Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers

We compare the sets of Calabi-Yau threefolds with large Hodge numbers that are constructed using toric hypersurface methods with those can be constructed as elliptic fibrations using Weierstrass model techniques motivated by F-theory. There is a close correspondence between the structure of"tops"in the toric polytope construction and Tate form tunings of Weierstrass models for elliptic fibrations. We find that all of the Hodge number pairs ($h^{1, 1},h^{2, 1}$) with $h^{1,1}$ or $h^{2, 1}\geq 240$ that are associated with threefolds in the Kreuzer-Skarke database can be realized explicitly by generic or tuned Weierstrass/Tate models for elliptic fibrations over complex base surfaces. This includes a relatively small number of somewhat exotic constructions, including elliptic fibrations over non-toric bases, models with new Tate tunings that can give rise to exotic matter in the 6D F-theory picture, tunings of gauge groups over non-toric curves, tunings with very large Hodge number shifts and associated nonabelian gauge groups, and tuned Mordell-Weil sections associated with U(1) factors in the corresponding 6D theory.


Introduction
While Calabi-Yau threefolds have played an important role in string theory since the early days of the subject [1], the set of these geometries is still relatively poorly understood. Following the approach of Batyrev [2], in 2000 Kreuzer and Skarke carried out a complete analysis of all reflexive polytopes in four dimensions, giving a systematic classification of those Calabi-Yau (CY) threefolds that can be realized as hypersurfaces in toric varieties [3]. For many years the resulting database [4] has represented the bulk of the known set of Calabi-Yau threefolds, particularly at large Hodge numbers. More recently, the study of F-theory [5,6] has motivated an alternative method for the systematic construction of Calabi-Yau threefolds that have the structure of an elliptic fibration (with section). By systematically classifying all bases that support an elliptically fibered CY [7][8][9][10] and then systematically considering all possible Weierstrass tunings [11,12] over each such base, it is possible in principle to construct all elliptically fibered Calabi-Yau threefolds. While there are some technical issues that must still be resolved for a complete classification from this approach, at large Hodge numbers this method gives a reasonably complete picture of the set of possibilities. One perhaps surprising result that has recently become apparent both from this work and from other perspectives [13][14][15][16] is that a very large fraction of the set of Calabi-Yau threefolds that can be constructed by any known mechanism are actually elliptically fibered, particularly at large Hodge numbers.
The goal of this paper is to carry out a direct comparison of the set of elliptically fibered Calabi-Yau threefolds that can be constructed using Weierstrass/Tate F-theory based methods with those that arise through reflexive polytope constructions. While the general methods for construction of elliptic Calabi-Yau threefolds can include non-toric bases [9,10], and even over toric bases there are non-toric Weierstrass tunings [11,12], we focus here on the subset of constructions that have the potential for a toric description through a reflexive polytope. In §2, we review some of the basics of F-theory and the systematic construction of elliptic Calabi-Yau threefolds through the geometry of the base and the tuning of Weierstrass or Tate models from the generic structure over each base. In §3, we review the Batyrev construction and reflexive polytopes, and the structure of elliptic fibrations in this context. In particular, in §3. 4 we describe the precise correspondence between a particular fibration structure for a reflexive polytope and Tate form Weierstrass models. In §4, we restrict attention to toric base surfaces B 2 and identify the set of tuned Weierstrass/Tate models over such bases that naturally correspond to a reflexive polytope in the Batyrev construction. This gives us a systematic way of constructing from the point of view of elliptic fibrations over a chosen base a large set of elliptic Calabi-Yau threefolds that are expected to be seen in the Kreuzer-Skarke database with a specific (P 2,3,1 ) fiber type. At large Hodge numbers, for reasons discussed further in §4. 8, we expect that this should give most or all elliptic fibrations that arise in the KS database; we find that this is in fact the case.
The main results of the paper are in §5 and §6, where we describe an algorithm to systematically run through all tuned Tate models over toric bases and we compare the results of running this algorithm to the Kreuzer-Skarke database. The initial result, described in §5, is that these simply constructed sets match almost perfectly in the large Hodge number regimes that we study: both at large h 2,1 and at large h 1,1 all the models constructed by an appropriate set of Tate tunings over toric bases appear in the KS database, and virtually all the Hodge numbers in the database are reproduced by elliptic Calabi-Yau threefolds produced using this approach. There is a small set of large Hodge numbers (18 out of 1,827) associated with toric hypersurface Calabi-Yaus, however, that are not reproduced by our initial scan. By examining these individual cases, as described in §6, we find that all these exceptions also correspond to elliptic fibrations though with more exotic structure, such as non-flat fibrations resolved through extra blow-ups in the base that take the base outside the toric class, and/or force Mordell-Weil sections on the elliptic fiber. The upshot is that when these more exotic constructions are included, all Hodge number pairs with either h 1,1 or h 2,1 at least 240 are reproduced by an elliptic Calabi-Yau over some explicitly determined base surface. We conclude in §7 with a summary of the results and some related open questions.
Note that in this paper the focus is on understanding in some detail the connection between elliptic fibration geometry and polytope geometry for these different approaches to construction of elliptic Calabi-Yau threefolds. In a companion paper [17] we will describe a more direct analysis of the polytopes in the KS database that also shows explicitly that there is a toric fiber associated with an elliptic fibration for every polytope in the database at large Hodge numbers. The principal class of Tate tunings that we consider in this paper have a complementary description in the language of "tops" [18]. The construction of many polytopes in the KS database through combining K3 tops and "bottoms" was accomplished in [13], and a systematic approach to constructing toric hypersurface Calabi-Yau threefold with a given base and gauge group using the language of tops is developed in [19], with particular application to models with gauge group SU(5) as also studied in e.g. [20]. One of the main results of this paper is the systematic relationship of such constructions with certain classes of Tate tunings. This leads in some cases to the identification of new Tate tunings from observed polytope structures, and the observation that some polytopes in the KS database have a more complex structure that does not admit a direct description in terms of standard tops. On the other hand, new structures of tops are also found through the construction of polytopes via the correspondence with Tate tunings.

F-theory physics and elliptic Calabi-Yau threefold geometry
We briefly summarize here how the massless spectrum of a six-dimensional effective theory from F-theory compactification is related to the geometric data of the internal manifold, which is an elliptically fibered Calabi-Yau threefold (CY3) over a two-dimensional base B 2 (complex dimensions). F-theory models can then be systematically studied by first choosing a base B 2 and then specifying an elliptic fibration in Weierstrass form over that base. Further background on F-theory can be found in for example [5,6,21,22]. F-theory compactified on a (possibly singular) elliptically fibered Calabi-Yau threefold X gives a 6D effective supergravity theory. Such a compactification of F-theory is equivalent to M-theory on the resolved Calabi-YauX in the decompactification limit of M-theory, where in the F-theory picture the resolved components of the elliptic fiber are shrunk to zero size. F-theory can also be thought of as a nonperturbative formulation of type IIB string theory. In this picture the type IIB theory is compactified on the base B 2 . In this F-theory description, spacetime filling 7-branes sit at the codimension-one loci in the base where the fibration degenerates. The non-abelian gauge symmetries of the 6D effective theory arise from the seven-branes and can be inferred from the singularity types of the elliptic fibers along the codimension-one loci in the base, according to the Kodaira classification (Table 2). At the intersections of seven-branes there are localized matter fields that are hypermultiplets in the 6D theory; the representations of the matter fields can be determined from the detailed form of the singularities over the codimension-two points in the base (see e.g. [23][24][25]). Therefore the physics data can be extracted by studying the singular fibers by means of the Weierstrass models (short form) or the Tate models (long form) of X that we review in §2.2 and 2.3. There can also be abelian gauge symmetries, which arise from additional rational sections of the elliptic fibration [6]. The study of u(1) symmetries is more subtle in that it relates to the global structure of the fibration, as opposed to non-abelian symmetries where we can just study singular fibers locally. We will see cases with abelian factors in §6, with a detailed example worked out in Appendix C. In §2.4, we review the Zariski decomposition, which allows us to determine the order of vanishing and consequent gauge group of a Weierstrass or Tate form description of an elliptic fibration, and in §2.5 we describe how this method can be applied systematically in the context of Tate tunings. In §2.6, we review the 6D anomaly cancellation conditions and their connection to the matter content of a 6D theory and the Hodge numbers of the corresponding Calabi-Yau threefold. In §2.7 we review the constraints imposed by global symmetry groups on the set of gauge groups that can be supported on curves intersecting a given curve, and we conclude the overview of F-theory in §2.8 with a summary of the systematic classification of complex surfaces that can support elliptic Calabi-Yau threefolds and can be used for F-theory compactification.

Hodge numbers and the 6D massless spectrum
By going to the 5D Coulomb branch after reduction on a circle, the F-/M-theory correspondence can be used to relate the geometry ofX to the associated 6D supergravity theory [26,27]. In particular, the Hodge numbers, h 1,1 and h 2,1 , ofX can be related to the (massless) matter content of the 6D theory: where T is the number of tensor multiplets, which is determined already by the choice of base and r = r abelian + i r i is the total rank of the gauge group, of the 6D effective theory. We also have h 2,1 (X) = H neutral − 1, (2.4) where H neutral is the number of hypermultiplets that are neutral under the Cartan subalgebra 1 of the gauge group G of the 6D F-theory. The spectra of 6D theories are constrained by consistency conditions associated with the absence of anomalies, which we describe in further detail in §2. 6. The gravitational anomaly cancellation condition (2.33) gives H − V = 273 − 29T , where V is the dimension of the gauge group G, and H = H charged + H neutral is the total number of hypermultiplets (separated into neutral and charged matter under the Cartan of the gauge group G). So we have another expression h 2,1 (X) = 272 + V − 29T − H charged .
(2.5) 1 In other words, this counts fields that are neutral matter fields in the 5D M-theory sense but may transform under the unhiggsed non-abelian factors of the 6D F-theory. Often, matter charged under the non-abelian factors is still charged under the Cartan subalgebra, but for certain representations of some non-abelian groups there can be charged matter that is neutral under the Cartan subalgebra.
Rank r Algebras 2 su(3), g 2 3 su(4), so(7) 4 so(8), so (9), f 4 r ≥5 so(r), so(r + 1) This is more useful for some of our purposes than equation (2.4). In particular, as we discuss in further detail in the following section, we are interested in studying various specializations (tunings) of a generic elliptically fibered CY3 over a given base B 2 . The number of tensors T is fixed for a given base. Thus, if we start with known Hodge numbers h 1,1 and h 2,1 for the generic elliptic fibration over a given (e.g. toric [8,28]) base, and specialize/tune to a model with a larger gauge group and increased matter content, then the Hodge numbers of the tuned model can be simply calculated by adding to those of the generic models respectively the shifts ∆h 1,1 = ∆r, (2.6) Such a specialization/tuning amounts physically to undoing a Higgsing transition, and the second of these relations simply expresses the physical expectation that the number of matter degrees of freedom that are lost ("eaten") in a Higgsing transition is equal to the number of gauge bosons lost to symmetry breaking. Note that the data on the right hand sides are associated in general with tuned non-abelian gauge symmetries but also in some special cases involve abelian factors. Note also that the right-hand sides of (2.6) and (2.7) are always nonnegative and non-positive respectively for any tuning. In most cases, the gauge group increases in rank and some of the h 2,1 moduli are used to implement the tuning. In rank-preserving tunings, however, the Hodge numbers do not change (see Table 1)h 1,1 of course does not change in a rank-preserving enhancement; h 2,1 does not change either in these tunings, as one can check by considering carefully the matter charged under the Cartan subalgebra (cf. footnote 1.)

Generic and tuned Weierstrass models for elliptic Calabi-Yau manifolds
An elliptic fibration with a section over a base B can be described by the Weierstrass model The Calabi-Yau condition on the total space X requires that f, g are sections of O(−4K B ), O(−6K B ), where K B is the canonical class of the base. More abstractly, we take the weighted projective bundle π : P = P P 2,3,1 [L 2 ⊕ L 3 ⊕ O B ] → B, (2.9) where L = O(−K B ) is required by the Calabi-Yau condition and x ∈ O P (2) ⊗ π * L 2 , y ∈ O P (3) ⊗ π * L 3 , z ∈ O P (1) and [x : y : z] can be viewed as weighted projective coordinates of the P 2,3,1 , while f and g are sections of, to be more precise, π * L 4 and π * L 6 respectively. Consider an elliptic Calabi-Yau threefold over a complex two-dimensional base B 2 , so the divisors in the base are curves. The elliptic fiber becomes singular over the codimension-one loci in the base where the discriminant ∆ = 4f 3 + 27g 2 (2. 10) vanishes. The type of singular fiber at a generic point along an irreducible component of the discriminant locus {σ = 0} is characterized by the Kodaira singularity type, which is determined by the orders of vanishing of f , g, and ∆ in an expansion in σ (see Table 2). The physics interpretation is that there are seven-branes on which open strings (and junctions) end located at the discriminant locus, and the resulting gauge symmetries can be determined (up to monodromies) by the type of the singular fiber. The gauge algebras that are further determined by monodromy conditions [24,29] are those of types I n , I * 0 , I * n , IV, IV * , where some factorizability conditions are imposed on the terms of f, g, ∆ of lowest degrees of vanishing order along {σ = 0}. We summarize these conditions in Table 3, in terms of the first nonvanishing sections f i (ζ), g j (ζ), ∆ k (ζ) in the local expansions f (σ, ζ) = f 0 (ζ) + f 1 (ζ)σ + · · · , (2.11) g(σ, ζ) = g 0 (ζ) + g 1 (ζ)σ + · · · , (2.12) ∆(σ, ζ) = ∆ 0 (ζ) + ∆ 1 (ζ)σ + · · · , (2. 13) where {ζ = 0} defines a divisor that intersects {σ = 0} transversely so that σ, ζ together serve as local coordinates on an open patch of base. A generic Weierstrass model (i.e. with coefficients at a generic point in the moduli space) for an elliptically fibered CY3 over a given base B 2 corresponds physically to a maximally Higgsed phase. In the maximally Higgsed phase over many bases the gauge group and matter content are still nontrivial. The minimal gauge algebras and matter configuration associated with a given base B 2 are carried by non-Higgsable clusters (NHCs) [7], which are isolated rational curves of self-intersection m, −12 ≤ m ≤ −3, and clusters of multiple rational curves of self-intersection ≤ −2: {−2, −3}, {−2, −2, −3}, and {−2, −3, −2}. The sections f, g, ∆ automatically vanish to higher orders along these curves in any Weierstrass model over the given base. This can be understood geometrically as an effect in which the curvature over the negative self-intersection curves must be cancelled by 7-branes to maintain the Calabi-Yau structure of the elliptic fibration. The orders of vanishing and the corresponding minimal gauge groups on these NHCs are listed in Table 11 in §4. 4.
Starting from the generic model over a given base B, we can systematically tune the Weierstrass model coefficients f and g to increase the order of vanishing over various curves beyond what is imposed by the NHCs, producing additional or enhanced gauge groups on some curves in the base. Many aspects of such tunings are described in a systematic fashion Type ord (f ) ord (g) ord (∆) singularity nonabelian symmetry algebra (8) or so (7) or g 2 I * n 2 3 n ≥ 7 D n−2 so(2n − 4) or so(2n − 5) does not occur in F-theory Table 2: Kodaira classification of singularities in the elliptic fiber along codimension one loci in the base in terms of orders of vanishing of the parameters f, g in the Weierstrass model (2.8) and the discriminant locus ∆.
in [12]. While over some bases there is a great deal of freedom to tune many different gauge group factors on various curves in the Weierstrass model, there are also limitations imposed by the constraint that there be no codimension one loci over which f, g vanish to orders (4,6).
In this paper we also avoid cases with codimension two (4,6) loci; while such singularities can be related to 6D superconformal field theories, in the geometric picture such singularities are associated with non-flat fibers and the resolution of the singularity proceeds by first blowing up the (4, 6) point in the base, which modifies the geometry of the base B, increasing h 1,1 (B) by one. While in many cases the extent to which enhanced gauge groups can be tuned in the Weierstrass model over any given base can be determined by considerations such as the low-energy anomaly consistency conditions, the precise set of possible tunings is most clearly described in terms of an explicit description of the Weierstrass coefficients. In the case of toric bases, the complete set of monomials in f, g has a simple description (see e.g. [8,12]) and we have very strong control over the parameters of the Weierstrass model.

Tate form and the Tate algorithm
The Tate algorithm is a systematic procedure for determining the Kodaira singularity type of an elliptic fibration, and provides a convenient way to study Kodaira singularities in the context of F-theory [24,29]. The associated "Tate forms" for the different singularities match up neatly with the toric construction that we focus on in this paper. We start with an equation for an elliptic curve in the general form y 2 + a 1 xyz + a 3 yz 3 = x 3 + a 2 x 2 z 2 + a 4 xz 4 + a 6 z 6 , (2.14) ord(f ) ord(g) ord(∆) algebra monodromy condition for some u 0 (ζ), which is a perfect square sp( n/2 ) otherwise for some a(ζ), b(ζ) so(7) is a perfect square for even n so(2n − 5) otherwise where for an elliptic fibration a n are sections of line bundles O(−nK B ). The general form (2.14) can be related to the standard Weierstrass form (2.8) by completing the square in y and shifting x, which gives the relations   (6) gives alternate exotic matter content; see text for further details. Groups and tunings marked with require additional monodromy conditions.
An advantage of the general form (2.14) is that by requiring specific vanishing orders of the a n 's according to Table 4, specific desired vanishing orders of (f, g, ∆) can be arranged to implement any of the possible gauge algebras. Moreover, the monodromy conditions in Table  3 imposed by some gauge algebras on f , g, or ∆ are also satisfied automatically by these "Tate form" models. For example, for tunings of fiber types I m or I * m where ∆ is required to vanish to a certain order while ord(f ) and ord(g) are kept fixed, the vanishing order of a n 's prescribed by the Tate algorithm immediately give the desired ord(∆). This makes the Tate form much more convenient for constructing these singular fibers by only requiring the order of vanishing of the a n 's to be specified, in contrast to the Weierstrass form (2.8) where it is necessary to carefully tune the coefficients of f and g to arrange for a vanishing of ∆ to higher order. The Tate forms described in Table 4 are also connected very directly to the geometry of reflexive polytopes. As we discuss in the subsequent sections, tuning a Tate form can be described by simply removing certain monomials from the general form (2.14), which corresponds geometrically to removing certain points from a lattice in the toric construction. We refer to tunings of this type as "Tate tunings" in contrast to tunings of the coefficients of f and g; when applied to the polytope toric construction, we refer to Tate tunings as "polytope tunings".
Note that Table 4 has incoporated some results of the present study into the Tate table originally described in the F-theory context in [24] and later modified in [29]. The most significant new feature is an alternate Tate form for the algebras su(2n), with a 2 vanishing to order 2. For n = 3, in particular, this Tate form gives a tuning with exotic 3-index antisymmetric SU (6) matter. An example of a polytope that realizes this tuning is described in §6.1.1. For higher n, in cases where a 1 is a constant -i.e. on curves of self-intersection −2 -this simply gives an alternate Tate tuning of SU(2n). On any other kind of curve, at the codimension two loci where a 1 = 0 there is a codimension two (4, 6) singularity when n > 3. This can immediately be seen from the fact that at the locus a 1 = 0, each a k vanishes to order k so that (2.15-2.21) give a vanishing of (f, g, ∆) to orders (4,6,12). Resolving this singularity generally involves blowing up a point on the base, so that the resulting elliptic fibration is naturally thought of as living on a base with larger h 1,1 , but this kind of Tate model for SU (8) and higher would be relevant in a complete analysis of all reflexive polytopes.
We have also identified Tate tunings of so(4n + 4), like those of so(4n + 2) that do not require an extra monodromy condition and only require the vanishing order of a i 's; this arises naturally in the context of the geometric constructions of polytopes. We discuss briefly how these two types of Tate tunings are relevant in the constructions of this paper. For so(4n + 4), if a 6 is of order 2n + 1, then the necessary monodromy condition is that [30,29] (a 2 4 − 4a 2 a 6 )/z 2n+2 | z=0 is a perfect square. This condition is clearly automatically satisfied if a 6 is actually of order 2n + 2, so can be guaranteed simply by setting certain monomials in the Tate coefficients to vanish (in a local coordinate system, which can become global in the toric context used in the later sections of the paper). On the other hand, if the leading terms in a 2 , a 4 , a 6 are each constrained to be powers of a single monomial m, m n+1 , m 2n+1 , then the monodromy condition will be automatically satisfied with a 6 of order 2n+1 without specifying any particular coefficients for these monomials. We encounter both kinds of situation in this paper. For so (8), the monodromy condition when a 6 is of order 4 is that (a 2 2 − 4a 4 )/z 2 | z=0 is a perfect square [29]. 2 This can be satisfied if a 2 , a 4 contain only a single monomial each 2 To relate this to the condition stated in Table 3, note that the leading term in the discriminant when that condition is satisfied becomes −(a − b) 2 (2a + b) 2 (2b + a) 2 , so that condition implies the perfect square condition. Going the other way, when the perfect square condition is satisfied we can determine a, b by noting that a2/3 is one of the roots a, b, −a − b of the cubic x 3 + f2x + g3, so without loss of generality we have m, m 2 at leading order, but cannot be imposed by simply setting the orders of vanishing of each a i . The situation is similar when a 6 is of order 3, though the monodromy condition is more complicated when a 2 , a 4 , a 6 are not single monomials m, m 2 , m 3 . This is the only gauge algebra with no monodromy-independent Tate tuning except through this kind of single monomial condition. Finally, for so(4n + 2), with a 6 of order 2n, the monodromy condition is that (a 2 3 + 4a 6 )/z 2n | z=0 is a perfect square, satisfied in particular if a 6 is actually of order 2n + 1 or if the leading terms in a 3 , a 6 are each a single monomial proportional to m, m 2 . We explore further, for example, in Section 4.6 for so (12) the subtleties in using the Tate tuning {1, 1, 3, 3, 5} described in [24], which requires an additional monodromy condition, vs. our alternative tuning {1, 1, 3, 3, 6}; In fact, analogous situations occur in tuning all gauge algebras with monodromies.

