 0 + s 7,0 x 2 0 = 0}, which are the two singular points of an I 2 Kodaira fiber as the discriminant enhances to vanishing order ∆(z 2 ). Indeed, looking at the equation of the fiber, we see that this splits in two components The two components indeed intersect in two different points, thus showing that this is an I 2 singular fiber. One of the sections intersects one component, while the two remaining sections intersect the other, so we will denote this fiber as These two fibers represent the starting points for the analysis to be carried out in the remainder of this section. Given the equation for the fibration, we can ask whether z divides any of the coefficients s i . Then we can conclude, inside our preferred lop-equivalence class, the following: , II (ijk) and II (i|jk) fibers.
• If z s 1 and z s 2 then the fiber over the locus {z = 0} is smooth.
• If z | s 1 , z | s 2 and z | s 3 then we can carry on the analysis as in the next section and check whether the singularity is simply I (01|2) 2 or some other enhanced kind.
• If z | s 1 , z | s 2 and z | s 5 we will instead start our analysis from an I It is important to notice that in this part of the algorithm we will not let z | s 3 as this case is covered in the previous branch.
Equivalently, because of the Z 2 symmetry explained in section 3.2, we cannot consider s 8,0 vanishing.
First let us consider the simple case where s 1,1 = 0, which is equivalent to stating that z 2 | s 1 . Then z 2 | ∆ and the singular fiber type, determined by resolving the singularity The discriminant can also be enhanced in order by allowing z to divide either of the two polynomials in (4.9). Let us first consider the situation where s 2 6,0 − 4s 3,0 s 8,0 vanishes. The solution to this equation over this unique factorization domain is given in appendix A and states that The discriminant then enhances so that z 2 | ∆. To determine the type of singular fiber here let us consider the equation of the single component of the I 1 fiber which is being enhanced (s 3,0 x 2 + s 6,0 xy + s 8,0 y 2 ) + xy(s 7,0 x + s 9,0 y) = 0 . ( . (4.13) from table 1.
Finally we can consider the singular fiber that occurs when the second polynomial in ∆ vanishes: s 2 7,0 s 8,0 − s 6,0 s 7,0 s 9,0 + s 3,0 s 2 9,0 = 0. Appendix A lists four generic solutions of this polynomial, three canonical and one non-canonical, which are: (4.14) Any of the three canonical solutions will remove us from our preferred lop-equivalence class and so we do not consider them as they will give rise to a redundancy of singular fiber types.
The only solution to consider therefore is the non-canonical one. The fiber found at this locus is another I 2 fiber, which can be written as .
(4.15) Table 1 is then complete up to second order, once we also include the I (01|2) 2 which was found in the previous section as one of the alternate starting points in the z | s 3 branch.

Enhancements from ord z (∆) = 2
We will now consider the enhancement of the four previously found fibrations which have a discriminant with vanishing order two in z. In this section we shall include the details only of those enhancements that have some non-standard behaviour.
The fibrations (2, 1, 0, 1, 0, 0, 0, 0) and (1, 1, 1, 0, 0, 0, 0, 0) can contain, respectively, in their discriminants polynomials with five and seven terms. These are not polynomials that are discussed in appendix A as their solutions are not known in full generality. In lieu of a complete solution we consider non-generic but canonical type solutions which allow us to obtain singular fibers of a particular type which would be unobtainable without determining a full, generic solution to these polynomials. The subbranches which follow from enhancements where it has been necessary to consider a non-generic solution will also therefore be non-generic, however all remaining branches are determined in full generality.
As a consequence z 3 | ∆ and we find an I The second non-general solution to the five-term polynomial we consider here is found by canonically setting s 1,2 = 0, and then the five term polynomial reduces to P | (s 1,2 =0) = s 2 2,1 s 8,0 − s 2,1 s 6,0 s 5,1 + s 2 5,1 s 3,0 . (4.18) We notice that we cannot set s 3,0 to zero because we are in the z s 3 part of the algorithm (and by Z 2 symmetry we cannot set to zero s 8,0 either). Moreover we just considered the canonical solution given by setting s 2,1 = s 5,1 = 0. We are then left with imposing the non-canonical solution given in appendix A The resulting singular fiber is then an I . (4.20)

Polynomial enhancement in the z | s 3 branch
The other relevant details we will provide concern enhancements from the singular I  We can also apply the non-canonical solution of appendix A to the three-term component

Enhancements from ord z (∆) = 3
We now proceed to consider enhancements of the discriminant starting from the fibers with ord z (∆) = 3, listed in table 1, and we report here the cases that deserve mention due to some peculiarity. In particular we will consider distinctions between split and non-split singular fibers and an instance where we will need to consider the structure of the algorithm in order not to reproduce singular fibers already obtained.