The Zariski decomposition
A central feature of the geometry of an F-theory base surface is the structure of the intersection form on curves (divisors) in B 2 . The intersection form on H 2 (B, Z) has signature (1, T ). Curves of negative self-intersection C ·C < 0 are rigid. A simple but useful algebraic geometry identity, which follows from the Riemann-Roch theorem, is that for any curve C of genus g. We are primarily interested in rational (genus 0) curves, for which therefore C · C = −K B · C − 2. All toric curves on a toric base B 2 are rational, and the intersection product of toric curves has a simple structure that we review in the following section.
To study the orders of vanishing of f , g and ∆ along some irreducible divisors in the base, aside from looking explicitly at the sets of monomials of f , g and ∆, it is convenient to consider the more abstract "Zariski decomposition", in which an effective divisor A is decomposed into (minimal) multiples of irreducible effective divisors C i of negative self-intersection and a residual part where Y is effective and satisfies Then the order of vanishing along the curve C i of a section of the line bundle corresponding to the divisor A must be at least c i = q i . Mathematically, the Zariski decomposition is normally considered over the rationals, so q i ∈ Q. Here, however, we are simply interested in the smallest integer coefficient of C i compatible with the decomposition over the ring of integers. For example, consider the decomposition The goal is to find the minimal set of integer values c i such that the conditions Y · C i ≥ 0 are satisfied. Taking the intersection product on both sides with C j , the conditions can be rewritten as the set of inequalities where M ji ≡ C j · C i are pairwise intersection numbers (non-negative for i = j) and selfintersection numbers M jj = C j · C j ≡ m j , and v j,n ≡ −nK B · C j . The Zariski decomposition of −4K B and −6K B was used in [7] to analyze the non-Higgsable clusters that can arise in 6D theories. More generally, we can use the same approach to analyze models where we tune a given gauge factor on a specific divisor beyond the minimal content specified by the non-Higgsable cluster structure. In such a situation, we would choose by hand to take some values of c i in (2.25) to be larger than the minimal possible values; this may in turn force other coefficients c j to increase. As a simple example, consider a pair of −2 curves (i.e. curves of self-intersection −2) C, D that intersect at a point (C·D = 1). The Zariski decomposition of the discriminant locus gives simply −12K B = Y , since K B · C = K B · D = 0 from (2.22), so the discriminant need not vanish on C or D. If, however, we tune for example an su(4) gauge algebra on D so that ∆ vanishes to order 4 on D then we have the Zariski decomposition −12K B − 4D = 2C + Y , since −4D · C = −4, implying that ∆ must also vanish to order 2 on C, so that C must therefore also carry at least an su(2) gauge algebra.

Zariski decomposition of a Tate tuning
A particular application of the Zariski decomposition that we use here extensively is in the context of a Tate tuning. In particular, assume that we have an elliptic fibration in the Tate form (2.14) over a complex surface base B, and we have a set of curves C j in the base that includes all curves of negative self-intersection. The parameter space of the elliptic fibration is given by the five sections a n ∈ O(−nK), n = 1, 2, 3, 4, 6. We denote by c j,n the order of vanishing of a n on C j . The minimal necessary order of vanishing of each a n on each curve C j can be determined by applying the Zariski decomposition for −nK. This gives rise to a set of vanishing orders c j,n associated with each non-Higgsable cluster, which we list in Table 5. These are the minimal values c j,n = c NHC j,n that satisfy the inequalities (2.26) for each value of n ∈ {1, 2, 3, 4, 6}. In doing a Tate tuning, we impose the additional condition that over certain curves C j , the vanishing order is at least some specified value that is higher than the minimum imposed by the NHCs, c j,n ≥ c tuned j,n ≥ c NHC j,n . We can then use the Zariski decomposition to determine the minimum values of the c j,n compatible with this lower bound that also satisfy the inequalities (2.26).
More concretely, to determine the unique minimum set of values c j,n that satisfy the inequalities (2.26), we proceed iteratively, following an algorithm described in appendix A of [7]. For each n, we begin with an initial assignment of vanishing orders c (0) j,n = c tuned j,n (2.27) when we are imposing a given tuning. When we are computing the minimal values from NHC's without tuning we simply use the minimal order of vanishing from the Zariski decomposition on each isolated curve of self-intersection m j = M jj , (2.28) We can then use the inequalities (2.26) to determine the minimal correction that is needed to each vanishing order (label n dropped for clarity of the notation), The second corrections are obtained similarly, replacing c (0) on the RHS with c (1) = c (0) + ∆c (1) j . We continue to repeat this procedure until the corrections in the f -th step all become zero, ∆c At each step this algorithm clearly increases the orders of vanishing in a minimal way, so when the algorithm terminates the solution is clearly a minimal solution of the inequalities (2.26). Note that in some cases, the algorithm leads to a runaway behavior when there is no acceptable solution without (4, 6) loci. When this occurs, or when one of the factors of the gauge algebra exceeds that desired by the tuning, we terminate the algorithm and do not consider this tuning as a viable possibility.
Note that a tuning beyond that shown in Table 5 does not necessarily increase the gauge group on any of the curves. In particular, for some gauge groups there are multiple possible Tate tunings. Both for the generic gauge group associated with the generic elliptic fibration over a given base and for constructions with gauge groups that are enhanced through a Tate tuning, this means that there may be distinct Tate tunings with the same physical properties.  {-3}  {{1, 1, 1, 2, 2}}  {-4}  {{1, 1, 2, 2, 3}}  {-5}  {{1, 2, 2, 3, 4}}  {-6}  {{1, 2, 2, 3, 4}}  {-7}  {{1, 2, 3, 3, 5}}  {-8}  {{1, 2, 3, 3, 5}}  {-12}  {{1, 2  As we will see later, these distinct Tate tunings can correspond through distinct polytopes to different Calabi-Yau threefold constructions. Note also that for the toric bases we are studying here, an essentially equivalent analysis could be carried out by explicitly working with the various monomials in the sections a n , which in the toric context are simply points in a dual lattice, as we discuss in the next section. We use the Zariski procedure because it is more efficient and more general; the results of this analysis should, however, match an explicit toric computation in each case.

Matter content from anomaly constraints in F-theory
Six-dimensional N = (1, 0) supergravity theories potentially suffer from gravitational, gauge, and mixed gauge-gravitational anomalies. We focus here primarily on nonabelian gauge anomalies, though similar considerations hold for abelian gauge factors. On the one hand, the anomaly information can be encoded in an 8-form I 8 , which is built from the 2-forms characterizing the non-abelian field strength F and the Riemann tensor R, and which has coefficients that can be computed in terms of T, V, H, and the explicit numbers of chiral matter fields in different representations. On the other hand, the anomalies can be cancelled through a generalized Green-Schwarz term if I 8 factorizes for some constant coefficients a α , b β i in the vector space R 1,T associated with self-dual and anti self-dual two-forms B µν in the gravity and tensor multiplets, Here Ω αβ is a signature (1, T ) inner product on the vector space, and λ i are normalization constants for the non-abelian gauge group factors G i . Then, using the notation and conven-tions of [31], the conditions for anomaly cancellation are obtained by equating the coefficients of each term from the two polynomials where A R , B R , C R are group theory coefficients 3 defined by

40)
x i R is the number of matter fields 4 in the representation R of the non-abelian factor G i , and x ij RS is the number of matter fields in the (R, S)-representation of G i ⊗ G j . For 6D theories coming from an F-theory compactification, the vectors a, b i are related to homology classes in the base B 2 through the relations where, again, K B is the canonical class of B 2 , and C i ∈ H 2 (B 2 , Z) are irreducible curves in the base supporting the singular fibers associated with the non-abelian gauge group factors G i . With this identification, the Dirac inner products between vectors in R 1,T are related to intersection products between divisors in the base. In principle, the matter content of a 6D theory can be determined by a careful analysis of the codimension two singularities in the geometry. In many situations, however, the generic matter content of a low-energy theory is uniquely determined by the gauge group content and anomaly cancellation simply from the values of the vectors a, b i . For example, a theory with an SU(N ) gauge factor associated with a vector b generically has g adjoint matter fields, (8 − N )n + 16(1 − g) fundamental matter fields, and (n + 2 − 2g) two-index antisymmetric matter fields, where n = b · b and g = 1 + (a · b + n)/2 (see e.g. [12]); this simplifies in the g = 0 case of primary interest to us here to a spectrum of n + 2 two-index antisymmetric matter fields and 16 + (8 − N )n fundamental fields. For most of the theories we consider here the matter content follows uniquely in this way from the values of a, b i . In some situations, however, more exotic matter representations can arise; we encounter some cases of this later in this paper, such as the three-index antisymmetric representation of SU (6).
In general, the anomaly constraints on 6D theories provide a powerful set of consistency conditions that we use in many places in this paper to analyze and check various models that arise through tunings; in particular, using the anomaly conditions to determine the matter spectrum gives a direct and simple way in many cases to compute the Hodge numbers of the associated elliptic Calabi-Yau manifold that can be matched to the Hodge numbers of a toric hypersurface construction.

Global symmetry constraints in F-theory
In many cases, the anomaly cancellation conditions impose constraints not only on the matter content of the theory but also on what gauge groups may be combined on intersecting curves, corresponding to vectors b i with non-vanishing inner products in the low-energy theory. For example, two gauge factors of g 2 or larger in the Kodaira classification cannot be associated with vectors b, b having b · b > 0; in the low-energy supergravity theory this is ruled out by the anomaly conditions while in the F-theory picture this would correspond to a configuration with a codimension two (4, 6) point at the intersection between the corresponding curves. In addition to these types of constraints, another set of constraints on what combination of gauge groups can be tuned on specific negative self-intersection curves in a base B 2 can be derived from the low-energy theory by considering the maximum global symmetry of an SCFT that arises by shrinking a curve C of self-intersection n < 0 that supports a given gauge factor G i [32]. While in most cases these global symmetry conditions simply match with the expectation from anomaly cancellation, in some circumstances the global symmetry condition imposes stronger constraints. For example the "E 8 rule" [33] states that the maximal global symmetry on a −1 curve that does not carry a nontrivial gauge algebra is e 8 ; i.e., the direct sum of the gauge algebras carried by the curves intersecting the −1 curve should be a subalgebra of e 8 . While the global symmetry constraints are completely consistent with F-theory geometry, they may not be a complete and sufficient set of constraints; for example a similar constraint appears to hold in F-theory for the algebras on a set of curves intersecting a 0 curve [12], though the low-energy explanation for this is not understood in terms of global constraints from SCFT's.
The maximal global symmetry groups realized in 6D F-theory for each possible algebra on a curve of self-intersection m ≤ −1 are worked out in [32]. We use their results in our algorithm to constrain possible gauge algebra tunings. More explicitly, given a gauge algebra on a curve, the maximal global symmetry on the curve is determined, so the direct sum of the algebras on the curves intersecting it should be a subalgebra of the maximal global symmetry algebra. For instance, consider a linear chain of three curves {C 1 , C 2 , C 3 } carrying gauge algebras {g 1 , g 2 , g 3 }. These can be either minimal or enhanced algebras, but they have to satisfy is the maximal global symmetry algebra given g 2 on the curve C 2 , as enumerated in the tables in [32]. This will be useful for us to constrain the possible tunings on a curve when the gauge symmetries on its neighboring curves are known, making our search over possible tunings more efficient. This is also convenient sometimes for us to determine the gauge algebras that have monodromy conditions without having to figure out the monodromy directly; the trick to doing this is described in §4.6. We also include the "E 8 rule" in our algorithm in §5.1, corresponding to the case where m = −1 and g 2 is trivial.

Base surfaces for 6D F-theory models
There is a finite set of complex base surfaces that support elliptic Calabi-Yau threefolds. It was shown by Grassi [34] that all such bases can be realized by blowing up a finite set of points on the minimal bases P 2 , F m with 0 ≤ m ≤ 12, and the Enriques surface. This leads to a systematic constructive approach to classifying the set of allowed F-theory bases. The structure of non-Higgsable clusters limits the configurations of negative self-intersection curves that can arise on any given base, so we can in principle construct all allowed bases by blowing up points in all possible ways and truncating the set of possibilities when a disallowed configuration such as a curve of self-intersection −13 or below arises. This was used in [8] to classify the full set of toric bases B 2 that can support elliptic Calabi-Yau threefolds (toric geometry is described in more detail in the following section). While further progress has been made [9,10] in classifying non-toric bases, we focus here primarily on toric base surfaces, as these are the primary bases that arise in the toric hypersurface construction of Calabi-Yau threefolds. Note, however, that as we discuss later in the paper, particularly in e.g. §4.7, §6.1.3, there are cases in the Kreuzer-Skarke database where a toric polytope corresponds to an elliptic fibration over a non-toric base. The primary context in which this distinction is relevant involves curves of self-intersection −9, −10, and −11. As discussed in [7], the Weierstrass model over such curves automatically has 1, 2, or 3 points on the curve where f, g vanish to degrees (4,6). Such points on the base must be blown up for a smooth Calabi-Yau resolution, so that the actual base supporting the elliptic fibration is generally a non-toric complex surface. 5 In the simplest cases, such as F 11 and F 10 , the blown up base still has a toric description; in other simple cases, such as F 9 , the resulting surface is a "semi-toric" surface admitting only a single C * action [9], but on surfaces with, for example, multiple curves of self-intersection −9, −10, −11, the blow-up of all (4, 6) points in the base gives generally a non-toric base that is neither toric nor admits a single C * action. Despite this complication, this blow-up and resolution process is automatically handled in a natural way in the framework of the toric hypersurface construction, so that (non-flat) elliptic fibrations over bases with these types of curves arise naturally in the Kreuzer-Skarke database. Thus, we include toric bases with curves of self-intersection −9, −10, −11 in the set of bases we consider for Tate/Weierstrass constructions. The complete set of such bases was enumerated in [8], where it was shown that there are 61,539 toric bases that support elliptic CY3's. Generic elliptic Calabi-Yau threefolds over these bases give rise to a range of Hodge number pairs that fill out the range of known Calabi-Yau Hodge numbers, in a "shield" shape with peaks at (11,491), (251, 251), and (491, 11) [28]. As we see in §6, in some cases the base needed for a tuned Weierstrass model to match a toric hypersurface construction is even more exotic than those arising from blowing up points on curves of self-intersection −9, −10, −11. In these more complicated cases as well, however, the general story is the same. The polytope construction gives rise to a non-flat elliptic fibration with codimension two (4, 6) points on the toric base. Blowing these curves up gives rise to another, generically non-toric, base with multiple additional curves. After these blow-ups, there is a tuned Weierstrass model giving a (flat) elliptic fibration over the new base. While the toric base is what arises most clearly from the polytope construction, the structure of the blown up base admitting the flat elliptic fibration is relevant when considering F-theory models and anomaly cancellation.
In §4 we consider Tate tunings over the toric bases and compare to the toric hypersurface construction of elliptic Calabi-Yau threefolds, which we now describe in more detail.
3 Elliptic fibrations in the toric reflexive polytope construction

Brief review of toric varieties
Following [36,37], we review some basic features of toric geometry. An n-dimensional toric variety X Σ can be constructed by defining the fan of the toric variety. A fan Σ is a collection of cones 6 in N R = N ⊗ R, each with the apex at the origin, and where N is a rank n lattice, satisfying the conditions that • Each face of a cone in Σ is also a cone in Σ.
• The intersection of two cones in Σ is a face of each.
Then X Σ can be described by the homogeneous coordinates z i corresponding to the onedimensional cones v i (also called rays) of Σ; X Σ may be constructed as the quotient of an open subset in C k (k is the number of rays), by a group G, where • Z(Σ) ⊂ C k is the union of the zero sets of the polynomal sets S = {z i } associated with the sets of rays {v i } that do not span a cone of Σ.
• G ⊂ (C * ) k is the kernel of the map where v j,l specifies the lth component of the ray v j in the coordinate representation in C n .
Toric divisors D i are given by the sets D i ≡ {z i = 0} associated to all the rays v i . The anti-canonical class −K of a toric variety is given by the sum of toric divisors Smooth two-dimensional toric varieties are particularly simple. The irreducible effective toric divisors are rational curves with one intersecting another forming a closed linear chain. This is easily seen from the 2D toric fan description, where each ray of the 2D fan corresponds to an irreducible effective toric divisor. The intersection products are also easy to read off from the fan diagram, where (including divisors cyclically by setting D k+1 ≡ D 1 , etc.) and the self-intersection of each curve is and zero otherwise. We will generally denote the data defining a smooth 2D toric base by the sequence of self-intersection numbers. (The 2D fan can be recovered given the intersections, up to lattice automorphisms.) In the context of this paper, toric varieties play two distinct but related roles. On the one hand, we can use toric geometry to describe many of the bases that support elliptically fibered Calabi-Yau threefolds. On the other hand, toric geometry can be used to describe ambient fourfolds in which CY threefolds can be embedded as hypersurfaces, as we describe in the next section.