Commutative enhancement structure of the algorithm
We consider enhancements from the III (1|02) nc 2 fiber type. This was found by applying twice the solutions in appendix A. Schematically (4.35) Noting that in the last step a coprimality condition had to be imposed, the discriminant of this singular fiber takes the form ). We can therefore conclude that all the enhancements would just reproduce singular fibers found in other parts of the algorithm. The order in which the enhancements are carried out is of no importance, but it is crucial, in particular with non-canonical fibers, to keep track of which enhancements would reproduce fiber types already obtained.

Enhancements from ord z (∆) = 4
In this section we will proceed with the algorithm by again mentioning only enhancements which require comment. In particular we will deal with the structure of obstructions to full generality due to the complexity of polynomials in the discriminant, we will encounter the distinction between split and semi-split fibers for I * 0 and we will provide details for one of the I 5,nc 3 , obtained by solving non-canonically polynomials in the discriminant three times.

Obstruction from polynomial enhancement
At vanishing order of the discriminant ord z (∆) = 4 we find again the two obstructions to full generality encountered at ord z (∆) = 2, i.e. the same five-term and seven-term polynomials.
These come up respectively in the discriminant of the singular fibers I . We therefore review the singular fibers that we obtain from the enhancements. More details can be found in section 4.3.

Split/semi-split Distinction
The split/semi-split distinction arises for singular fibers of Kodaira type I * 0 . The example we provide concerns the possible enhancement of the canonical type IV s(01|2) , which was found schematically by  The enhancement we will consider here is when s 2,1 = 0. As a consequence z 2 | s 2 and z 6 | ∆. This way we have found the semi-split I * ss(01|2) 0 I * ss(01|2) 0 : (2, 2, 1, 1, 1, 0, 0, 0) . Note that this solution implies that (σ 2 , σ 3 ) are coprime. This gives an I s(0|1|2) 3,nc . Looking at the discriminant of this singular fiber we see that one of the components is ∆ ⊃ (σ 2 3 s 1,1 − σ 2 σ 3 s 2,1 + σ 2 2 s 3,1 ). We apply again the same solution to this three-term polynomial σ 3 = ξ 2 , σ 2 = ξ 3 , s 1,1 = ξ 3 ξ 4 , s 3,1 = ξ 2 ξ 5 , s 2,1 = ξ 2 ξ 4 + ξ 3 ξ 5 . (4.49) Where we used that (σ 2 , σ 3 ) are coprime to set ξ 1 = 1. We have now enhanced the singular fiber to an I s(0|1|2) 4,nc 2 . To obtain the thrice non-canonical I 5 we now consider the two-term polynomial contained in the discriminant at fourth order: ∆ ⊃ (σ 4 ξ 4 − σ 5 ξ 5 ). Applying the non-canonical solution in appendix A (4.50) We have now reached the singular fiber I s(0|1||2) 5,nc 3 I * s(0|2||1) 5,nc 3 : . the same results that were found from the study of toric tops. Details of the relationship between the canonical models and the SU (5) top models and their charges as found in [38] are given. In the algorithm a number of non-canonical models which, as far as the authors are aware have not been seen before, were found, some of which can realize two or three distinctly charged 10 matter curves, potentially a desirable feature, also some models realize as many as seven differently charged 5 matter curves, which are of some interest in light of the phenomenological study in [33].

Canonical I 5 Models
The U (1) charges of the canonical models are found in table 5. Models with these particular U (1) charges are well-studied in the literature. In this subsection we provide a short compar-  ison to the known toric constructions from tops [43] , which were constructed with two extra sections in [30,34,36,38].

Non-canonical I 5 Models
Listed in tables 8 to 10 are the U (1) charges of the, respectively once, twice, and thrice, non-canonical I 5 models found in the algorithm. The U (1) charge generators are given by the Shioda map, as described in section 3.3, where the zero-section of the fibration corresponds to the divisor l 1 = 0 after the P 2 fibration ambient space has been blown up into dP 2 . As opposed to the canonical models the majority of the models tabulated in this section were previously unknown. Some of these models appear to have interesting properties for phenomenology, such as the above noted multiple differently charged 10 and 5 curves. While Tate's algorithm provides a generic procedure there are some caveats that were introduced in the application of it studied in this paper. There are situations where we were not able to solve for the enhancement locus in the discriminant to a reasonable degree of generality. In these cases we have sometimes, as discussed in section 4, used a less generic solution where it was obtainable in such a way that it did not lead to obvious irregularities with the model. In cases where no such solution was obtained we have left that particular subbranch of the Tate tree unexplored.
Throughout the application of Tate One may point out the surprising paucity of non-minimal matter loci in these models with highly specialised coefficients. In the fibrations which are at least twice non-canonical there can occur polynomial enhancement loci where some of the terms in the solutions (as given in appendix A) are fixed by a coprimality condition coming from a previously solved polynomial. Were these terms not fixed to unity by the algorithm then they would contribute non-minimal loci to the fibrations.
In [30,38] there were listed tops corresponding to an SU (4) non-abelian singularity with two additional rational sections, and it was noted that one expects multiple 10 matter curves where these tops are specialized with some non-generic coefficients of the defining polynomial, and such a model, which realizes multiple 10 curves, was constructed there from the SU (4) tops. Included in table 7 are the relations (via the lops) between these SU (4) tops and the SU (4) canonical models which underlie the once non-canonical SU (5) models obtained in the algorithm.