Batyrev's construction of Calabi-Yau manifolds from reflexive polytopes
Given a lattice polytope, which is the convex hull of a finite set of lattice points (in particular, the vertices are lattice points), we may define a face fan by taking rays to be the vertices of the lattice polytope, and the top-dimensional (n-dimensional) cones to correspond to the facets of the polytope. By including more lattice points in addition to vertices of the polytope as rays, and thus subdividing ("triangulating") the facets of the polytope into multiple smaller top-dimensional cones, we can refine the fan to impose further properties such as simpliciality or smoothness. 7 In this way, a lattice polytope can be associated with a toric variety. In general, a given lattice polytope can lead to many different varieties, depending upon the refinement of the face fan. Even for a given set of additional rays added, there can be many different triangulations of the fan.
We will be interested in particular in the fans from reflexive polytopes, which are defined as follows. Let N be a rank n lattice, N R ≡ N ⊗ R. A lattice polytope ∇ ⊂ N containing the origin is reflexive if its dual polytope is also a lattice polytope. The dual of a polytope ∇ in N is defined to be where M = N * = Hom(N, Z) is the dual lattice. If ∇ is reflexive, its dual polytope ∆ = ∇ * is also reflexive as (∇ * ) * = ∇. We call the pair of reflexive polytopes a mirror pair. Both of them contain the origin as the only interior lattice point. Calabi-Yau manifolds in Batyrev's construction [38] are built out of reflexive polytopes. Given a mirror pair ∇ ⊂ N and ∆ ⊂ M , the (possibly refined) face fan of ∇ describes a toric ambient variety, in which a Calabi-Yau hypersurface is embedded using the anti-canonical class of the ambient toric variety, so that the hypersurface itself has trivial canonical class. Explicitly, a section of the anti-canonical bundle is given by where c i are generic coefficients taking values in C and each monomial m i is given by an associated lattice point where z j is the homogeneous coordinate associated with the ray v j in the fan associated to ∇. The well-definedness of each m i in terms of the homogeneous coordinates z j is guaranteed by the linear equivalence relations among the divisors associated to v j 's, and holomorphicity in the z j s by the reflexivity of ∇. We can check that Equation (3.7) indeed defines a section of the anti-canonical class, so that a CY hypersurface is cut out by p = 0. We can determine the class by looking at any one of the monomials; we pick the origin since we know it is always an interior point. Its associated monomial by equation (3.8) is simply the product of all homogeneous coordinates associated to all toric divisors # rays j=1 z j , which immediately we see by equation (3.2) is a section in the anti-canonical class. For the smoothness of the Calabi-Yau, as mentioned previously, there exists a refinement 8 of the face fan of ∇ such that 7 A fan is simplicial if all its cones are simplicial. A cone is simplicial if its generators are linearly independent over R. A fan is smooth if the fan is simplicial and for each top-dimensional cone its generators generate the lattice N . 8 Appropriate subdivisions of the face fan of the toric ambient variety by additional lattice points in the facets of the polytope give the resolved description of the embedded Calabi-Yau, where extra coordinates in the fan is simplicial so the ambient toric variety will have at most orbifold singularities. In the case of n ≤ 4, with the anti-canonical embedding, a hypersurface will generically avoid these singularities and therefore is generically smooth. M. Kreuzer and H. Skarke have classified all 473,800,776 four-dimensional reflexive polytopes for the Batyrev Calabi-Yau construction [4,39]. A pair of relexive polytopes in the KS database are described in the format: M:# lattice points, # vertices (of ∆) N:# lattice points, # vertices (of ∇) H: h 1,1 , h 2,1 .
We will refer to ∇ as the (fa)N polytope and ∆ as the M(onomial) polytope to remind ourselves that ∇ gives the fan of the ambient toric variety for the CY anti-canonical embedding and ∆ determines the monomials of the anti-canonical hypersurface. In many cases, it is sufficient to specify polytopes with the information given in the format above, but sometimes there can be distinct polytopes with identical information of this type, in which case we will either give further the vertices of the N polytope to specify the polytope more precisely, or indicate its numerical order as it appears among those with identical data in the KS database website (http://hep.itp.tuwien.ac.at/˜kreuzer/CY/) with a superscript, e.g., M:165 11 N:18 9 H:9,129 [1] or M:165 11 N:18 9 H:9,129 [2] .
Note that conversely, we can start from ∆ and associate it with the polytope that defines the fan of the ambient space, and calculate monomials associated with lattice points in ∇. Then the hypersurface CY is mirror to the previous one. The Hodge numbers of mirror pairs are related by h p,q (CY ∇ ) = h d−p,q (CY ∆ ), where d = n − 1 is the complex dimension of the CY; in particular, we will look at 4 dimensional reflexive polytopes for CY threefolds, where the only non-trivial Hodge numbers are h 1,1 and h 2,1 , and mirror CY hypersurfaces have exchanged values for h 1,1 and h 2,1 .
For the CY associated with ∇, the Hodge numbers are given by where θ are faces of ∆,θ are faces of ∇, F ∇/∆ l denotes the set of l-dimensional faces of ∇ or ∆ (l < n), and pts(∇/∆) := number of lattice points of ∇ or ∆, int(θ/θ) := number of lattice points interior to θ orθ. As ∇ and ∆ are a pair of 4D reflexive polytopes, there is a one-to-one correspondence between l-dimensional faces θ of ∆ and (4 − l)-dimensional facesθ of ∇ related by the dual operation θ * = {y ∈ ∇, y, pt = −1| for all pt that are vertices of θ}, which makes the duality between the Hodge number formulas manifest. Note that the Hodge numbers depend only on the polytope and not on the detailed refinement of the fan.

Fibered polytopes in the KS database
For the purpose of studying (often singular) elliptically fibered Calabi-Yau threefolds that arise in the KS database, we will be interested in 4D reflexive fibered polytopes [40][41][42]35]. A fibered polytope ∇ is a polytope in the N lattice that contains a lower-dimensional subpolytope, ∇ 2 ⊂ N 2 = Z 2 , which passes through the origin. We are interested in the case where ∇ 2 is itself a reflexive 2D polytope, containing the origin as an interior point. Such a fibered polytope ∇ admits a projection map π : ∇ → N B such that π −1 (0) = ∇ 2 , and N B = Z 2 is the quotient of the original lattice N by the projection. We can construct a set of rays v in N B that are the primitive rays with the property that an integer multiple of v arises as the image π(v i ) of a ray in ∇. (A primitive ray v ∈ N is one that cannot be described as an integer multiple v = nw of another ray w ∈ N , with n > 1.) We define the base B 2 to be the 2D toric variety given by the 2D fan Σ B with v (B) i taken to be the 1D cones; the 2D cones are uniquely defined for a 2D variety. Note that in higher dimensions, the base of the fibration is not uniquely defined as a toric variety since the cone structure of the base may not be unique.
In the toric geometry language, a fan morphism is a projection π : Σ → Σ B with the property that for any cone in Σ the image is contained in a cone of Σ B . Such a fan morphism can be translated to a map between toric varieties π : X Σ → B 2 . Such a map is a toric morphism, which is an equivariant map with respect to the torus action on the toric varieties that maps the maximal torus in X Σ to the maximal torus in B 2 . As far as the authors are aware, it is not known whether in general every fibered polytope admits a triangulation leading to a compatible fan morphism and toric morphism. Note, however, that the elliptic fiber structure of the polytope does not depend upon the existence of a triangulation with respect to which there is a fan morphism π : Σ ∇ → Σ B . Thus, to recognize an elliptic Calabi-Yau threefold in the KS database, it is only necessary to find a reflexive subpolytope ∇ 2 ⊂ ∇. The Calabi-Yau manifold defined by an anti-canonical hypersurface in X Σ through the Batyrev construction with reflexive polytopes will then be an elliptically fibered Calabi-Yau threefold over the base B 2 [41]. A primary goal of this paper is to relate reflexive polytopes in the Kreuzer-Skarke database that have this form to elliptic fibrations of tuned Weierstrass models as described in §2.
There are in total 16 2D reflexive polytopes, which give slightly different realizations of an elliptic curve when an anti-canonical hypersurface is taken [35,19,43]. The hypersurface equations p = 0, with p given in (3.7), of all 16 types of fibered polytopes can be brought into the Weierstrass form (2.8) by the methods described in Appendix A in [35]; this gives an equivalent description of the same Calabi-Yau as long as the fibration has a global section. The Kreuzer-Skarke database of reflexive polytopes and associated Calabi-Yau hypersurface constructions contains a wide range of polytopes that include fibered polytopes with many different examples of the 16 fiber types.
For a given base B 2 and a given fiber type, there can be a variety of different polytopes corresponding to configurations with different "twists" of the fibration, associated with different embeddings of the rays v i defining the base B 2 with respect to the fiber subpolytope ∇ 2 . For example, the Hirzebruch surfaces F m are each associated with fibered polytopes with fiber and base P 1 , distinguished by the different twists of the fibration. For a fibered polytope ∇ with a reflexive subpolytope ∇ 2 , the dual ∆ admits a projection to ∆ 2 = (∇ 2 ) * .
One of the findings of this paper is that the bulk of KS models with large Hodge numbers appear to have a description in the form of a standard P 2,3,1 -fibered type, with a specific form for the twist of the fiber over the base surface B 2 ; these models can be connected directly to the Tate form for elliptic fibrations, and in fact can be constructed from that point of view directly. On the one hand, we describe the structure of this type of standard polytope in §3.4, with the result that the anti-canonical hypersurface equations from (suitably refined) standard P 2,3,1 -fibered polytopes are in the form of equation (2.14). On the other hand, we describe the direct construction of polytopes by carrying out Tate tunings on the effective curves in the toric bases in §4, and develop an algorithm in §5 to systematically classify models of this type that give polytopes and elliptic Calabi-Yau threefolds with large Hodge numbers; these models are all expected to have a corresponding standard P 2,3,1 -fibered polytope, and we compare the two constructions in the remainder of §6. For a given base B 2 there are generally many distinct polytopes that have the standard P 2,3,1 -fibered structure; as we describe in the following section, these correspond to different Tate tunings over the same base.

Standard P 2,3,1 -fibered polytopes and corresponding Tate models
The fiber polytope ∇ 2 that provides a natural correspondence with the Tate form models (2.14) is associated with the toric fan giving the weighted projective space P 2,3,1 ; this is a toric variety given by the rays v x = (−1, 0), v y = (0, −1), v z = (2, 3) (see Figure 1a). Given a P 2,3,1 -fibered polytope ∇ over a toric base B 2 , where the fiber is defined by three rays satisfying 2v x + 3v y + v z = 0, we can always perform a SL(2, Z) transformation to put the rays in the fiber into the coordinates v x = (0, 0, −1, 0), v y = (0, 0, 0, −1), v z = (0, 0, 2, 3) . (3.12) We can define a standard P 2,3,1 -fibered polytope over the base B 2 as one where there is a coordinate system after an SL(4, Z) transformation such that the vectors are contained within ∇ for every ray v Note that in fact, these lattice points are all on the boundary of ∇ since the only interior point of a reflexive polytope is the origin. This particular choice of fiber and twist geometry represents a very specific class of fibered polytopes that produce elliptically fibered Calabi-Yau threefolds as hypersurfaces. These standard P 2,3,1 -fibered polytopes play a central role in the analysis of this paper, and are a generalization of the well-studied 3D reflexive polytope for a K3 surface that is an (a) The toric fan for P 2,3,1 . The convex hull of P 2,3,1 plays the role of the reflexive subpolytope ∇2 for standard P 2,3,1 -fibered polytopes ∇ in the N lattice. The rays vx, vy, vz are associated with the homogeneous coordinates x, y, z, respectively, in the hypersurface equation.
lattice. Projection onto the M2 plane projects the lattice points in ∆ into seven lattice points in ∆2. These lattice points correspond to the five sections a1,2,3,4,6 in the Tate form of the Weierstrass model, indicated in the figure by xyz, x 2 z 2 , yz 3 , xz 4 , z 6 , respectively, and to the coefficients of the remaining two terms x 3 , y 2 in the hypersurface equation. elliptic fibration over a P 1 base [42]. As mentioned above, these polytopes appear to be highly prevalent in the Kreuzer Skarke database at large Hodge numbers. This seems to occur for several reasons. The P 2,3,1 fiber is the only one of the 16 reflexive 2D polytopes that is possible in the presence of curves of very negative self intersection in the base (see discussion in §4.8). And the natural correspondence between tuned Tate models and the particular twist structure defined by (3.13) makes this twist structure particularly compatible with the reflexive polytope Calabi-Yau construction. We do, however, encounter some specific examples of other twists in later sections.
For a standard P 2,3,1 -fibered polytope, the lattice points of ∆ ⊂ M in this coordinate system organize into the following sets of points: (3.14) The elliptically fibered CY hypersurface equation p = 0 with p from (3.7) then takes precisely the Tate form (2.14). The sets of points in (3.14) are associated with the monomials y 2 , x 3 , xy, x 2 , y, x, 1 respectively; y 2 and x 3 have a single overall coefficient, and the monomials in the base associated with the other five sets of points correspond precisely to monomials in the five sections {a 1 , a 2 , a 3 , a 4 , a 6 } (see figure 1.) In particular, the condition that ∆ is the dual polytope of ∇ precisely imposes the condition that a n ∈ O(−nK B ). For example, for a 6 we have the condition on the monomial associated with the point (m 1 , m 2 , 1, 1) that v ) in the fan of the base B 2 , so (m 1 , m 2 ) represents a section of −6K B 2 , in much the same way that the monomials in (3.8) represent sections of −K of the ambient toric variety. A similar computation for each a n confirms that the corresponding monomials satisfy v An analogous computation shows that for the points associated with y 2 and x 3 the condition is v 2 m 2 ≥ 0; for any compact base this implies that m 1 = m 2 = 0, so the first two points in (3.14) are the only points of the form (m 1 , m 2 , 1, −1) and (m 1 , m 2 , −2, 1) and are associated with constant functions on the base. This matches with the fact that these are sections of the trivial bundle O over the base, and the fact that the only global holomorphic functions on any compact base are constants. This proves that for any standard P 2,3,1 -fibered polytope, the lattice points in ∆ are associated precisely with the Tate form of a Weierstrass model over the base, as stated above.
In the simplest cases, all the lattice points of the polytope ∇ are simply given by the vectors (3.12) and the vectors of the form (3.13). This corresponds to the generic elliptic fibration over a toric base B 2 without non-Higgsable clusters. In other cases, however, there are lattice points in ∇ other than those given by (3.12) and (3.13). This corresponds to situations with NHCs or gauge groups tuned over curves in B 2 by removing Tate monomials. The set of monomials in ∆ completely span the set of sections of the appropriate line bundles O(−nK B ) for the generic elliptic fibration over a given base. In the case of NHCs, in particular, the monomials in ∆ span the appropriate set of sections, while in the case of gauge group tunings, some of these monomials are set to zero. From the point of view of the Calabi-Yau geometry, the lattice points in ∇ other than those given by (3.12) and (3.13) reflect the singular nature of the resulting Calabi-Yau hypersurface. Up to some monodromy subtleties that we discuss further in §4, the set of new lattice points introduced together with v [18,44,45], which forms the extended Dynkin diagram of the gauge algebra of the singular fiber over the associated divisor D the affine root (this is the only inverse image when the fiber is smooth). In section 4 we describe in more detail the dictionary between Tate tunings and toric/polytope geometry for specific gauge groups on particular local curve configurations in the base geometry.

A method for analyzing fibered polytopes: fiber types and 2D toric bases
Our primary approach in this paper is to systematically construct Tate tunings that should have counterparts as reflexive polytopes in the Kreuzer-Skarke database. Thus, we start from the F-theory construction and match the results with the known data in the KS database. This gives us something like a "sieve" that leaves behind a set of special cases of KS data not produced by our algorithm. After implementing this sieve, we have then considered separately those few examples in the KS database in the range of interest that were not found by our F-theory construction. We have found that there are a few polytopes in the KS database that can be described in terms of the standard P 2,3,1 -fibered type; i.e., have Tate forms, but were nonetheless not found with the initial sieve. This turns out to be because they involve such extensive tunings that the starting bases needed are outside the range we considered. There are also data in the KS database that we did not identify in the original sieve because they are accompanied by more sophisticated constructions involving u(1) tunings, novel su(6) tunings associated with exotic matter representations, or tunings of generic models over nontoric bases, which we had not considered. Moreover, we encounter a type of novel models that did not arise from our systematic construction because they are involved with tunings on non-toric curves in the base; they turn out nonetheless to also be described by reflexive polytopes with toric fibers associated with elliptic fibrations.
We had to explicitly study these specific polytopes with Hodge numbers that we did not immediately identify from Weierstrass/Tate tunings to determine whether these polytopes give hypersurfaces that are actually elliptically fibered. We provide here a summary of our algorithm to analyze reflexive polytopes. We can learn from this analysis whether one of the 16 reflexive fiber types is a fiber of the polytope in question; we then define the 2D toric base from the fibered polytope. As we describe later in the paper, we can thereby determine the singularities of the elliptic fiber over the curves in the base, and then we check that the Hodge numbers of the inferred tuned model are consistent with those of the polytope model. Here we briefly summarize the first piece of this analysis: the algorithm to determine if a given reflexive 2D polytope is a fiber of a 4D polytope. There are also software programs like Sage [46] with built in routines to identify the reflexive subpolytopes of a given polytope.
1. We assume that we are interested in a fiber described by the 2D reflexive polytope ∇ 2 .
To increase the efficiency of the algorithm in the case that the number of lattice points in ∇ is large (which is true in the case of large h 1,1 that we are focusing on), we begin by focusing on only a subset of these lattice points that can possibly play a role as the points in a fiber ∇ 2 . As mentioned in §3.3, the presence of a fiber subpolytope ∇ 2 ⊂ ∇ implies that there is a projection from ∆ → ∆ 2 . Let us call the maximum value of the inner product for any pair of vectors in the fiber and its dual For example, for P 2,3,1 , M max = 5, and for P 1,1,2 , M max = 3. We can then check for each lattice point v ∈ ∇ whether there exists a vertex w in ∆ with v · w > M max . If there is, then v cannot be a ray in a fiber ∇ 2 . We collect the subset of rays in ∇ that are not ruled out by this condition: 2. We then look for a subset of rays of S that satisfy the necessary linear relations to be elements of the fiber ∇ 2 . For example, for P 2,3,1 , we want to find rays {v In this case we can look at all pairs of rays v, v in S, and check to see if 2v + 3v is also an element of S. If so, we can then check that the intersection of ∇ with the plane spanned by v, v precisely contains the 7 points in the polytope ∇ 2 shown in Figure 1a.
If this is the case than ∇ has a fiber ∇ 2 . The other fiber types can be checked in a similar fashion.
By equations (3.8), (3.15) and the projection ∆ → ∆ 2 , the maximum exponent of all monomials in the variables associated with the rays in the fiber should be M max + 1, and the monomials can be grouped according to the powers of the fiber coordinates into sets that are in one-to-one correspondence with the lattice points in ∆ 2 . For example, for P 2,3,1 -fibered polytopes (see Figure 1b), we have the maximum exponent in z among all fiber coordinates; M max + 1 = 6, and the lattice points in ∆ 2 are in one-to-one correspondence with the sections Note that, following the definition of a standard P 2,3,1 -fibered polytope from §3.4, the lattice points in ∆ 2 are in one-to-one correspondence with the sections of the line bundles O(−nK B ), and the monomials x 3 and y 2 are the only two independent of the base coordinates. Similarly, the sections of the P 1,1,2 -fibered polytope (see Figure 5) are when the associated rays are such that and M max +1 = 4. The first step in the algorithm above is only used to speed up the algorithm, but particularly when the number of lattice points in ∇ is large, this speedup is significant. For example, for the polytope associated with the Calabi-Yau with Hodge numbers H:491, 11, the number of lattice points in ∇ is 680, while the number in S is only 9. Since the second step of the algorithm is quadratic in the number of lattice points considered, this represents a speedup by a factor of hundreds or thousands of times in many cases. While in this paper we are only considering a few examples, such a speedup is useful when considering larger datasets.
In the companion paper [17] we will describe the systematic application of this algorithm to all elements of the KS database with large Hodge numbers. Once we have determined the fiber, we can then compute the base B 2 of the fibration. We define the set of rays of the fan describing B 2 to be i,2 ) and π is the projection along the fiber subpolytope (π(∇ 2 ) = 0). The division by GCD(v i,2 ) is done to restrict to primitive rays in the image, as discussed in §3.3. Given the rays v (B) i , we associate a 2D cone with each pair of adjacent rays, giving a unique toric structure to the base geometry B 2 . 9 Note that the base defined this way gives a flat toric fibration, but not necessarily a flat elliptic fibration [35]. We discuss this point in more detail in later sections.
In the regions of the Hodge numbers that we study in this paper, we also encounter polytopes that have no standard P 2,3,1 fiber. These polytopes can be described using two different types of models. One of these other types of model that we encounter is very similar to the standard P 2,3,1 -fibered polytopes, but has a fiber that is a single blowup of P 2,3,1 . This Bl [0,0,1] P 2,3,1 fiber, which is one of the other 16 reflexive 2D fiber types, is shown in Figure  7. The corresponding fiber subpolytope ∆ 2 is identical to that for the P 2,3,1 fiber except that it has an additional vertex at (−1, −1), so that the number of lattice points in the plane of the fiber subpolytope is 8 rather than 7. From the Tate point of view, such a fiber occurs when all the monomials in the coefficient a 6 are taken to vanish. This vanishing of a 6 forces a global u(1) symmetry that we mentioned earlier [19,48]. We describe an explicit example of this type of model in Appendix C. Models with this fiber can be treated in essentially the same fashion as standard P 2,3,1 -fibered polytopes.
The other unusual kind of fibration that we encounter in a few models is a fibered polytope with fiber ∆ 2 given by the usual P 2,3,1 polytope, but with a different "twist" to the P 2,3,1 bundle over the base. In other words, while there is a projection of ∆ to the dual polytope ∆ 2 of the P 2,3,1 fiber, the base rays in ∇ do not all lie in a plane that contains the vector v z ; i.e., the base of the polytope defined in (3.21) can not be constituted by a set of rays all in the form (3.13). The consequence of this is that the hypersurface equations (3.7) for these Calabi-Yau threefolds do not take on the Tate form (2.14). In particular, there is generically more than one lattice point projected to the points in ∆ 2 associated with y 2 and/or x 3 . To determine the Weierstrass form (2.8) for the models of this type that we found and analyze their structure, we found that it was useful to view them as essentially "P 1,1,2 -fibered polytopes" (or more precisely, P 1,1,2 with two more blowups) rather than the standard P 2,3,1 -fibered polytopes (see figure 5 for comparison). This allows us to follow the method for analyzing P 1,1,2 -fibered models described in Appendix A of [35] to bring them into Weierstrass form. This type of novel model gives rise to an enhancement over non-toric curves as we mentioned earlier.
We refer to this type of models as non-standard P 2,3,1 -fibered polytopes, and describe their analysis in more detail in section 6.2. The treatment of non-standard P 2,3,1 models in terms of models with a blow-up of P 1,1,2 as a fiber is closely analogous to the analysis of models with a Bl [0,0,1] P 2,3,1 fiber as special cases of P 2,3,1 Weierstrass/Tate models.