Exceptional Singular Fibers
In this section the algorithm is continued up to the exceptional singular fibers. In determining the exceptional fibers we recall that the sections can only intersect the fiber components of multiplicity one, which means that there is a very restricted number of singular fibers.
For what concerns the type IV * singular fiber there are three different ways in which the sections can intersect the multiplicity one components. These are the types IV * (012) , IV * (01|2) and IV * (0|1|2) . As can be seen from figure 6 the three multiplicity one components of the IV * singular fiber appear symmetrically, and so sections separated by a slash merely indicates that they do not intersect the same multiplicity one component.
Regarding the singular III * fibers, the possible ways the sections can intersect the components restrict the range of singular fibers to III * (012) and III * (01|2) . The different singular fibers can be seen in figure 7.
Finally it is clear that the only type II * fiber one could find (since there is only one multiplicity one component) is the II * (012) . This fiber is also shown in figure 7.
It was also possible to obtain the singular fibers corresponding to gauge groups G 2 and F 4 which come from, respectively, the non-split singular fiber types I * ns(012) 0 and IV * ns(012) .
Proceeding through these subbranchs of the Tate tree will involve the I * n fibers corresponding to Dynkin diagrams of D-type in the split case. There fibers are composed of a chain of multiplicity two nodes with two multiplicity one nodes connected to each end of the chain.
As the rational sections can only intersect the multiplicity one nodes they are constrained to lie of these outer legs. The notation of these fibers shall be (01) represents two sections on the same leg, (0|1) represents two section intersecting two of the outer legs attached to the same end of the chain, and (0||1) will represent two sections sitting on multiplicity one component separated by the length of the chain. Figure 6: The type IV * s fibers. The sections, which intersect the components of the IV * s fiber represented by the blue nodes, are seen to intersect only the external, multiplicity one components. Because of the S 3 symmetry we write these as IV * s(ijk) , IV * s(ij|k) , and IV * s(i|j|k) respectively.

Non-canonical Enhancements to Exceptional Singular Fibers
In this section the remaining exceptional fibers are obtained through non-canonical enhance-

A Solving Polynomial Equations over UFDs
In this appendix details are included of how to solve polynomial equations in the sections s i given that they belong to a unique factorization domain [60]. These solutions were repeatedly used in the algorithm to enhance the vanishing order of the discriminant. For convenience a part of this section will be a summary of the details given in the appendix A of [32], however there are polynomials specific to the case of two additional rational sections and the derivation of the solution for these is provided here. For more details on polynomial equations over UFDs that arise in the application of Tate's algorithm the reader is referred to appendix B of [46].
In [32] solutions were obtained for a three-term polynomial of the form The non-canonical solution is when where σ 2 and σ 3 are coprime over this UFD. The non-canonical solution of a two-term polynomial was also needed s 1 s 2 − s 3 s 4 = 0 : With this solution σ 2 and σ 3 are coprime, and so are σ 1 and σ 4 .

A.1 Two Term Polynomial
We now look at the polynomial P = s 2 1 − 4s 2 s 3 . (A.5) Setting P = 0 imposes the following conditions: • There is an equality between the irreducible components of s 2 1 and the product of the irreducibles of s 2 and s 3 .
• Write µ for the irreducible components common to all the three terms.
• Write σ 1 for the irreducible components common to s 1 and s 2 .
• Write σ 2 for the irreducible components common to s 1 and s 3 .
Note that no conclusion is drawn about irreducibles shared only by s 2 and s 3 . Then the most general solution takes the form Since µ is the greatest common divisor of s 2 and s 3 we have that σ 1 and σ 2 are coprime.

A.2 Perfect Square Polynomial
The first perfect square polynomial is given by From the first two of these equations, one finds the generic form of s 1 So the general solution to the perfect square condition is It follows from the solution of (A.4) that σ 2 and σ 3 are coprime, as are σ 1 and σ 4 .