Tate tunings and the Kreuzer-Skarke database
We want to understand how the set of Calabi-Yau threefolds produced by toric hypersurface constructions through reflexive polytopes in the Kreuzer-Skarke database can be related to 9 Note that in higher dimensions, the cone structure of the fan is not uniquely determined by the rays. the general construction of elliptic Calabi-Yau threefolds through tuned Weierstrass models. The approach we take is to identify a specific subclass of tuned models that match with toric hypersurface constructions. In particular, we begin with the set of toric bases identified in [8] and consider Tate tunings over these bases.
In principle, to find all the elliptically fibered threefolds in the Kreuzer-Skarke database we might want to consider a variety of tunings and singularity structures that correspond to all 16 of the toric fiber types mentioned in §3.3. To simplify the set of possibilities, however, we focus on a region of Hodge numbers where we expect a single toric fiber type to dominate. A generic Tate-form elliptic fibration over a given toric base can always be constructed starting from the "simple stacking" procedure as we will describe in §4.1 and §4.3; this procedure uses the standard P 2,3,1 fiber type. Tuning the resulting generic Tate model by removing monomials in the dual polytope then leads to a set of possible tunings corresponding to further reflexive polytopes that can appear in the database; we describe this process in §4.3.3 and §4.5, and give an example in Appendix A. Such a construction can be carried out for any base. The gauge symmetries associated with the tunings can be read off from the tops [18,42,44,45] of the polytopes. We review polytope tops in §4.2, where we also give new results on tops from tunings of algebras su(n) and so(n) of rank higher than six; we also address some subtle issues about the multiple tops of a gauge algebra in §4.4, which is related to the monodromy choices of the Tate tunings of the algebra that we have discussed in §2.3.
The other 15 fiber types, however, implicitly constrain the Weierstrass model associated with an elliptic fibration. We explain in §4.8 some constraints on the other 15 fiber types, which are related to the structure of the base. Based on these constraints, we expect that when we confine the range of Hodge numbers to relatively large values, as we do in section 5, the simplest P 2,3,1 fiber type will dominate the set of polytopes. 10 By focusing on this simple class of constructions, therefore, we realize almost all the Hodge numbers in the range of interest with a single class of Tate-tuned elliptic fibration models. Although we will not deal with the matching of the multiplicity in KS database of a given Hodge pair with our systematic tuning construction, we will explore a bit more this aspect in sections 4.6 and 4.7.

Reflexive polytopes from elliptic fibrations without singular fibers
In §3.4 we defined a standard P 2,3,1 -fibered polytope, and showed that there is always a corresponding Tate model. Now we are trying to do the converse -given a toric base and a corresponding Tate model, we wish to construct a corresponding reflexive polytope. As alluded to earlier, the recipe for the construction of a 4D standard P 2,3,1 -fibered polytope for an elliptically fibered threefold is the natural generalization of the 3D reflexive polytope for a K3 surface that is an elliptic fibration over a P 1 base as described in e.g. [42].
To construct a 4D P 2,3,1 -fibered polytope, we start with the 2D P 2,3,1 fiber and a 2D base, and we construct the polytope in a straightforward way to have the desired fibration Fan Table 6: Non-convexity of NHCs: The rays corresponding to an NHC cannot be a vertices; hence, the vertex contribution from the base can only come from curves of self-intersection ≥ −1 (isolated -2 curves will be on a 1D face, and also cannot be vertices).
structure over the base. We denote the toric fan associated with the base B by Σ B , with the set of rays being {v Taking the fan of P 2,3,1 to be the ambient space of the elliptic fiber, we can embed this in the 4D coordinates such that the three rays are {v Since in the Weierstrass or Tate model framework of equation (2.9) the fiber coordinate z is associated with the trivial bundle over the base, the lattice point associated with the ray v z = (0, 0, 2, 3) should be in the plane of the base. Thus, we define a polytope∇ to be the convex hull of the set i,2 are the first and the second components of the 1D ray v in the smooth 2D toric base B. From the definition in the previous section, this is a standard P 2,3,1 -fibered polytope; we refer sometimes to this construction as the "simple stacking" approach to construction of a polytope. Note that the 4D rays Table 6). We now wish to check that∇ is reflexive, so it can be used as the reflexive polytope ∇ in Batyrev's construction of a Calabi-Yau threefold. In some cases∇ is immediately reflexive, and in other more complicated cases it must be modified to make it reflexive.
We start with the simplest case, in which we have a generic elliptically fibered Calabi-Yau over a toric base B that contains no non-Higgsable clusters (i.e., no curves with self-intersection less than −2). In this case, the Weierstrass/Tate model of the Calabi-Yau is smooth and there is no gauge group in the 6D supergravity theory. In this context, lattice points associated to curves of self-intersection −2 lie on the 1D faces of∇ that are boundaries of the 2D face θ B , which is the 2D face associated with the base; and there are no interior points in θ B other than (0, 0, 2, 3). We can now check directly that in these simple cases∇ is reflexive without further modification. The vertices of the polytope dual to the convex hull of the set of vertices (4.1), in any case, are where (i, j) are taken to run over all pairs of labels of base rays that correspond to adjacent vertices of θ B . The vertices in (4.2) will lie on the M lattice only when the denomina- j ] are cleared so that all entries are integers. For a smooth 2D base fan, i+1 ] = 1, so we have a lattice point whenever j = i + 1 (including the boundary case j = 1, i = n); i.e., we get lattice points as long as there are no non-convex base rays, which would be skipped. We also get a lattice point as long as v j ] = k, so we again have a cancellation and the vertex of the dual polyhedra is an integral lattice point. Thus, as long as the base B contains no non-Higgsable clusters, the set of vertices (4.1) immediately provides a reflexive polytope. 11 Simple examples of polytopes realized in this way are the elliptically fibered Calabi-Yau threefolds over the toric bases P 2 , F n=0,1,2 , whose vertex sets of the M polytopes ∆ are (4.2), with the first set of vertices respectively being {(−6, −6, 1, 1), (12, −6, 1, 1), (−6, 12, 1, 1)}, {(−6, −6, 1, 1), (6(1 + n), −6, 1, 1), (6(−n + 1), 6, 1, 1), (−6, 6, 1, 1)}, given the respective base The P 2 model gives the only polytope (up to lattice automorphism) with Hodge numbers H:2,272 in the KS database and the F n=0,1,2 models give exactly the three data points with Hodge numbers H:3,243.
The bases described by toric varieties with no curves of self-intersection less than −2 are weak Fano varieties, and correspond to reflexive 2D polytopes, as we have just verified explicitly. We now want to describe the generalization of this construction to situations where there is a gauge group arising either from a non-Higgsable cluster in the base or a Tate tuning. The realization of reflexive 4D polytopes in these cases arises from a general relationship between Tate tunings and "tops" in the toric language.

Tate tuning and polytope tops
We saw in §3.4 that for a standard P 2,3,1 -fibered polytope, the lattice points of ∆ that project to each of the different lattice points of ∆ 2 (figure 1) correspond precisely to the sets of monomials in the coefficients of the Tate form (2.14). The lattice points of ∆ are thus divided into 5 groups corresponding to the 5 sections a n ∈ O(−nK B ) and another 2 points corresponding to the constant coefficients of y 2 and x 3 . In the previous subsection we described generic elliptic fibrations over weak Fano bases, where the "simple stacking" procedure immediately gives a reflexive 4D polytope, and no additional rays are needed in ∇, corresponding to a physics model with no nonabelian gauge group. We now wish to consider how this story changes when there is a nontrivial nonabelian gauge group due either to an NHC in the base or a Tate tuning of the monomials in the Tate form.
The presence of an NHC in the base or an explicit Tate tuning can force some of the coefficients in the a n s to vanish to some specified order along a particular base divisor D (B) i . This absence of monomials in ∆ gives rise to a corresponding enlargement of ∇ from the simple stacking. The additional lattice points in the fan polytope ∇ correspond to the exceptional divisors that resolve the singularities of the associated fibered geometry. These additional lattice points form the "top" [18,42,44,45] of the enhanced gauge symmetries over D is the associated 2D ray, and l ∈ N specifies the "level" of the point away from the fiber plane (see figure 2). We adopt the shorthand notation pt (l) j or pt ··· j , where the number of primes specifies the level parameter l. When we denote a top, the points with fewer than the maximal number of primes over each point are omitted and implied by the point of most primes with the same index; e.g.
(0, 0, 1, 1, 2), associated with the absence of a certain set of lattice points in ∆, can the point pt 2 appear in ∇, and indeed if all the a n s vanish to these orders then the point pt 2 must appear in the polytope ∇ dual to ∆. Thus, there is a precise local correspondence between Tate tunings of the a n coefficients over a certain ray in the base, associated with lattice points absent from ∆, and the toric top in ∇ over that ray. We tabulate a few examples of this correspondence in Table 9. Note that just as multiple Tate tunings can correspond to the same gauge algebra, the corresponding multiple tops also correspond to the same gauge algebra. The multiplicity of constructions for a given gauge algebra was studied from the point of view of tops in [19]. This correspondence leads to a natural association of reflexive polytopes with elliptic fibrations over toric bases that have Tate forms. Over a given base, various gauge groups can arise from a combination of non-Higgsable clusters and Tate tunings. The interplay between extra vertices in ∇ over nearby divisors and the absence of monomials in ∆ leads to local interactions between the sets of lattice points in the polytope that are affected by adjacent rays in the base. We consider more explicitly in the following section how this leads to consistent reflexive polytopes in both the NHC and Tate tuning cases.

Reflexive polytopes for NHCs and Tate tunings
In this subsection we describe the construction of reflexive polytopes from elliptic fibrations corresponding to F-theory models with gauge groups from non-Higgsable clusters or tuning. We give the construction of generic models over bases with NHCs in §4.3.1 and §4.3.2, and constructions with tunings in §4.3.3.

NHCs with immediately reflexive polytopes
Now consider models where the base has a non-Higgsable cluster. We begin with the simplest cases, where the NHC contains a single curve of negative self-intersection −m, and m|12. In these cases, the simple stacking construction described in section 4.1 leads directly to a reflexive polytope. This can be understood from several points of view. Due to the factor 6 in the numerators of the first two coordinates in (4.2), those cases where a ray is skipped and Det[v j ] = 3, or 6 also give lattice points; i.e., when the skipped rays are NHCs −3 and −6; furthermore, the NHCs −4 and −12 are fine as well because of extra factors of 2 that arise from the difference terms in the numerators. Therefore the set (4.2) should also  Table 11) forming the extended Dynkin diagram of the gauge algebra with pt 1 the affine node.
be sufficient to give the ∆ polytopes of the models with the NHCs −3, −4, −6, −12, so that the simple stacking polytope ∇ defined through (4.1) is reflexive. The values of m compatible with the simple stacking can also be understood from the bounds on the set of monomials in a 6 controlled by the −m curve. Other than the vertices from x 3 , y 2 , all vertices of ∆ come from lattice points associated with monomials in a 6 . Choosing local toric coordinates for a set of adjacent rays v and is the only reflexive polytope in the M lattice (up to lattice automorphism) associated with the Hodge pair H:11,491 in the KS database. We can understand the reflexive polytopes formed in this way in terms of the dual Tate tunings and tops described in the previous subsection. For example, consider the case of the −3 curve NHC. Using again the local toric coordinates (4.6) with m = 3, the polytope ∇ has vertices from (4.1), (1, 0, 2, 3) and (−1, −3, 2, 3). Considering a 3D slice of ∇ that contains the fiber polytope ∇ 2 and the ray v  Figure), so the reflexive polytope associated with a simple stacking from a base with a −3 curve automatically has the point pt 2 = (0, −1, 1, 2) in the top in ∇. Using the same methodology as in the n = 6 example above, we see that the orders of vanishing of the a n s in the dual polytope are (1, 1, 1, 2, 2). From Table 4, we see that this is a type IV singularity; in this case the monodromy condition for the gauge algebra su(3) is automatically satisfied, so this actually corresponds to an su(3) top, as indicated in the first line of Table 11.

Other NHCs: reflexive polytopes from the dual of the dual
The rest of the NHCs have the issue that there are fractions in the vertices of the dual polytope described by (4.2). Let us denote the convex hull of the set of vertices defined by (4.1) by∇, and its dual by by∆. If∆ is not a lattice polytope then∇ is not a reflexive polytope. We have to supply∇ with additional lattice points to make it into a reflexive polytope ∇ so that ∆ = ∇ * is a lattice polytope.
We can turn∇ into a reflexive polytope in a minimal fashion by taking the "dual of the dual". We begin by defining the lattice polytope ∆ • = convex hull(∇ * ∩M ) to be the polytope defined by the convex hull of the set of integral points of∆; the polytope ∆ • then has itself a dual ∇ = (∆ • ) * . This gives us the minimal reflexive polytope ∇ ⊃∇ in the N lattice that we are looking for; for any base with NHCs, as we have confirmed by explicit computation in each case, the resulting ∇ indeed has a dual ∆ = ∇ * that is a lattice polytope.
This "dual of the dual" procedure adds points in the N lattice that are needed to complete the tops associated with the gauge symmetries coming from the NHCs in all cases other than those of a single curve with self-intersection n|12. For example, take the generic model over is {pt 1 , pt 1 , pt 2 , pt 3 }, it is {pt 1 , pt 1 , pt 1 , pt 2 , pt 2 , pt 3 , pt 4 } in ∇; the latter is exactly the f 4 top as described in earlier literature, which is obtained explicitly via the ∆ • construction we just described above.
For each of the NHC's, Table 11 describes the tops that arise over the divisors supporting the NHC, the corresponding Tate forms, and the vanishing orders of f, g, ∆ along with the resulting gauge algebra. The minimal top associated with the ∆ • construction of ∇ as the dual of the dual is in each case the first top listed. In a number of cases there are other higher Tate tunings that give different tops but the same gauge algebra, as discussed further in §4.4. The global models describing generic elliptic fibrations over the Hirzebruch surfaces that incorporate each of the single-curve NHC's are also described explicitly in Table 10, showing how this construction works in the context of the global polytopes. While in this paper we focus on the systematic construction of polytopes through tuning of Tate forms (corresponding to the structure of ∆), one could also construct general polytopes by considering the different tops over each base and thus classifying polytopes ∇; in Table 11 we also list the possible new vertices that may arise in the polytope ∇ for each top.

Reflexive polytopes from Tate tunings
We can understand Tate tunings in the polytope in a similar fashion. Consider starting with the reflexive polytope ∇ associated with the generic elliptic fibration over a given toric base B, constructed as above using the simple stacking procedure and the dual of the dual if needed for NHC's. We take ∆ to be the dual polytope of ∇, which is also a lattice polytope. We can produce an additional gauge group beyond the minimum imposed from the NHC's by performing a tuning in the Tate description of the model, which corresponds to removing certain vertices from the polytope ∆. Using a Tate tuning from Table 4 gives us the set of lattice points that should be removed from ∆ associated with certain coefficients in the a n s over the divisor(s) in B. Calling the new M polytope that results from removing these lattice points∆, we get an enlarged N polytope∇ = (∆) * , which has extra lattice points. In general, each Tate tuning in ∆ gives a corresponding top in ∇, giving a new reflexive polytopê ∇. This gives a large class of constructions for Tate tunings that should have reflexive polytope analogues in the KS database.
As a simple example, consider the polytope ∇ associated with the generic elliptic fibration over F 2 . As discussed in §4.1 this polytope follows from the simple stacking procedure and has vertices given by  Now consider a Tate tuning of the algebra su(2) over the −2 curve C in the base, which corresponds to the 2D toric vector (0, −1). This is achieved by setting a 1 , a 2 , a 3 , a 4 , a 6 to vanish to orders {0, 0, 1, 1, 2} in the coordinate associated with C, which is the second coordinate in ∆.
The set of the lattice points that have to be removed from ∆ to achieve the required vanishing orders is {(−6, 5, 1, 1), (−6, 6, 1, 1), ( This gives the reflexive polytope∇ given by augmenting ∇ from (4.8) with the additional lattice point (0, −1, 1, 2), which gives the su(2) top over C, as described in Table 9. The resulting polytope corresponds to the example M:329 7 N:12 6 H:4,238 in the KS database. The Hodge numbers from the polytope data are consistent with those from the anomaly cancellation calculation in equations (2.6) and (2.7) with a tuning of su(2) on the isolated −2 curve: ∆h 1,1 = rank(su(2)) = 1, ∆h 2,1 = dim(su (2) In general, we find that the correspondence described in the last few subsections between Tate tunings and tops immediately provides reflexive polytopes for most Tate tuning constructions. There are several subtleties in this construction, which we elaborate in the remainder of this chapter.

Multiple tops
One thing that we have found in considering the variety of Tate tunings and the corresponding models in the KS database is that for many gauge algebras there are multiple distinct tops that can arise in the N -polytope ∇. This multiplicity of tops was also discussed in [19]. These different tops correspond to distinct Tate tunings of the same gauge algebra. In many cases these arise in situations where the gauge algebra in the Weierstrass model depends upon some monodromy condition, which may be satisfied automatically in certain cases by the Tate tuning.
As an example of this phenomenon, consider the generic model over the F m=3 base,  Table 9, this might appear to be an "su(2)" top; however looking explicitly at the Tate form associated to the polytope ∆, the vanishing orders along the −3-curve are {1,1,1,2,2} in terms of the five sections a n , and {2, 2, 4} in terms of {f, g, ∆}, and the su(3) monodromy condition is satisfied -hence the gauge algebra is indeed su(3) (indeed, we know from the presence of the −3 NHC that su(2) is not possible in this geometry.) In §2.3 (see in particular Table 4), we described two distinct Tate tunings for su (3). In this case, the geometry matches the alternate Tate form for IV s associated with vanishing of a 6 to order 2 and an additional monodromy condition, and the "top" is a non-standard su (3) top. There also exists a polytope model with the "usual" su(3) top: adding pt 3 ((0, −1, 1, 1)) to the top gives another polytope model M:347 7 N:13 6 H:5,251, which has the standard su(3) top; on the Tate side this model can be obtained by the reduction of the M polytope such that the vanishing orders along the −3-curve become {1, 1, 1, 2, 3} -the standard Tate form for IV s . Analogous situations arise for the NHCs −4 and −6 as well: in these cases, as discussed above, ∇ in equation (4.11) is already a reflexive polytope model of the generic CY over F m=4,6 . The tops over the −m curves in these cases look like those appearing in the literature for gauge algebras g 2 , f 4 respectively, and the vanishing orders along the −m-curves are {1, 1, 2, 2, 3}, {1, 2, 2, 3, 4} for m = 4, 6, which would naively be tunings for g 2 , f 4 . In these cases, however, the gauge algebras are actually so(8), e 6 with monodromy conditions satisfied. Just like the case for F 3 , there are also generic polytope models over F 4,6 that have the usual so(8), e 6 tops and Tate vanishing orders of so (8), e 6 . The extra lattice points in the tops of these ∇ polytopes precisely correspond to the reduction in Tate monomials of the M polytope ∆.
In addition to multiple tops associated with monodromy conditions in Tate tunings, there are also other Tate tuning/top combinations that can arise for certain gauge groups. We have not attempted a systematic analysis of all possibilities, but we have encountered a range of possibilities simply in analyzing the polytopes of the KS database with fixed Hodge numbers and associated Tate tunings for the dual polytopes. To give a sense of the possibilities that arise, we list the structures of the polytopes in the KS database that have the Hodge numbers of generic elliptically fibered CYs over F m bases for 0 ≤ m ≤ 12 in Table 10. The details of the corresponding Tate forms for the −m NHCs are given in Table 11. Note in particular, that in addition to those models mentioned above, there is a third polytope model associated with the Hodge numbers of the generic elliptic fibration over F 6 in addition to the monodromy construction and the standard construction discussed above. This third possibility involves a further specialization of the vanishing orders of the standard construction along the −6curve, giving a further reduced M polytope ∆. Another interesting case of multiple tops that arises in these tables is the possibility of a second type of Tate tuning/top for e 7 on a −8 curve. In this case there is no monodromy issue 12 , but a second Tate tuning where the degree of vanishing of a 6 is enhanced, associated with a second e 7 top and corresponding reflexive polytopes.
In the analysis in the remainder of this paper we focus on classifying the possible elliptic fibrations constructions through the set of Tate tunings. One could also, however, imagine classifying different reflexive polytopes by considering all ways of augmenting the set of vertices (4.1) associated with the "simple stacking" with all possible tops. Proceeding in this fashion would require a systematic way of identifying the complete set of tops for each possible tuned gauge group.
We will not deal systematically with the explicit triangulation of ∇, corresponding to the resolution of the Calabi-Yau threefold, but make some comments here on the relationship between extra rays in ∇ and the resolution of the singular fiber associated with a tuned or 1 , pt 2 , pt 3 , pt 4 , pt 5 } non-Higgsable gauge group. Many of the details of this correspondence were worked out in [44,45]. When the gauge algebra is non-trivial over a divisor D (B) , there are lattice points in the top over v (B) in addition to just pt 1 . Specifically, in the cases where there are no lattice points in the top lying in the interior of the 2-dimensional faces of ∇, the lattice points in the top that do not lie in the interior of the 3-dimensional faces of ∇ form the Dynkin diagram of the gauge algebra. These correspond to the exceptional divisors that arise in the resolution of the corresponding singularities. However, when there are lattice points lying in the interior of the 3-dimensional and the 2-dimensional faces of ∇, they contribute to the second and third terms, respectively, in Batyrev's h 1,1 formula (3.10). The second term corresponds to components that miss the hypersurface, and contributions to the third term arise when the singularity is not resolved by a toric divisor but rather by a non-toric deformation, so the Dynkin diagram is not fully visible from the top. This happens exactly in those gauge algebras with an additional monodromy condition that is automatically satisfied. In summary, ∇ models are divided into two types according to whether there is a nonzero third term in the h 1,1 formula (3.10): (1) Trivial third term: There is no lattice point lying in the interior of any two-dimensional face. Gauge algebras can be read off directly from tops (the nodes of the Dynkin diagram are given by the lattice points in the top that do not 1 , pt 2 , pt 3 , pt 4 , pt 5 } pt  lie in interior of facets), which are those in the literature. The Tate forms are those with no additional monodromy condition, which again match those in the literature. The nodes also correspond to exceptional divisors resolving the singular fiber.
(2) Non-vanishing third term: There are lattice points lying in the interior of two-dimensional faces. These cases give rise to the additional Tate forms we have described. For example, in the gauge algebras involved with monodromy conditions, there are Tate forms of lower degrees, which achieve the gauge algebras by satiesfying the additional monodromy conditions automatically. The singular fiber is (partially) resolved by deformation. Therefore, there are fewer exceptional divisors in the top, in which the "Dynkin diagram" would seem to be the lower rank gauge algebra counterpart. Finally, recall from Table 1 that there are rank-preserving tunings of certain gauge algebras that leave the Hodge numbers of an elliptic Calabi-Yau unchanged. These are also associated with further Tate tunings on ∆ and additional tops in ∇ that do not change the Hodge numbers from the generic elliptic fibration over a given base. The polytopes listed in Table 10 include rank-preserving tunings of g 2 over the −3 curve in F 3 , and f 4 , so(9) over the −4 curve in F 4 . A detailed example of the polytopes associated with different tunings of su 3 and g 2 over F 2 is given in Appendix B.

Combining tunings
A final important issue that we must consider in attempting to systematically construct global models associated with polytopes is whether given a generic model over a given base, all combinations of Tate tunings that are each possible locally can be combined into an allowed global model. This depends on the global structure of the base and can be tested by the Tate-Zariski decomposition discussed in §2.5. As discussed there, we can perform a Zariski decomposition, with the initial values of {c j,n |n} over each curve set to be the initial values we want in Table  11. We then carry out the Tate-Zariski iteration procedure and if the Zariski decomposition with the desired vanishing values and corresponding gauge groups does not exist, there will not be a corresponding polytope model. In general, if the Zariski decomposition works out, there is a corresponding polytope. We do not have a proof of this in general but as we see later, at least the Hodge numbers of every elliptic Calabi-Yau threefold constructed in this way arise from a polytope in the KS database. This analysis of combined tunings through Tate-Zariski is the essential analysis we carry out in our systematic enumeration of Tate tunings that should have corresponding polytopes. To illustrate the issues that can arise we give  Table 11 and the discussion in the preceding subsection, we see that there are three different possible Tate tunings over the −6 curve: {1, 2, 2, 3, 4}, {1, 2, 2, 3, 5}, {1, 2, 2, 4, 6}. We wish to know which of these three tunings leads to a consistent Tate-Zariski decomposition, and which corresponding polytopes exist.
For the polytope ∇ in each of these three cases, we have the vertices from the fiber We now consider each of the tunings in turn over the −6 curve: 1. Minimal tuning {1, 2, 2, 3, 4}, corresponding to no additional top vertex from Table 11. This construction leads to a consistent Zariski decomposition, which gives rise to the generic polytope model M:148 11 N:33 11 H:23,107 [2] : we start with the initial configuration   Note that these models also illustrate another point: a vertex of the base can only come from curves with self-intersection number m greater than −2, but all curves with m > −2 will not necessarily be vertices. Though this generally is the case for small h 1,1 , exceptions increase as h 1,1 increases, since additional rays can expand the convex hull of the base polytope. Also, a vertex associated with a tops can only come from those possibilities listed in the third column of Table 11, but the entries in that column are not always vertices, though they are always lattice points in the N polytope ∇. This fact can be seen in the first model in the second example: pt 4 over D B156 ([-8]) is not a vertex point.

Tate tunings and polytope models of so(n) gauge algebras
As described in §4.4, for some gauge algebras such as su 3 and e 6 there are multiple tops associated with distinct Tate tunings, where one tuning involves an additional monodromy condition. This also occurs for the gauge algebras so(n). We discuss in this subsection some particular aspects of so(n) tunings and the associated reflexive polytopes, which have some unique features.
As can be seen from Table 4, for each of the so(n) gauge algebras with n even, starting with n = 8, there are two distinct Tate tunings that realize the algebra, with one (both in the case of so(8)) involving a monodromy condition. (Note that these forms in the table expand on earlier versions of the table appearing in the literature, which did not include all these possibilities.) As discussed in §4, the monodromy condition for the weaker Tate tuning can be realized automatically when the leading terms in certain a i s are powers of a single monomial, corresponding in the polytope language to a condition that the associated set of lattice points contain only a single element with appropriate multiplicity properties.
As for su 3 , e 6 , we find that both kinds of Tate tunings of the so(2n) gauge algebras can arise in corresponding polytopes in the KS database, corresponding to the usual condition that a global Tate-Zariski decomposition is possible. We also note, however, that when the algebra so(2n − 1) can be realized on one polytope over a given curve, then the monodromy realization of so(2n) is generally not possible, though the higher Tate tuning generally is. This basically corresponds geometrically to the question of whether the minimally tuned Tate model with the weaker vanishing condition has the appropriate single monomials in the a i s, or not. By the same token, the gauge algebra so (8), which has only monodromy realizations, can only be realized when neither g 2 or so(7) is possible over a given curve, which essentially reduces the appearance of this algebra to the NHC structure of −4 curves.
To illustrate these points we give     monodromy condition is automatically satisfied. This can also be understood from the perspective of global symmetry constraints [32]; when the gauge algebra is so(9) on a −4-curve, the maximal global symmetry algebra is sp(1), so it is not possible for so (9) to appear on D B4 next to two sp(1)'s. Thus, D B4 indeed must carry the gauge algebra so(10), for which the maximal global symmetry algebra is sp(2) ⊃ sp(1) ⊕ sp(1). In the examples just given, on certain curves the so(2n−1) gauge algebra cannot arise, and the lower Tate tuning with the monodromy condition is realized. As mentioned above, when the so(2n − 1) tuning is allowed, there is not generally a polytope in the KS database with the same Tate tuning and the monodromy condition automatically satisfied, and one has to use the higher Tate tuning to guarantee the condition. These facts can be seen in contrasting the polytope models, for example, of so(9) and the two realizations of so (10) (10). In this case, the weaker tuning satisfies the monodromy condition automatically, which is expected as {· ⊕ so(9) ⊕ sp(1) ⊕ so(9) ⊕ sp(1) ⊕ so(9) ⊕ ·} is not allowed as mentioned.
There is a similar story between so(11) and so (12). For example, we can tune an so(11) on the −3-curve of the generic model over F 3 by requiring Tate vanishing orders of {1, 1, 3, 3, 5}, which gives rise to M:328 8 N:18 7 H:8,242 in KS database. Then to get a polytope corresponding to a tuning of so(12), we need to use {1, 1, 3, 3, 6}, which has a good Zariski decomposition, and therefore a corresponding reflexive polytope exists, M:318 10 N: 19 (8), and this algebra only arises over the NHC −4. In the case of the NHC −4, so(8) is the minimal gauge algebra, so either vanishing orders {1, 1, 2, 2, 3} or {1, 1, 2, 2, 4} will automatically satisfy the so(8) monodromy condition in any Tate tuning over a base with a −4 curve. This unique aspect of so(8) matches with the observation that a tuned so(7) cannot be ruled out through the global symmetry group since the global symmetry group on a tuned so(7) curve contains that on a tuned so(8) curve. Thus, any Tate tuning of {1, 1, 2, 2, 3} or {1, 1, 2, 2, 4} over a curve with self-intersection greater than −4 will lead to a model with, if not g 2 , so(7) enhancement.

Multiplicity in the KS database
Given a pair of Hodge numbers h 1,1 , h 2,1 , there are in general many distinct polytopes in the KS database. There are many ways in which such a multiplicity may arise. Of course, generic or tuned elliptic fibrations over distinct bases may coincidentally give the same Hodge numbers. As discussed above, however, there are also many closely related constructions that give identical Hodge numbers. Different realizations of the same gauge algebra through different Tate tunings may contribute, often related to monodromy tunings as discussed in the preceding subsections. There are also rank-preserving tunings that change the gauge algebra but not the Hodge numbers. And in some cases there are non-toric deformations that can give additional multiplicity. A complete analysis of the KS database that accounts for these multiplicities exactly would require a complete and systematic tracking of all distinct possible Tate tunings for each gauge algebra combination and a clear and systematic analysis of the non-toric deformation possibilities. We have not attempted such a systematic analysis here. Rather, in the analysis in the remainder of the paper we focus on constructing distinct possible gauge groups through Tate tunings and identifying the distinct Hodge numbers that can arise for reflexive polytopes in this way. In this section we discuss in a bit more detail some aspects of the multiplicity question. To systematically analyze multiplicities of different Tate tunings of the same algebra, we would need to consider all combinations of monodromy and non-monodromy tunings of algebras like su 3 , e 6 , so(n) etc. Over bases with many curves allowing such tunings this could give a large combinatorial multiplicity. For example, consider the two polytope models in the first block of Table 12. We start with the minimal {1, 1, 2, 2, 3} Tate vanishing orders for all three −4 curves, which together do have a corresponding Tate-Zariski decomposition, so there is a corresponding polytope construction. Then we tune the vanishing orders on the middle −4-curve alone to be {1, 1, 2, 2, 4}. After iteration, the other two −4-curves are forced to also have {1, 1, 2, 2, 4} vanishing, giving the second generic model with all −4 curves reaching the second realization. This exhausts the possibilities. So from what might appear to in principle be 8 possible combinations of tunings, only two are actually consistent. It can also happen that only the lower-order realization exists, while the higher-order realization does not have an acceptable Zariski decomposition and there is no corresponding polytope, as we have seen for example in the failure to realize the third model of H:23,107 with the generic gauge group over a −6 curve in section 4.5. In general, the realization of any given combination must be checked by performing a global Tate-Zariski decomposition, as local information may not be completely adequate to rule in or out a possible tuning. An example is given by the models in Table 13, where there is no global Zariski decomposition of the {1, 1, 3, 3, 6} realization of so (12), and the reflexive polytope model does not exist over the given global base, though it would seem to be fine if we were to analyze the tuning pattern with the focus on the local sequence {−1, −3, −1, −4, −1, 0} only.
Note also that further Tate tunings of a given algebra may not give rise to a new reflexive polytope, even if the higher vanishing orders still have a valid Zariski decomposition. We describe briefly several examples here: There is only one polytope in the KS database with H:4,226, which corresponds to the type I 2 su(2) tuning {0, 0, 1, 1, 2} on the −2-curve of the F 2 base, but there is no polytope that corresponds to the type III su(2) {1, 1, 1, 1, 2}. It is even more interesting to compare the H:5,233 models discussed in Appendix A and H:5,251 in table 10: there is no IV su(3) for the former since it is just a specialization of the type I 3 su(3) tuning, while there are two different IV su(3) realizations for the latter; and both of these sets have the rank-preserving tuning g 2 model. Similarly, type I ns 2n and type I ns 2n+1 sp(n) tunings do not give rise to different polytopes. Also for three different types I 0 , I 1 , II of the trivial algebra, only the one with the lowest vanishing orders that has a Tate-Zariski decomposition has a reflexive polytope construction. An amusing exercise is illustrated in Table 14, where we can see the changes in three different types of trivial algebra under various tunings.
Another source of multiplicity comes from tunings of rank-preserving type as described in the end of §2.  (7) and e 6 are not subalgebras of each other, and the tunings give different h 2,1 s. Lastly, multiplicity can come from situations where the elliptic fibration over a toric base has (4, 6) points that must be blown up. As discussed in §2.8, over toric bases containing curves with self-intersection number −9, −10, −11 the generic elliptic fibration is non-flat and the base must be blown up at the (4, 6) points to give −12-curves, over which there is a flat elliptic fibration. In general the base resulting from these blow-ups will be non-toric, and the blowups give extra tensor multiplets contributing to anomaly cancellation [8,9]. In some cases, however, the base is still toric after blowing up one or more of the (4, 6) points; in such cases there will be multiple entries in the KS database associated with these distinct bases. In general we expect that these all represent the same smooth Calabi-Yau threefold, with different choices of flat and non-flat elliptic fibrations over different bases, though we have not checked explicitly that this is true in all cases. Examples of some non-flat elliptic fibrations of this type are analyzed in [35]. To illustrate this structure, in Appendix B we analyze the non-flat elliptic fibration structure of the toric hypersurfaces associated with (flat) toric fibrations of the reflexive fibered polytopes over the Hirzebruch surfaces F 9 , F 10 , F 11 . In these cases, we see explicitly that the fiber over the (4, 6) points in the −9, −10, −11-curves contains extra irreducible components that may naturally be associated with divisors in the blown up space. The multiplicity with which the Hodge pairs for the generic elliptic fibration models over the suitably blown up Hirzebruch surfaces F 9/10/11 are listed in Table 15, the tops over the -9/-10/-11-curves are listed in the second block in Table 11. This illustrates the way in which the same smooth Calabi-Yau threefold can be realized as a non-flat elliptic fibration over one or more toric bases as well as sometimes a flat elliptic fibration over another toric base, with each fibration structure realized in a different polytope in the KS database. For example, as illustrated in the table there are 6 distinct polytopes at Hodge numbers H:14,404, which correspond to toric realizations of elliptic fibrations over different "semi-toric" bases that admit only a single C * action (including various limits in which −2 curves arise).   6) singularities on the −9, −10, or −11 curve are at non-toric points and the elliptic fibration is non-flat. In these cases the blow-ups are handled automatically by the resolution of the toric geometry, giving a resolved model corresponding to a flat elliptic fibration over a "semi-toric" base. There are also toric bases that arise by blowing up one or more of the (4, 6) points at toric points, giving polytopes with toric fibrations over blow-ups of the Hirzebruch surfaces. When multiple (4,6) points coincide this corresponds to a limit with a −2 curve in the base. For each polytope the base of the fibration is a toric surface given by the curves on the outside of the diagram, with self-intersections as labeled.