C Resolution of Generic Singular Fibers
In section 2 a table (table 4) of canonical forms for many of the different fiber types as originally denoted by Kodaira was presented. In this section is is shown by explicitly constructing the resolution that each of the forms is the stated fiber. Given the set of resolutions and the canonical vanishing orders, the resolved geometry is uniquely determined and the form of the resolved geometry will not be written explicitly. For the Cartan divisors the equations are given after the resolution process and they will intersect according to the fiber type of the singularity under consideration.
The generic form for the singular fibers of type I s(0| n 1| m 2) 2k+1 with section separation of the form m + n ≤ 2 3 (2k + 1) is given by (2k + 1 − (m + n), m, m, n, 0, 0, n, 0), where it is assumed that m ≥ n. In order to resolve the geometry the following set of resolutions is used Notice that the first three sets of resolutions produce 2m + n + 1 Cartan divisors. The fourth set of resolutions is then necessary if 2k − 2m − n − 1 = 0. The Cartan divisors in the most general case are

D Determination of the Cubic Equation
In this appendix a non-singular elliptic curve with three marked points is constructed following [74,75] and it is embedded into the projective space P 2 . This non-singular elliptic curve is then fibered over some arbitrary base, B 3 , to create a non-singular elliptic fibration.
Begin by considering a genus one algebraic curve, X, with three Function Order P Q R 1 0 0 0 x 1 1 0 y 1 0 1 xy 2 1 1 x 2 2 2 0 y 2 2 0 2 x 2 y 3 2 1 xy 2 3 1 2 is identified with the vector space of meromorphic functions on X, with poles of at worst order one at the points P , Q, and R, and regular elsewhere. The Riemann-Roch theorem for algebraic curves fixes the dimension of such vector spaces. Any divisor in an algebraic curve X can be written as a formal sum over the points of X: D = P ∈X n P P , where n P = 0 for all by finitely many P .
The Riemann-Roch theorem then states that for any such divisor where deg(D) is the sum over the n P associated to D. Thus it follows that the vector space O(P + Q + R) has dimension 3. Let the three generators of this space be denoted by the functions 1, x, and y. We can determine the pole structure of these functions. Consider first the vector space O(P ), which has dimension 1 for any P ∈ X, and which must contain the one dimensional space of constant functions. As it has dimension 1 it can only contain these holomorphic functions, and therefore there are no functions with a pole of order one at any single point of X. The pole structure of 1, x, and y can then be determined to be as given in Similarly one can consider the vector space O(2(P + Q + R)) which has degree, and thus dimension, 6. Clearly 1, x, and y are generators of half this space, and the other three generators can be written as x 2 , y 2 and xy, which have the pole structures given in table D.
Finally consider O(3(P + Q + R)) which has dimension nine. Out of the six generators for O(2(P + Q + R)) one can construct ten meromorphic functions inside O(3(P + Q + R)), which must be linearly dependent for the space to be of dimension nine. We write this relation as A 1 + A 2 x + A 3 y + A 4 xy + A 5 x 2 + A 6 y 2 + A 7 x 2 y + A 8 xy 2 + A 9 x 3 + A 10 y 3 = 0 . (D. 2) The right-hand side of this equation is the zero function, which does not have poles anywhere. It must then be the case that the left-hand side must not have any poles for such a relation to hold. There are two terms with poles of order three at the points Q, R, which are the x 3 and y 3 terms respectively. There is no other term which contributes a pole of these orders and so could be tuned to cancel it off, therefore the only solution is to set the coefficients, A 9 and A 10 , to zero.
This leaves exactly two terms with a pole of order three at P and, by the same argument as above, if either of these coefficients vanish then the other must also vanish. Let us follow this line of argument and demonstrate that it leads to a contradiction. If A 7 = A 8 = 0 then it is clear that both A 5 = 0 and A 6 = 0 as these are the only terms remaining with a pole of order two in Q, R. Further if these terms are vanishing the arguments above lead us to conclude that A 4 = A 3 = A 2 = A 1 = 0. If this is the case then this is not a non-trivial relation among these ten meromorphic functions, and so the relation cannot have either of After the embedding of the elliptic curve into projective space the relation defines the curve by a hypersurface equation which we write as s 1 w 3 + s 2 w 2 x + s 3 wx 2 + s 5 w 2 y + s 6 wxy + s 7 x 2 y + s 8 wy 2 + s 9 xy 2 = 0 , (D. 3) where [x : y : w] are the coordinates of a P 2 and s i lie in some base coordinate ring R. This will be taken as the defining equation of our elliptic fibration.
The cubic equation (D.3) can always be mapped into the form of a Weierstrass model using Nagell's algorithm [76,77]. For the convenience of the reader we write here only the f and g of the corresponding Weierstrass model. The complete derivation of the Weierstrass model from the cubic (D.3) is given in [34,35,38,39] and we do not repeat it here. The Weierstrass equation is where f and g are given in terms of the coefficients of (2.1) as f = 1 48 (−s 4 6 + 8s 2 6 (s 5 s 7 + s 3 s 8 + s 2 s 0 ) − 24s 6 (s 2 s 7 s 8 + s 3 s 5 s 9 + s 1 s 7 s 9 ) + 16(−s 2