Bases with large Hodge numbers
In this work we have confined our study to the simplest P 2,3,1 fiber type polytopes. In part this is because the standard fiber type matches with the Tate structure of the Weierstrass model as discussed previously. Also, however, this fiber type dominates the structure at large Hodge numbers. In particular, we can explicitly identify constraints on the bases that can be used for the other 15 fiber types. These constraints are such that the other fiber types all lead to problematic codimension one (4, 6) singularities on some divisor in the base when the base contains curves of sufficiently negative self-intersection. In particular, none of the other 15 fibers can be supported over any base that contains a curve of self-intersection less than −8. This immediately constrains the set of constructions at large Hodge number, since the generic elliptic fibrations with the largest Hodge numbers almost always involve −12 curves in the base (though there are notable exceptions to this general principle, including the other one of the two fibrations of the H:491:11 polytope). We leave a more detailed analysis of the constraints on different fiber types for future work, but briefly outline the issue that arises for other fiber types besides the P 2,3,1 fiber. Consider for example the P 2 fiber type. Carrying out the analogue of the simple stacking procedure for a P 2 fiber, we find that there are 10 dual monomials analogous to the coefficients a 1 , . . . , a 6 . These 10 monomials are sections of line bundles O(−K), O(−2K) and O(−3K). Any section of a line bundle −nK must vanish over a −12 curve to at least degree n when n < 5 by the Zariski decomposition. This immediately leads to the presence of a codimension one (4, 6) singularity over any −12 curve in the base. Similar issues arise for the other fiber types.
Considering the toric bases, we can simply consider the complete enumeration carried out in [8] and identify the bases with largest Hodge numbers that have curves of self-intersection no smaller than −8. The base of this type with the largest h 2,1 for the generic elliptic fibration is F 8 , over which the generic elliptic fibration has Hodge numbers (10, 376). Even over F 8 , the largest h 2,1 value that can be achieved for a tuning with any fiber other than P 2,3,1 is quite restricted; over this base, for example, there are 5 other fiber types including P 1,1,2 that are possible; the generic fibration with each of these fiber types gives an elliptic Calabi-Yau threefold with Hodge numbers (11,227). Any other fibration with these or any other fibers other than P 2,3,1 over any base would seem to give a Calabi-Yau threefold with an even smaller value of h 2,1 . Thus, by restricting to Hodge numbers above h 2,1 ≥ 240, we can expect that the threefolds in the KS database that admit elliptic fibrations should be all or almost all described by the P 2,3,1 fiber type.
Similarly, the largest value of h 1,1 that can arise for a base with no curves of selfintersection below −8 is 224. The corresponding base has a set of toric curves of self- where // denotes the sequence −1, −2, −3, −2, −1 associated with E 7 chains (see e.g. [8]), and a generic elliptic fibration with Hodge numbers (224,18). There is nothing that can be tuned over this base without producing a curve of self-intersection below −8 so it seems that confining attention to threefolds with h 1,1 ≥ 240 should again restrict us to primarily P 2,3,1 fiber types. As we see in §6, however, there are a few unusual cases in which bases that have generic elliptic fibrations with rather small values of h 1,1 admit extreme tunings that dramatically increase the value of h 1,1 without producing curves of highly negative self-intersection. In a companion paper [17], we study the fibration structure of the hypersurface models in the KS database more directly, and confirm both the prevalence of P 2,3,1 fibers at large Hodge numbers and the existence of exceptions involving extreme tunings.

Systematic construction of Tate-tuned models in the KS database
Kreuzer and Skarke have classified all 473,800,776 4D reflexive polytope models, which give 30,108 distinct Hodge pairs. It was found in [28] that the set of Hodge pairs {h 1,1 , h 2,1 } of all generic elliptically fibered CYs over toric bases is a subset of all the Hodge pairs in the KS database. We gave in §4.3.2 a general construction of reflexive polytope models of these generic elliptic fibrations over toric bases with NHCs, and we expect that all generic elliptic fibration models over toric bases have these corresponding reflexive polytope models in the KS databse. We wish to carry out a more comprehensive comparison by matching tuned Weierstrass models of CYs over 2D toric bases with 4D reflexive polytope models of Calabi-Yau hypersurfaces at large Hodge numbers.
Of the 30,108 distinct Hodge pairs in the KS database, 1,827 have either h 1,1 ≥ 240 or h 2,1 ≥ 240 (only the Hodge pair {251, 251} satisfies both inequalities). To compare the two constructions at large Hodge numbers, the next step would be to construct roughly this number of distinct Weierstrass models of tuned CYs in these regions. Not all Weierstrass models correspond to reflexive polytope constructions, however. Nonetheless, as discussed in §4, there is a close relation between Tate-tuned models and P 2,3,1 -fibered polytopes, which dominate at large Hodge numbers as argued in §4.8. Therefore, our approach is to construct systematically all Tate-tuned models via tunings of generic Tate models over 2D toric bases. As a preliminary to this analysis, however, we begin with a simpler systematic analysis of which gauge group tunings may be possible based on more general Weierstrass tunings, and then we refine the analysis to Tate tunings. We describe the logic of this analysis in more detail in §5.1.
All Hodge pairs of the Tate-tuned models from this algorithm fall within those in the KS database. However, there are certain Hodge numbers in the KS database in the regions of interest at which our initial analysis identified no matching Tate-tuned model. We therefore have analyzed directly, via the method described in §3.5, the polytope models with the Hodge numbers that were not found in our systematic tuning construction; the analysis of these cases is described in §6. It turns out that all these remaining polytope models can be described as somewhat more exotic Weierstrass or Tate tunings over bases that are either toric bases or blow-ups thereof. This completes the comparison of the two constructions at the level of Hodge numbers. At a basic level the result of this analysis is that in the regions of interest all the Hodge pairs in the KS database are realized through generic or tuned elliptic fibrations. This matches with the results through a direct analysis of the fibration structure in the companion paper [17]. The more detailed analysis we carry out here, however, gives much more insight into the structure of these fibrations and the complex variety of Weierstrass tunings and geometries that are realized through simple toric hypersurface constructions.
We also discuss briefly the limits of Tate tuning in section 5.4, where we collect some results on tunings that are compatible with the global symmetry constraints but can't not be realized by Tate tunings. These tunings may be realized by Weierstrass models and in such cases give new Hodge pairs outside the KS database.

Algorithm: Global symmetries and Zariski decomposition for Weierstrass models
We give an algorithm in this section to systematically construct all tunings of enhanced gauge groups over a given 2D toric base, starting with the generic model. Our goal is to construct all Tate-tuned models over toric bases that give elliptic Calabi-Yau threefolds with Hodge numbers in the regions h 1,1 ≥ 240 or h 2,1 ≥ 240. As we saw in §2.7, global symmetry constraints on each curve put upper bounds on the gauge algebras that can be tuned on intersecting curves. On the other hand, as discussed in §4.5, there is an issue of whether local tunings on subsets of curves can be combined into a global model over some toric base B 2 . This can be tested by the Zariski decomposition. More specifically, our goal is to carry out explicitly arbitrary combinations of the Tate tunings from §2.3 on the curves in the base, applying the variant of the Zariski decomposition described in §2.5 to determine which combinations are globally compatible. While in principle we could simply iterate over all possible gauge algebra combinations, using the global symmetry constraints on what gauge algebras can arise on the curves intersecting a curve of negative self intersection helps prune the tree and make the algorithm more efficient. Global symmetries are also helpful in limiting the set of possible monodromy-dependent gauge groups that can arise on sequences of intersecting curves in ways that are not apparent at the level of the Zariski decomposition.
Although ultimately we wish to analyze Tate tunings, we perform an initial analysis of Weierstrass tunings using global symmetry constraints and the Zariski decomposition. This gives us a set of possible tunings that we expect may be possible at the level of the gauge algebras. Not all these constructions, however, are compatible with Tate tunings and with polytopes. We begin the discussion by focusing on the Weierstrass tunings and then in §5. 4 we use the results of the Weierstrass tunings to check which Tate tunings are possible.
Given a 2D toric base B, which is represented by a set of K irreducible toric curves {C j , j = 1, 2, . . . , K} intersecting each other in a linear chain, we first obtain for the generic model over the base B the orders of vanishing {c j,4 , c j,6 , c j,12 } of f , g, and ∆ along each curve. The sets of values {c j,4 }, {c j,6 }, and {c j,12 } can be determined by the Zariski decomposition via the procedure described in equations (2.27)-(2.30) with n = 4, n = 6, and n = 12, respectively, or can be directly read off from the non-Higgsable cluster structure of the curves {C j }. Now let us consider all possible (Weierstrass) tunings of the generic model. We describe a procedure to determine an allowed pattern {g j } of tuned algebras g j on each curve C j in the base. Note that in this algorithm we assume that there are no toric (4, 6) points in the base, even after the tuning; such a point would be blown up to form a different toric base, and the tunings over the blown up base would be found directly by tunings over that base. We do allow non-toric (4, 6) points in the case where the base contains −9, −10 or −11 curves; in these cases we essentially treat the curve as a −12 curve supporting an e 8 , understanding that the polytope hypersurface construction will automatically resolve these singularities and effectively blow up the non-toric points in the base, in accord with the discussions in §2.8 and §4.7.

Main structure of the algorithm: bases with a non-Higgsable e 7 or e 8
We consider first the simplest cases, where there is at least one curve in the toric base of selfintersection m ≤ −9; such a curve necessarily carries a non-Higgsable e 8 gauge algebra. We start the procedure by choosing a specific curve with a non-Higgsable e 8 and first considering the possible tunings on one of the adjacent curves. Let us label the curve with the e 8 using the index j = 1, the curve we attempt the first tuning on by j = 2, the subsequent curve by j = 3, etc. This choice of the initial configuration is convenient to serve as the starting point of a branching algorithm because an e 8 algebra cannot be further enhanced; moreover, nothing can be tuned next to an e 8 , without producing a (4, 6) singularity at a toric point, which we are assuming does not happen as discussed above. Therefore, the gauge algebras on C 1 and C 2 are fixed: g 1 = e 8 and g 2 has to be a trivial algebra.
We then pass to tunings g 3 on C 3 . The possible tunings on C 3 are constrained by the global symmetry group g (glob) 2 on C 2 , which is determined by the self-intersection number of C 2 and the gauge algebra g 2 on C 2 . Let the set {g 3,α } be the set of algebras that satisfy the constraint g 1 ⊕ g 3,α = e 8 ⊕ g 3,α ⊂ g (glob) 2 . For the global symmetries, we used the results in Table 6.1 and Table 6.2 in [32] for the maximal global symmetry group g (glob) j on a curve C j of negative self-intersection m carrying a gauge algebra g j . Additionally, the curves of negative self-intersection that do not support an NHC can carry trivial gauge algebras, of types I 0 , I 1 , II; therefore in such cases when g j = ·, we use g (glob) j = e 8 and g (glob) j = su(2) for m = −1 and m = −2, respectively. We used the results tabulated in [47] for the subgroups of a global symmetry group g (glob) j−1 to obtain the restricted set of algebras {g j,α } satisfying the constraint g j−2 ⊕ g j,α ⊂ g (glob) j−1 . We attempt tunings one-by-one for each g 3,α . For each possible algebra we replace the original orders of vanishing {c 3,4 , c 3,6 , c 3,12 } with the desired orders of vanishing corresponding to g 3,α using Table 2. We then perform the Zariski iteration procedure on all curves with the new {c 3,4 , c 3,6 , c 3,12 } for n = 4, 6 and 12, respectively. If all the gauge algebras on the curves prior to and including C 3 stay unchanged after the iteration, tuning g 3,α is not ruled out. If any of the gauge algebras on the curves prior to C 3 have changed, or the vanishing orders {c 3,4 , c 3,6 , c 3,12 } do not produce the desired gauge algebra g 3,α in the new configuration after the iteration, tuning g 3,α is not allowed on C 3 ; in such cases we terminate the procedure with this g 3,α branch, and attempt the next tuning g 3,α+1 on C 3 . Note, however, that the fact that the gauge algebras stay unchanged does not mean that the set of values {c j≤3,4 }, {c j≤3,6 }, {c j≤3,12 } stay unchanged under the iterations. Indeed, often it is the case that the orders of vanishing on curves near C 3 are increased, but without modifying the gauge algebra on C 2 . In other words, in this case g 2 should be the trivial algebra, but it may be type I 0 , I 1 , or II (cf. also examples in Table 14.) Note that the vanishing orders {c j>3,4 , c j>3,6 , c j>3,12 } can obtain new values after the initial set of iterations just described. If these increase beyond those determined by the initial NHC configuration, we use the larger vanishing orders as the starting points in further iterations of the tuning. We denote by g j[i] the gauge algebra on curve j after the iteration procedure associated with curve C i . For i = j, g j[j] denotes a choice of gauge algebra in a branch, g j[j] ∈ {g j,α }, and g j[j] ⊇ g j[j −1] . Note that we must have g j[k] = g j[j] for all k > j as we require in the branch that the gauge algebra on C j stays unchanged in tuning gauge algebras on C k>j , but the orders of vanishing realizing the gauge algebra may be different. We can proceed with the new configuration to the next step of tuning algebras on C 4 , as long as g 4 [3] ⊕ g 2 [3] ⊂ g (glob) 3 [3] is satisfied, where now g 4 [3] is the gauge algebra on C 4 in the new configuration and g (glob) 3 [3] is determined by the self-intersection of C 3 and the gauge algebra g 3 [3] ∈ {g 3,α }. For example, let us start with g 3 [3] = g 3,1 . We terminate the procedure with the g 3,1 branch and attempt the next branch of tuning g 3,2 on C 3 if g 4 [3] ⊕ g 2 [3] ⊂ g (glob) 3,1 is violated in the new configuration.
The procedure continues similarly until the second to the last curve C K−1 is met. As the last curve C K is connected back to the first curve C 1 , we need to consider also the global symmetry constraint on C K to close the tuning pattern. The set of possible tunings 2] . First, the usual two conditions have to be satisfied for g K−1,γ to be allowed: in the new configuration after the iterations associated with the tuning of g K−1,γ (1) the prior gauge algebras are held fixed, and (2) the global symmetry constraint on C K−1 is satisfied. Moreover, there is the additional third condition: (3) the global symmetry constraint on the curve C K has to be satisfied; i.e., ] , where g 1 is held fixed and is e 8 in the simplest cases, and g is determined by the self-intersection of the curve C K and the gauge algebra g K after the Zariski interations for the tuning g K−1,γ . In fact, g K is only allowed to be a trivial algebra in the simplest cases as C 1 carries an e 8 algebra, so no tuning is allowed on C K . Hence, if the global symmetry constraint on C K is satisfied, we are basically done to this point in the procedure searching for a tuning pattern. In this case, we obtain a tuning pattern We check all g K−1,γ 's in order similarly to complete the scan through all possible tuning patterns compatible with the initial viable possibility for g 3 [3] , . . . , g K−2[K −2] . After all g K−1,γ 's are processed, we proceed iteratively with a nested loop, continuing with the next possible value of g K−2 , etc. so that all possible combinations of gauge group tunings are considered.
All tunings increase h 1,1 and decrease h 2,1 with respect to the generic model over a given base. Thus, to classify all tuned models of h 2,1 ≥ 240, we need only consider toric bases for which the generic elliptic fibration has h 2,1 ≥ 240. In our initial scan, we also restricted to bases that have generic models with h 1,1 ≥ 220. As we describe in more detail in the following section, this misses a few cases where there is a large amount of tuning that significantly changes h 1,1 . On the other hand, as bases associated with generic models having h 1,1 > 224 always contain at least one curve carrying an e 8 algebra, the algorithm as described above is quite effective in dealing with tunings in the large h 1,1 region as we always have a simple starting point for the iteration. In fact, the algorithm actually can work in the same way for tunings of generic models with a curve carrying e 7 in the base; i.e., generic models with a curve of self-intersection m ≤ −7 in the base. This is because e 7 algebras also cannot be further enhanced without modifying the base -an enhancement to an e 8 algebra would give additional (4, 6) points that must be blown up. And no non-trivial algebra can be tuned next to an e 7 algebra. Thus, in this case we similarly can make the convenient choice that the initial configuration is fixed to be g 1 = e 7 , g 2 = ·.
We make some final comments on two technical issues in the tuning procedure. As mentioned above, in tuning the curve C j , not only do the orders of vanishing on C j+1 (and in some cases on further curves C j+2 , . . .) also change in general, but the new vanishing orders {c j+1,4 , c j+1,6 , c j+1,12 } can in some cases correspond to a different gauge algebra. However, because the three Zariski iterations were performed independently, sometimes these vanishing orders do not correspond to any algebras in the Kodaira classification. We encountered a few cases of this type, for example where {c j+1,4 , c j+1,6 , c j+1,12 } = {1, 2, 4}; this can happen for example if a previous su(n) tuning ({0, 0, n}) pushes up the order of vanishing of ∆ more significantly than f, g (where some required orders of f, g already imposed on the curve); however, note that, this can never happen in a real ∆ as calculated in a complete model from f and g in equation (2.10). In such situations, we modify the orders of vanishing on C j+1 to fit with those that correspond to the gauge algebra that arises by increasing the values c j+1,4 , c j+1,6 minimally. Then we perform the iteration again after the modification, and use the resulting configuration to test the conditions (1) and (2).
Another detail to take care is the tuning of algebras only distinguished by monodromy conditions. For those cases where there are distinct algebras associated with different monodromy conditions, we retain all the possibilities allowed by global symmetries; in the list of possible tunings we attach an additional label to the orders of vanishing using a fourth entry {c j,4 , c j,6 , c j,12 , algebra} to ensure that all possible tunings are considered.

Special cases: bases lacking curves of self-intersection m ≤ −7 and/or having curves of non-negative self intersection
The algorithm described in the preceding subsection relies on the presence of a curve of self-intersection m ≤ −7 in the base, where we can begin the iteration process in a simple fashion as the gauge algebra on the initial curve is fixed. In the regions we are considering, there are very few bases that lack such curves; we describe here briefly how these cases are handled. Of course, one could simply use a brute force algorithm of choosing an arbitrary starting point and looping over all tunings on the initial curve C 1 . In principle, however, for efficiency we would like to choose the curves C 1 , C 2 such that there are fewer allowed combinations {g 1 , g 2 }. Note that in this case there cannot be any enhancement on C 2 as the global symmetry algebra is always the trivial algebra on −6-curves without an further enhancement to e 7 , so no tunings are allowed on any intersecting curves. In fact, in the Hodge number regions we are considering, there are very few cases that lack non-Higgsable e 7 or e 8 gauge algebras. Every base with a generic elliptic fibration having h 1,1 ≥ 220 has a curve of self-intersection −7 or below. In the region h 2,1 ≥ 240, there are 14 generic models that contain no curve in the base carrying an an e 7 or e 8 algebra; the generic models over bases F 0≤m≤6 and P 2 compose 9 of these 14 models, and are discussed further below. In the remaining cases, there is no choice of the initial configuration that uniquely determines the initial configuration, and we have to enumerate and specify different initial configurations {g 1 , g 2 } over a curve of minimal self-intersection to perform the algorithm. A further issue arises for bases that have curves of non-negative self intersection. On such curves, there is no global constraint on the adjacent algebras from the SCFT point of view. While there are some analogous constraints in the case of curves of non-negative self intersection [12], the constraints are weaker and less completely understood. So we do not impose global constraints in these cases. In principle this can be handled by simply iterating over all gauge groups, however in practice the number of cases where this issue is relevant is rather limited and can be handled efficiently using more specific methods. From the minimal model point of view we can fairly easily classify the types and configurations of non-negative self intersection curves that can arise. The minimal model bases P 2 and F m have three consecutive curves of non-negative self intersection. Any blow-up of one of these bases has either only one such curve or two adjacent such curves, since blow-ups reduce the self-intersection of curves containing the blow-up point and do not introduce new curves of non-negative self intersection. Blow-ups of the resulting bases again have at most two curves of non-negative self intersection and when there are two they are always adjacent. So the possibilities are actually quite limited.
In general, the way we deal with the cases having one or two non-negative curves for bases with large h 2,1 is by performing the algorithm separately in both opposite directions from a good starting point (curve of maximally negative self intersection) to get two "halfpatterns" of tunings, and connect them appropriately. In other words, we start from a chosen curve C 1 , run the algorithm in both directions, and stop the tuning procedures when the first non-negative curve is met in both directions. We do not impose any global conditions for the curves of non-negative self intersection. The combination of the two sets of the halfpatterns connected in this way gives all tuning patterns of a generic model with one or two non-negative curves in the base. For bases with large h 1,1 , there is generally at most one nonnegative self intersection curve and the nearby gauge group is generally constrained by global symmetries and nearby large negative self-interactions; in some of these cases we have used simpler heuristics to complete the analysis in the presence of non-negative self-intersection curves.
For the cases P 2 and F 0≤m≤12 that have three non-negative curves, most tunings in fact decrease the Hodge number h 2,1 below the value 240 of interest. For example [12], tuning an su(2) on a +1 curve of P 2 changes the Hodge numbers from (2, 272) to (3,231). There are some exceptions: for example tuning an su(2) on the +12 curve of F 12 gives a Calabi-Yau with Hodge numbers (12,318). But it turns out (as we see explicitly from the analysis of the following section) that all these cases with h 2,1 ≥ 240 are also realized in other ways by generic or tuned models over other toric bases. So we do not need to explicitly include these in our analysis since we are not trying to reproduce the precise multiplicity of models at each Hodge number pair.
Although we have only focused on tuning models in the large Hodge number regions, one can in principle classify all allowed tuning patterns of non-abelian gauge algebras on any toric base with the algorithm described here; though slightly different methods are needed for tunings over the bases P 2 and F 0≤m≤12 , an exhaustive search is straightforward in these cases as there are only a few curves in these bases (three curves in P 2 and four curves in F m ).

Tate-tuned models
The analysis described so far in terms of Weierstrass models gives a large collection of possible gauge algebra tunings over each toric base. Not all of these gauge algebra combinations correspond to hypersurfaces in reflexive polytopes. There are several reasons for this. First, not every Weierstrass tuning can be realized through a Tate form, so some of these tunings on toric curves will not have standard P 2,3,1 -fibered polytope constructions. Further, some of the combinations of gauge groups that are allowed by the Zariski analysis and global constraints still cannot be realized in practice in a global model -we alluded for example at the end of §5.2 to the fact that monodromy conditions are not really taken care of properly in the Zariski decompositions of n = 4, 6, 12. Indeed, an explicit check shows that not all the Hodge pairs calculated via equations (2.6) and (2.7) from the Weierstrass tuning patterns we got from §5.2 and §5.3 lie in the KS database. We are interested in constraining to a subset of tuning constructions for which we expect direct polytope constructions. Hence, for each gauge algebra tuning combination that satisfies the Weierstrass Zariski analysis and global conditions, we attempt to construct an explicit Tate-type model by specifying Tate orders of vanishing according to Table 4 for each tuning in a tuning pattern. We then perform the Zariski decomposition of the Tate tunings described in §2.5. A tuning pattern gives a genuine Tate-tuned model if it has the Zariski decompositions of Tate tunings. In performing this analysis, we used in our systematic analysis only the stronger version of the Tate forms for the algebras with multiple realizations and/or monodromy conditions. In particular, we did not use any of the tunings marked with • or in Table 4. The second version of the I s 2n Tate tuning (marked with •) was in fact previously not known and was identified through the analysis of the next section. For the so algebras, some of the alternate monodromy tunings were not previously known (for example, the non-version of so(4n + 4) algebras); also, we wished to restrict attention to cases where the algebra is guaranteed simply by the order of vanishing of the Tate coefficients. In general, as we have noted in the examples in §4.5 and §4.6, the polytope constructions do not satisfy the monodromy conditions for the higher rank gauge algebras in these cases.
These principles give us a set of gauge group and Tate tunings over each toric base that we believe should have direct correspondents in the KS database through standard P 2,3,1fibered polytopes, given the correspondence that we established in §4. We have carried out an explicit comparison of these two sets, and indeed the Hodge numbers of this more limited class of Tate-tuned gauge groups all correspond to values that appear in the KS database. Furthermore, the Hodge pairs from the original Weierstrass analysis that are not in the KS database are exactly those of the tuning patterns that can not be realized by Tate tuning. In fact, given this restricted set of tunings we reproduce almost all of the 1,827 distinct Hodge pairs in the range h 1,1 ≥ 240 or h 2,1 ≥ 240. Only 18 of the Hodge pairs in this range were not found by a "sieve" using the Tate constructions described above. In the next section we consider the analysis of these 18 outlying polytope constructions.
A question that we do not explore further here, but which is relevant to the more general problem of understanding the full set of Calabi-Yau threefolds and the classification of 6D F-theory models, is the extent to which tunings are possible that look like they should be allowed from the Weierstrass Zariski analysis and anomaly cancellation conditions, but do not correspond to Tate constructions. Various aspects of this "Tate tuning swampland" were analyzed in [12]. In the context of this project, we did a local analysis of the Weierstrass tuning patterns that are not Tate tuning patterns. We reproduced some parts of the known e8 Tate swamp su(3) ⊕ sp(3), su(3) ⊕ sp(4), g 2 ⊕ so(10), so(9) ⊕ su(4), so(9) ⊕ sp(2), so(10) ⊕ su(4), so(10) ⊕ sp(2), so(11) ⊕ su(3), so(11) ⊕ sp(2), so(13) ⊕ sp(1), so(13) ⊕ su(2)
6 Polytope analysis for cases missing from the simple tuning construction, and other exotic constructions As discussed above, there are only 18 Hodge pairs in the regions h 1,1 ≥ 240 or h 2,1 ≥ 240 in the KS database that are not produced by our Tate tuning algorithm. One of these missing 18 Hodge pairs is in the large h 2,1 region, {45, 261}, and the other 17 (see Table 17) are in the large h 1,1 region. In this section we analyze the polytopes in the Kreuzer-Skarke database associated with these 18 Hodge number pairs. By studying these 18 classes of Calabi-Yau manifolds, we have identified new tuning constructions that we had not known previously; the KS database provides us with global models utilizing these constructions that we did not expect a priori in our original analysis. We study the fibration structure of the 18 outstanding classes by analyzing the polytopes in the way described in section 3.5. All the polytopes associated with these 18 Hodge pairs have a P 2,3,1 fibered polytope structure (though in some cases it is really the more specialized Bl [0,0,1] P 2,3,1 fiber that occurs), but not all of them are the standard P 2,3,1 -fibered polytopes that we have defined in §3.4. In particular, the CY hypersurface of a standard P 2,3,1 -fibered polytope (or Bl [0,0,1] P 2,3,1 -fibered polytope) has a Tate form, while this is not for other fibration structures that use the same fiber but a different "twist". We analyze the two different types of polytopes arising from the 18 Hodge pairs separately. In §6.1 we analyze the standard P 2,3,1fibered polytopes in the KS database that we have not obtained in our systematic construction  Table 17: The Hodge number pairs in the KS database at large h 1,1 that we did not obtain from straightforward Tate-tuned models. However, all these can be reproduced by some flat elliptic fibrations that we discuss in this section: The standard P 2,3,1 models, which have a Tate form, are studied in §6.1, and the non-standard P 2,3,1 models, which involve tunings on non-toric curves in the base, are studied in §6.2.
of Tate-tuned models. In §6.2 we analyze the polytopes that do not have the standard P 2,3,1fibered structure. We also include in §6.2 some further examples in the KS database that are outside the range of focus of this paper but that illustrate some further interesting exotic structure associated with gauge groups on non-toric curves in the base.

Fibered polytope models with Tate forms
Of the 18 missing Hodge pairs, there are 1 + 9 Hodge pairs in the large h 2,1 , h 1,1 regions respectively in which there is a standard P 2,3,1 -fibered polytope (or a standard Bl [0,0,1] P 2,3,1fibered polytope), which has a Tate form. Therefore, we analyze the Tate models explicitly from these polytopes to learn about the Tate tunings that we missed in our initial construction.
The Hodge pair in the large h 2,1 region, {45, 261}, has only one polytope. This polytope reveals a second tuning of the type I s 2n singular fiber that is not just a specialization of the known Tate tuning. We also find that applying this novel Tate tuning su(6) on a m ≥ −1-curve gives models with the three-index antisymmetric representation as opposed to the generic fundamental and two-index antisymmetry representations. We describe this analysis in detail in §6.1.1. The polytopes of the nine missing Hodge pairs at large h 1,1 with the standard fibration structure are either extremely tuned models, with bases having generic elliptic fibrations with h 1,1 < 220 (described in §6.1.2), or are non-flat elliptic fibration models over a toric base (described in §6.1.3). In the non-flat elliptic fibration cases, as we have discussed at the end of §4.7, the CY resolution of (4,6) singularities in terms of the polytope model produces irreducible components of the ambient toric fiber (as the hypersurface equation restricted to the components is trivially satisfied over the (4, 6) points). Therefore, at these points the dimension of the fiber jumps to two giving the non-flat elliptic fibration structure. Associating the additional divisors with blow-ups in the base allows us to describe the resulting Calabi-Yau threefolds alternatively as flat elliptic fibrations over the blown up base. The resulting models in the cases found here give rise to e 8 tunings or so(n ≥ 13) tunings on −3-curves, and are also involved with tuned Mordell-Weil sections, which are associated with U(1) factors and U(1)-charged hypermultiplets.  Table 4, indicated by su(2n) • .
As this is the only polytope associated with the Hodge pair {45, 261}, it seems that the traditional su(2n) tuning is somehow not allowed in this configuration. We checked explicitly by performing a tuning where we substitute in the vanishing order {0, 1, 3, 3, 6} over D 1 , and perform the Tate-Zariski decomposition. The vanishing orders after iteration become which is problematic as the global symmetry constraint on the −6-curve D 4 is violated. This confirms again that there has to be a Tate-tuned pattern that is consistent under the Tate-Zariski decomposition for a corresponding polytope to exist. And we cannot obtain a polytope of these Hodge numbers using the standard tuning methods because the su(2n) • tunings, {0, 2, n − 1, n + 1, 2n}, are not specializations of the standard su(2n) tunings,{0, 1, n, n, 2n}.
In the case of a −2 curve, as in the example encountered at large h 2,1 , the matter content associated with the physics of the exotic su(6) • tuning is equivalent to that of a standard su(6) tuning over a −2 curve. After incorporating these alternative su(2n) • -tunings into our algorithm, however, we discovered that this second Tate realization of {f, g, ∆} = {0, 0, 2n} gives rise to the non-generic three-index antisymmetric (20) representation of su(6) when the tuning is performed on curves of self-intersection m ≥ −1. We describe an example of this explicitly, in the context of a global model that lies outside the regions of primary interest h 1,1 , h 2,1 ≥ 240.
The There is an su(6) • tuned on the −1-curve D 1 and an su(3) tuned on the −2-curve D 2 . Interestingly, by explicit analysis, we find that the f, g from the polytope data automatically satisfy the conditions for the codimension-two singularity on D 1 to support the three-index antisymmetric representation of su (6), as described in [25]. To see this, we fix the complex structure moduli of f, g to some general enough Z values to avoid accidental cancellations, expand f and g in terms of σ ≡ b 1 where the coefficients are in terms of a second local coordinate that we choose to be b 2 then we find (following the notation in [25]) • ∆ 0 = 0 : f 0 ∼ − 1 48 φ 4 0 and g 0 ∼ 1 864 φ 6 0 ; we choose to set φ 0 = 57 + 46b 2 .
The conclusion that the su(6) • -tuning on the −1-curve leads to this exotic matter representations is not particular to this specific global model. Following the same steps, we    (see table 18).

Large Hodge number shifts
Four of the "extra" Hodge number pairs in the region h 1,1 ≥ 240 turn out to come from standard Tate tunings of generic models that have h 1,1 < 220, outside the region we considered for starting points. These are listed as "huge tunings" in Table 17. These models each contain a chain of {−1, −4}s, which allows so(n) with n very large to be enhanced on the −4-curves, producing huge shifts of the Hodge numbers. While there are only four specific models of this type among the 18 Hodge pairs in the region of interest not found by Tate tunings, it seems that this large tuning structure on chains of −1, −4 curves is a common feature and there are many other examples of this in the database, both increasing multiplicities at large Hodge numbers in cases that also have Tate tuned realizations, and also occurring at Hodge numbers outside the range of interest here.
We work out one example here in detail; the others have similar structure. The example with the largest h 1,1 (from the four "extra" models of this type) is the polytope M: 20  The gauge algebras on −4 and −1 curves are only determined from this analysis up to monodromies. We can, however, determine the algebras without explicitly analyzing monomials. First, from the anomaly constraint analyzed in [12], su cannot be adjacent to so, so the algebras on −1 curves have to be sp. The choice so(2n − 5) or so(2n − 4) on −4 is determined from global symmetry constraints. For example, it has to be so (20) rather than so (19) between two sp(6) algebras for the global symmetry on the −4 curve to be satisfied; while the lower rank so (17), so (19) has to be chosen for two −4's connecting to sp ( which give the correct Hodge number shifts (in particular, one can quickly check that according to the rank of the gauge algebras ∆h 1,1 = 126 as expected above).

Tate-tuned models corresponding to non-toric bases
We have not considered tuning an e 8 algebra on any curve of self-intersection m ≥ −8, as it leads to a violation of the anomaly conditions that corresponds to the appearance of a (4, 6) singularity. Similarly, tunings of so(n ≥ 13) on −3-curves are also ruled out by anomaly cancellation. Nonetheless, there are polytope models in the KS database that appear to contain these tunings, which give rise to Hodge pairs that we have not obtained in Tate tunings of Kodaira type. This set of tunings can be understood as more complicated generalizations of the non-flat structure we have already described for fibrations over −9, −10 and −11 curves. As we discussed already in that context, over (4,6) points the resolved fiber in the polytope model is two-dimensional, but we can understand the Calabi-Yau geometry by resolving the base at these points to obtain a corresponding flat elliptic fibration model over a blown up base that is generally non-toric. In this section we describe models that involve e 8 algebras tuned on −8 curves and models involving tunings of so(n ≥ 13) on −3-curves. In the latter case, the "extra" models in the KS database in our region of interest involve a further complication in which a nontrivial Mordell-Weil group is generated associated with an abelian U(1) factor in the F-theory gauge group; a detailed example with that additional structure is relegated to Appendix C. The remaining four Hodge pairs corresponding to standard P 2,3,1 -fibered polytopes at large h 1,1 that were missed in our Tate tuning set have a combination of two novel features: they have apparent so(n ≥ 13) tunings on −3 curves, and also have extra sections associated with a nontrivial Mordell-Weil rank and corresponding U(1) factors in the F-theory physics. For clarity, we delegate a complete example of one of the "extra" models of this type to Appendix C, and focus in the rest of this section on the issue of so(n ≥ 13)-tunings on −3-curves in the context of simpler models with relatively small h 1,1 that do not also involve the U(1) issue.
As mentioned above, so(n)-tunings on −3-curves give rise to (4, 6) singularities and twodimensional resolved fibers when n ≥ 13. While the anomaly conditions impose an upper  curve with 13 ≤ n < 26. These are non-flat elliptic fibration models. The last column gives the (4, 6) points and the corresponding 2D toric fiber components contained in the hypersurface CY (see Table 8 for pt in tops). The Hodge numbers can be calculated from the associated flat elliptic fibration model over the non-toric base where the (4, 6)-point are blown up.
would be a rank-preserving tuning, and, as for the so(10) to so (11) tuning, in the absence of other issues these should have the same Hodge numbers, but they clearly do not. An explicit analysis shows that over a generic point on the −3-curve, the hypersurface equation intersects with seven components associated with the seven lattice points {pt 1 , pt 1 , pt 2 , pt 3 , pt 4 , pt 6 , pt 6 } in the so(13) top in a locus containing P 1 's which form the so (13)  The Hodge numbers of the polytope models from the succesive tunings can be calculated this way up to so (26), at which point all monomials in a 6 are tuned off, and a U(1) global factor comes into play. See Appendix C for an explicit analysis of one of the models associated with the missing Hodge pairs where such a U(1) becomes relevant. We list the non-flat polytope models of tuning so(n), 13 ≤ n < 26, over the −3 curve of F 3 in Table 20.
6.2 Weierstrass models from non-standard P 2,3,1 -fibered polytopes For the remaining eight Hodge pairs with large h 1,1 in the KS database that were missed by our Tate construction (see table 17), the CY hypersurface equations (3.7) with suitable homogeneous coordinates cannot be in Tate form, although the ∇ polytopes are still P 2,3,1 fibered. The failure to be in the Tate form arises from the feature that there are lattice points in ∆ that give rise to non-trivial base dependence in the coefficients of the monomials x 3 or y 2 ; i.e., these should be sections of non-trivial line bundles over the base.
We would like to have a Weierstrass description of the non-standard P 2,3,1 -fibered polytopes so that we can use the methodology of F-theory to understand and analyze the geometry. To this end, we treat the P 2,3,1 fiber as a twice blown up P 1,1,2 fiber, as depicted in figure 5a; following the procedure in appendix A of [35] to obtain the Weierstrass model of the associated Jacobian fibration model of a P 1,1,2 -fibered polytope, we can obtain similarly that of the blownup P 1,1,2 -fibered polytope. Note that because even non-standard P 2,3,1 -fibered polytopes give elliptic Calabi-Yau hypersurfaces that have a global section, the Jacobian fibration should have the same geometry as the original Calabi-Yau hypersurface; this would not be true for example if the original elliptic fibration had no section [49]. Explicitly in coordinates, instead of treating the elliptic fiber as being embedded in the P 2,3,1 ambient fiber with v x = (0, 0, −1, 0), v y = (0, 0, 0, −1), v z = (0, 0, 2, 3), we treat the P 2,3,1 fiber as a blownup P 1,1,2 , and embed the elliptic fiber in this blownup P 1,1,2 ambient fiber with v x = (0, 0, −1, 0), v y = (0, 0, 0, −1), v z = (0, 0, 1, 2). The blowup rays of P 1,1,2 reflect the fact that two of the nine sections of a P 1,1,2 -fibered polytope model are completely tuned off (see figure 5b) -the hypersurface equation of a non-standard P 2,3,1 -fibered polytope is a specialization of that of a generic P 1,1,2 -fibered polytope, and the blowups of the P 1,1,2 fiber resolve the singularities of the tunings.
The Weierstrass models obtained in this way from non-standard P 2,3,1 -fibered polytopes (a) Comparing the polytopes for P 2,3,1 (red) and P 1,1,2 (blue). The red triangle ∇231 comes from blowing up the (fiber) fan of ∇112 twice (cf. indicated rays in figure 1a for standard P 2,3,1 -fibered polytopes.) (b) The blue triangle ∆112 is the dual of ∇112 and the red triangle ∆231 is the dual of ∇231.
The monomials in different sections are categorized by projection to the different lattice points in ∆112, labeled in terms of the homogeneous coordinates x, y, z in the fiber. The equations describing the hypersurface Calabi-Yau for a non-standard P 2,3,1 polytope can be characterized as tunings for a P 1,1,2 -fibered polytope, in which there are no nonzero monomials in the sections labeled x 4 and x 2 y. This interpretation allows the possibility of having sections in y 2 or x 3 . Figure 5 have the novel feature that they can have gauge groups tuned over non-toric curves in the base. Moreover, unlike the toric curves, which are always genus zero curves (isomorphic to P 1 ), non-toric curves can be of higher genus, and this class of global Weierstrass models gives examples of tunings of gauge groups over higher genus curves in the bases. As a check on this picture, we can verify that the Hodge numbers of these Weierstrass models calculated from anomaly cancellation match with those of the polytope data.
We give some examples of Weierstrass models from non-standard P 2,3,1 -fibered polytopes in the following subsections. In §6.2.1, we give a simple example that illustrates the nontoric curve enhancement feature. In §6.2.2, we analyze the eight remaining polytope data with large h 1,1 from the KS database that were missing in the Tate-tuned construction. We also give some further examples of interesting geometries from the KS database at smaller Hodge numbers to illustrate the unusual nature of the non-standard P 2,3,1 -fibered polytope construction. In §6.2.3, we give a model with an su(2) tuning on a non-toric curve of genus one in the base.
The first ∇ 1st polytope is, on the other hand, of a non-standard P 2,3,1 -fibered form. The data of this polytope can can be obtained by removing the vertex (1, 0, 2, 3) from ∇ 2nd or equivalently by adding the lattice point (−1, 0, −2, 1) to ∆ 2nd , which becomes a vertex of ∆ 1st . The one lattice point reduction of ∇ 2nd corresponds to the one lattice point enhancement of ∆ 2nd . Let us now show explicitly that ∇ 1st is a non-standard P 2,3,1 -fibered polytope and check that it satisfies each of the two equivalent conditions (i.e., the absence of an appropriate preimage of the base in ∇ and the condition that ∆ has lattice points associated with monomials in x 3 or y 2 that have base dependence): The base rays of ∇ 1st are the same as those of ∇ 2nd , but the ray (1, 0) lacks the preimage (1, 0, 2, 3) that we have removed; instead, the base ray (1, 0) comes from the projection of the 4D ray (1, 0, 1, 2); nonetheless ∇ 1st still has P 2,3,1 as a subpolytope, and therefore ∇ 1st is a non-standard P 2,3,1 -fibered polytope. The equivalent condition for a non-standard P 2,3,1 -fibered polytope is also satisfied from the ∆ point of view: let us associate base coordinates Although we do not have a Weierstrass model from a Tate form for this polytope, we instead have a Weierstrass form for the hypersurface in the Bl 2 P 1,1,2 -fibered polytope (where we have substituted some generic Z values in the complex structure moduli): According to this analysis, there is an so(9) enhancement on the −4-curve (b 3 = 0) and an sp(1) enhancement on the non-toric 0-curve {35b 2 + 24b 4 = 0}. Note that this non-toric curve is a (rational) 0-curve because it is in the same class as the two toric 0-curves. The curve supporting the sp(1) algebra intersects both the −4-and the 4-curve at one point. This is essentially the same configuration with the second model, so the Hodge numbers from anomaly calculation also give the same result, {8, 250}, in both cases. While in this case, the non-toric curve supporting the sp(1) can be trivially transformed into a toric curve by a simple linear change of variables, this is not the case in the more complicated examples that we consider in the later subsections.

6.2.2
The eight remaining missing cases at large h 1,1 Now let us come back to the polytopes of the eight Hodge pairs in the large h 1,1 region that we did not obtain through Tate tunings and that have non-standard P 2,3,1 fibration structure. We go through one example in detail; the others have similar structure. where both the lattice points in the second line contribute to a y 2 term but with base dependence. Performing the projection we find that ∇ is a non-standard P 2,3,1 -fibered polytope over the base There are in total 102 base rays, and all rays but v i=98,99,100,101 have a preimage of the form (_, _, 2, 3). The generic Weierstrass model over this base has the Hodge numbers {257, 77}, so the tunings must be such that the shifts are {4, −32}. We analyze the Weierstrass model of the nonstandard P 2,3,1 polytope; as in the preceding example we treat ∇ as a Bl 2 P 1,1,2 -fibered polytope (in particular, the fiber coordinates are associated to {v x , v y , v z } = {(0, 0, −1, 0), (0, 0, 0, −1), (0, 0, 1, 2)}), and find the associated tuned Weierstrass model. The resulting computation of {f, g, ∆} shows that • Over the toric curve D 100 ≡ {b 100 = 0} the vanishing order is enhanced to {0, 0, 2}, which corresponds to an su(2) gauge symmetry on the −2-curve.
• Over the non-toric curve D non-toric ≡ {b non-toric = 0}, where , and there are no intersections between the divisors associated with the variables in the first and second terms). As in the preceding example, this non-toric curve is a 0-curve, and is linearly equivalent to the combination of curves D 98 + D 99 + D 100 + D 101 , as can be seen from the first term in (6.8). The complicated combination of powers in the second term in (6.8) arise from the structure of the toric rays and the sequence of blow-ups needed to build those rays from a fiber of a minimal model Hirzebruch base.
• Over the curve D 97 ≡ {b 97 = 0} (a −9-curve), there is a two-dimensional resolved fiber; due, however, to the enhancement over the non-toric curve intersecting the −9-curve, there are some differences between the fiber structure over this −9-curve and the one in an isolated −9-curve such as we discussed in §4.7 (see also Appendix B): The top is the same as that in equation (B.2), so the 9 components that are the boundary of the 3dimensional face intersect with the CY over a generic point in the −9-curve in a locus of which is a −1-curve. The three curves, −2, −2, −1, connectingD 97 andD non-toric are respec-tivelyẼ Now we figure out the gauge symmetries on these divisors. There was an su(3) on the 0-curve, D non-toric , which is now on the −1-curve,D non-toric . This forces an su(2) on the −2-curve, E 1 , connecting toD non-toric . The configuration of the intersecting curves and the symmetry enhancements are drawn in Fig. 6.
• su(2) ⊕ su(3) on {Ẽ 1 ,D non-toric }: ∆h 1,1 = +1 + 2 and ∆h 2, The total gives the desired Hodge number shifts {∆h 1,1 , ∆h 2,1 } = {4, −32}. Note that the final Hodge numbers correspond to the flat elliptic fibration over the non-toric base, and correctly reflect the associated contribution of three tensor multiplets from the three blow-ups of the base. 15 The correspondence between the non-flat and the flat models may be considered as that the four irreducible components of the 2-dimensional fiber over the (4, 6) point transform to the three divisors resolving the −9-curve in the base and one divisor in the fiber to resolve the forced su(2) onẼ 1 . We now consider the remaining non-standard P 2,3,1 fibered "extra" cases at large h 1,1 . The model associated with Hodge numbers {245, 57} has gauge symmetry enhanced on the non-toric curve, but there are no (4, 6) singularities involved, which is similar to the example in 6.2.1; we must, however, consider be careful to properly include the shared matter contribution to the matter multiplets, as a curve intersecting the non-toric curve also carries a gauge symmetry.  • one (4, 6) point on the −7-curve D 96 . 15 The Hodge numbers denoted in a generic model containing a −9/ − 10/ − 11-curve are understood to be those of the flat elliptic fibration over the base that has been resolved at the (4,6) points.

Summary of results
In this paper we have carried out a systematic comparison of elliptic Calabi-Yau threefolds with large Hodge numbers that are realized by tuning Tate-form Weierstrass models over toric bases and those that are realized as hypersurfaces in toric varieties through the Batyrev construction. Specifically, we have considered a class of Tate-tuned models over toric bases that have nonabelian gauge groups tuned over toric divisors. These tunings give a specific class of "standard" P 2,3,1 -fibered reflexive polytopes, all of which give Calabi-Yau threefolds with Hodge numbers that appear in the Kreuzer-Skarke database.
• Almost all Hodge number pairs of known CY3's in the regime studied come from elliptically fibered Calabi-Yau threefolds associated with polytopes constructed in this fashion that are associated with an explicit Tate/Weierstrass construction of the restricted class that we considered in our initial analysis.
• We have explicitly analyzed the structure of the Calabi-Yau threefolds in the Kreuzer-Skarke database for the 18 Hodge number pairs not found in our initial analysis from Tate constructions. All of these admit elliptic fibrations of slightly more complicated forms.
• Thus, we have found explicit realizations of elliptic Calabi-Yau threefolds that produce all Hodge number pairs with h 1,1 ≥ 240 or h 2,1 ≥ 240 that are known to be possible for Calabi-Yau threefolds. This matches with the results of a companion paper [17] showing that all polytopes in the KS database giving Calabi-Yau threefolds with h 1,1 ≥ 150 or h 2,1 ≥ 150 have a genus one fibration, and have an elliptic fibration whenever h 1,1 ≥ 195 or h 2,1 ≥ 228. These results provide additional evidence that virtually all known Calabi-Yau threefolds with large Hodge numbers are elliptically fibered, building on a variety of other recent work that has led to similar observations [13,14,11,15,16,12]. Since the number of elliptic Calabi-Yau threefolds is finite, this in turn suggests that the number of distinct topological classes of Calabi-Yau threefolds is in fact finite.
• In the course of this analysis we have encountered some novel structures in the Tate/Weierstrass tunings needed to reproduce certain CY3's associated with polytopes in the KS database. This has led to new insights into the Tate algorithm as well as in the structure of fibrations that may occur through polytopes.
-A novel Tate tuning of SU(6) gives rise to exotic 3-index antisymmetric matter, of a form recently studied in [25,50].
-Some polytopes in the KS database correspond to tunings of very large gauge algebras with components like so (20).
-Polytopes in the KS database include non-flat elliptic fibrations over toric bases that resolve into flat elliptic fibrations over more complicated non-toric bases including not only blow-ups of −9, −10, −11 curves, but also more exotic structure such as an e 8 over a −8 curve that must be blown up four times, or tunings of so(n), n ≥ −13 on −3 curves that must be blown up to −4 curves to satisfy anomaly conditions. In some of the so(n) cases the resolved geometry also gives rise to a nontrivial Mordell-Weil group associated with a U(1) factor in the gauge group.
-Some polytopes in the KS database have elliptic fibrations over toric bases in which nonabelian gauge algebras are tuned over non-toric curves in the base.
-We worked out the tops associated with the gauge algebras so(n), 13 ≤ n ≤ 25, as well as the tops associated with gauge algebras su(n), 7 ≤ n ≤ 13. The tops associated with I n and I * n types have the feature that they develop along the fiber direction, and the projection to the fiber plane falls outside the P 2,3,1 fiber subpolytope, which is a novel feature not found in the previously known tops.

Possible extensions of this work
In the companion paper [17], we carry out a complementary analysis to that of this paper.
Here we have started from the Tate tuning picture and matched to data in the Kreuzer-Skarke database. One can instead start with the polytopes in the database and try to derive the elliptic fibration structure. This is essentially the approach taken by Braun in [35], in which the database was scanned for elliptic fibrations over the base P 2 . In [17], we take that point of view and analyze the fibration structure of the polytopes in the KS database directly. The approach taken in this paper, however, shows that at large Hodge numbers most Calabi-Yau threefolds have a standard elliptic fibration structure; the "sieve" approach taken here enables us to identify some of the most interesting cases that present novel features. There are several closely related analyses that could be carried out that we have not done here or in [17]; each of these represents an opportunity for further work that would give increased understanding of the set of Calabi-Yau threefolds, the role of elliptic fibrations, and the landscape of 6D F-theory models.
First, we have started from the point of view of tuning Tate models and used the output of that analysis to match Hodge numbers in the KS database. In principle, we could have tried to reproduce all the polytopes in the database, i.e. included multiplicity information. For reasons discussed in §4.7, this would be a more complicated analysis. In many cases there are multiple local Tate tunings that give equivalent gauge groups, and we have in each case systematically taken only the lowest possible choice for NHCs and the lowest order choice with no further monodromy condition required for a given gauge group tuning. For bases with many toric divisors, the number of combinatorial possibilities of local tunings can become quite large. There are also many equivalent models that correspond to carrying out explicitly different subsets of toric blow-ups to partially resolve (4, 6) singularities. We have checked in some cases that the multiplicity of Hodge numbers in the KS database is reproduced by distinct Tate/Weierstrass tunings of elliptic fibrations, but we have not approached this systematically. This would be a natural next step for this kind of analysis, and might reveal additional novel structures for the elliptic fibrations found in the KS database.
Second, we have restricted to large Hodge numbers in part because we have only focused on Tate models associated with the most generic P 2,3,1 fiber structure for the polytope. There are 16 distinct possible toric fibers, analyzed in detail in the F-theory context in [35,19,43], each leading to a distinct class of Weierstrass tuning types with characteristic nonabelian and abelian gauge structure, and in principle we could systematically analyze all tunings that correspond to each of the different fiber types. This would be necessary to extend the analysis of this paper systematically to smaller Hodge numbers, where the other fiber types become prevalent [17]. We leave such an endeavor for future work. It would also be interesting to see whether the more general class of fibers considered in [53] may give further insights into other Weierstrass tuning types that may be possible with complete intersection fibers.
In this Appendix, we go through the details of a standard P 2,3,1 -fibered polytope tuning with an example of polytopes for elliptic fibrations with tuned su 3 , g 2 over the curve of self-intersection −2 in the Hirzebruch surface base F 2 .
Standard P 2,3,1 -fibered polytopes naturally correspond to Tate (tuned) models. In principle, as long as the Tate tunings on adjacent curves do not lead to (4, 6) singularities 16 , and are not merely further specialization of existing tunings that do not change the gauge algebra 17 , removing the lattice points corresponding to a given tuning gives a different reflexive polytope, associated with the resolved CY of the tuned singular model. The Hodge numbers of the new resolved polytope model can be computed either directly from the polytopes or through F-theory by anomaly cancellation.
As an example, consider tuning a type I s 3 su(3) gauge algebra on the −2 curve in the base B Elliptic fibrations over the bases F 9 , F 10 , and F 11 The "simple stacking" construction ( §4) of a polytope for a standard P 2,3,1 -fibered model over a base surface containing −9, −10, −11-curves produces a flat toric fibration that leads to a hypersurface that is a non-flat elliptic fibration. There are (4, 6)-points in the −9, −10, −11curves where the fiber becomes two-dimensional; the singular fiber is resolved into an irreducible component of the non-generic toric fiber, which is two-dimensional, as the hypersurface CY equation restricting to the component is trivially satisfied over these points. For a flat elliptic fibration, the (4, 6)-points in the base must be blown up, which in general leads to a non-toric base. Note that in the Calabi-Yau hypersurface picture, some flops may be necessary before the blow-ups can be done in the toric picture [35]. Nonetheless, this provides a clear correspondence between the non-flat elliptic fibrations associated with polytopes leading to (4, 6) points in the base and flat elliptic fibrations over blown up bases, which provide Calabi-Yau threefolds with the same Hodge numbers. In this Appendix we go through the details of these constructions for the Hirzebruch surface bases F m , m = 9, 10, 11. give non-flat elliptic fibration models over the toric bases F m=10 and F m=11 , respectively. Both vertex sets are to those in equation (B.1), and the tops over the −m-curves are the same as that over the −9 curve in equation (B.2). We know that a −10-curve (resp. a −11-curve) would need two blowups (resp. one blowup) to become a −12-curve, so we expect there are two (4, 6) points (resp. one point) in the −m-curve over which the resolved fiber is two-dimensional. Indeed, we calculate the CY hypersurface in equation in the case of m = 11. Over a generic point in the −m-curve, p| I is non-vanishing, and p intersects with the nine components in (B.3), each giving a P 1 that corresponds to a node in the extended E 8 Dynkin diagram. The correspondence between the non-flat and the flat models may be thought of as encoding the relationship between the irreducible component of the 2-dimensional fiber over a (4, 6) point and divisors that resolve the −m-curve to a −12-curve in the base.
C An example with a nonabelian tuning that forces a U(1) factor.
In this Appendix, we work through the details of an example of the missing Hodge pairs in the last part of  a blow-up from an so(n) tuning on a −3 curve, and the further feature of a forced nontrivial Mordell-Weil group giving a U(1) factor. After describing the geometry, we do a detailed calculation of the Hodge numbers through the associated flat elliptic fibration model.
The rational sections of an elliptic fibration form the Mordell-Weil group, which is a finitely generated group of the form Z rank × (torsion subgroup). If an elliptically fibered Calabi-Yau has a non-trivial Mordell-Weil rank, the F-theory compactification on it has an abelian sector U (1) rank [26]. The Weierstrass model of an elliptically fibered Calabi-Yau automatically comes with a zero section z = 0. Additional sections can be produced through constraints in the toric geometry [19]. For instance, an abelian global u(1) symmetry is forced when we set all the monomials in the section a 6 to vanish (the condition a 6 = 0 in [48].) While this can be simply imposed as a constraint to tune a U(1) factor, this condition can also be imposed when we tune a large enough set of nonabelian gauge algebras on the toric curves. The lack of the monomials in a 6 occurs in this way in the four missing Hodge pairs {263, 32}, {251, 35}, {247, 35}, {240, 37} in Table 17, which are therefore Bl [0,0,1] P 2,3,1 -fibered polytope models (see Figure 7).
The The non-flat fiber results from the so(92) on the −3-curve, as it exceeds the upper bound so(12) associated with anomaly conditions. As the non-abelian tuning uses all of the monomials in a 6 , the dual fiber subpolytope ∇ 2 becomes a blowup of P 2,3,1 , Bl [0,0,1] P 2,3,1 (see Fig. 7). Now we compute the Hodge numbers from the associated flat elliptic fibration model over the resolved base  where the 1/2 factors come from the group theoretic normalization constant of so(n). Hence, ∆H non-abelian charged = (sum of all terms in (C.5) − sum of all terms in (C.6)) = 14830. Note that all representations of a forced non-abelian gauge group are shared: 1/2(92) = 46 on the exceptional −1-curve are shared. All representations on the blown up −4-curve are also shared: 1/2(38 + 58 + 72) = 84, so the gauge symmetries can not be enhanced further on the three intersecting −1-curves.
The final piece needed is the U(1) charged matter. These fields are not charged under the non-abelian group, and therefore have not yet been taken into account in our computations. These matter fields are localized at codimension two on the I 1 component (away from the non-abelian components) of the discriminant locus (equation (2.10)), and the number of the U(1) charged matter fields corresponds to the number of the nodes, over which the fiber is Kodaira I 2 , on the I 1 component [51]. Concretely, as described for example in [52], we calculate the discriminant locus of the I 1 with respect to one of the two local coordinates, which we choose to be b 1 associated with the 2-curve and b 2 associated with the 0-curve; then the I 1 discriminant locus factors into where p 1 is a polynomial of degree 76 in b 2 , p 2 is a polynomial of degree 9 in b 2 , and p 3 is a polynomial of degree 63 in b 2 . The degrees of the polynomials p 2 , and p 3 correspond to the number of nodes and cusps on the I 1 , respectively. The hypermultiplets charged only under the U(1) are localized at the nodes, and therefore ∆H abelian charged = 9 in this example. Summing up all the pieces, we obtain ∆h 1,1 = ∆T + ∆r non-abelian + ∆r abelian = 1 + (251 − 18) + 1 = 235 and ∆h 2,1 = (∆V non-abelian + ∆V abelian ) − 29∆T − (∆H non-abelian charged + ∆V abelian charged ) = (14739 + 1) − 29 − (14830 + 9) = −128, which agrees with the differences in Hodge numbers from the polytopes: {263, 32} − {28, 160} = {235, −128}.