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F-Theory Compactifications with Multiple U(1)-Factors: Constructing Elliptic Fibrations with Rational Sections

We study F-theory compactifications with U(1)xU(1) gauge symmetry on elliptically fibered Calabi-Yau manifolds with a rank two Mordell-Weil group. We find that the natural presentation of an elliptic curve E with two rational points and a zero point is the generic Calabi-Yau onefold in dP_2. We determine the birational map to its Tate and Weierstrass form and the coordinates of the two rational points in Weierstrass form. We discuss its resolved elliptic fibrations over a general base B and classify them in the case of B=P^2. A thorough analysis of the generic codimension two singularities of these elliptic Calabi-Yau manifolds is presented. This determines the general U(1)xU(1)-charges of matter in corresponding F-theory compactifications. The matter multiplicities for the fibration over P^2 are determined explicitly and shown to be consistent with anomaly cancellation. Explicit toric examples are constructed, both with U(1)xU(1) and SU(5)xU(1)xU(1) gauge symmetry. As a by-product, we prove the birational equivalence of the two elliptic fibrations with elliptic fibers in the two blow-ups Bl_(1,0,0)P^2(1,2,3) and Bl_(0,1,0)P^2(1,1,2) employing birational maps and extremal transitions.


Introduction and Summary of Results
In recent years significant progress has been made on the construction of phenomenologically appealing models of particle physics from F-theory compactifications, initiated by the local model approach of [1,2,3,4]. The road to the construction of semi-realistic compact F-theory GUTs models with SU (5) or SO(10) gauge group has been paved in [5,6,7,8,9], although a model passing most phenomenological tests has yet to be constructed. Our understanding of both the local approach as well its embedding into global compactifications has improved vastly over the last years [10,11,12]. Since the development of F-theory [13,14,15] the construction of non-Abelian gauge symmetry in F-theory compactifcations is well understood due to the full classification of codimension one singularities of elliptically fibered Calabi-Yau manifolds [16,17,18]. A recent reconsideration and careful analysis has closed remaining gaps in the understanding of singularities at higher codimension [19,20,21]. Constructive algorithms for the explicit construction of non-Abelian gauge symmetries in toric Calabi-Yau manifolds are developed [22,23] and have recently been employed for an exhaustive exploration of 6d gauge symmetries from F-theory on toric elliptic fibrations over P 2 [24].
In contrast to this, the construction of Abelian gauge symmetries in F-theory is much less understood. One reason is the dependence of this question on the global geometry of the elliptic fibration. However, aspects of the physics of U(1)-symmetries can already be studied in local F-theory models employing spectral cover methods [25,26,6,27,28,29,30,31]. F-theory compactifications with an Abelian gauge theory sector arise from compactifications on elliptic Calabi-Yau manifolds X with general fiber being an elliptic curve with rational points. Elliptic curves E with a so-called non-trivial Mordell-Weil group of rational points are a classical subject in mathematics [32,33,34,35,36]. These rational points lift to rational sections of the fibration X, that contribute new harmonic two-forms to the cohomology that support Abelian gauge fields in the F-theory effective action. The number of Abelian gauge fields is set by the rank of the Mordell-Weil group of the elliptic curve and its torsion subgroup gives rise to non-simply laced groups [15,37,38]. Rank one Mordell-Weil groups in compact elliptic fibrations have been studied recently in the F-theory literature in varying contexts [39,40,41,42,43,44,45,46]. The gauge theory arising from an F-theory compactification changes significantly in the presence of U(1)-gauge symmetries. Additional codimension two singularities support singlet fields only charged under the U(1)-gauge fields and the charged matter fields of the non-Abelian sector split into multiple fields differing by their U(1)-charges. Furthermore, the structure of Yukawa-couplings are dominated by U(1)-selection rules. This additional structure has rich phenomenological implications in Beyond the Standard Model (BSM) model building, where it can be used to avoid dangerous proton decay operators, engineer realistic Yukawa textures or provide a solution to the µ-problem [6,27,47,39,48,30].
To broaden the applicability of F-theory to BSM physics it is desirable to construct Abelian sectors of higher rank in F-theory. For example, conservative SM-extensions by only a single U(1), e.g. in models G SM ×U(1) with a Z -boson, have an Abelian sector U(1)×U(1) Y of rank two and would require the understanding of rank two Mordell-Weil groups in F-theory. In addition, a classification of the possible Abelian gauge sector in F-theory analogous to the well-studied non-Abelian sector is lacking. However, see [49] for a systematic study of rational sections on toric K3-surfaces. We make some progress in this direction by constructing F-theory compactifications with a U(1)×U(1)-sector 1 .

Summary of Results
In this work we construct for the first time an elliptic curve E and a resolved elliptic Calabi-Yau fibrationX with Mordell-Weil (MW) group of rational sections of rank two. We systematically derive by purely algebraic methods the concrete presentation of the elliptic curve E as the Calabi-Yau onefold in the toric del Pezzo surface dP 2 2 . An Ftheory compactification onX will have two U(1)×U(1)-gauge symmetry and charged matter. We determine the full matter spectrum including matter multiplicities for the first time and show consistency with six-dimensional anomaly cancellation. The general procedure employed here can readily be generalized to higher rank MW-groups, and might be relevant for the classification of the possible Abelian sectors in F-theory.
It is the pivotal result of our analysis that it is necessary to depart from the Weierstrass and Tate form to study the curve E and its corresponding elliptic fibrations. The reasons for this can be summarized by the following key findings of our analysis: i) Abandoning the paradigm of a holomorphic zero section: General models with two rational sections require that the zero section is only a rational, not a holomorphic section.
ii) Departure from Tate/Weierstrass form: The zero section in the Tate or Weierstrass model is always holomorphic. Thus we have to work in a realization of elliptic fibrations that naturally incorporate a rational zero section such as toric geometry.
iii) Toric elliptic curves: Toric geometry greatly facilitates the construction of some higher rank MW-groups. Rational sections are directly realized as toric divisors 3 . iv) Birational maps: Due to i) the map of the toric elliptic curve iii) with rational zero section to the Weierstrass model ii) with holomorphic zero section is rational.
We emphasize that it is also imperative to present E as the hypersurface in dP 2 because the Weierstrass model of the elliptic fibrationX will be generically singular. The birational map from the Weierstrass form to dP 2 will automatically resolve all singularities in any codimension, which is in accord with the smoothness of generic toric Calabi-Yau hypersurfaces [50]. Thus, the understanding of the rational map involved is also the key to analyze higher codimension singularities which determine the F-theory matter spectrum.
In the following we summarize the key points of our program leading to the construction of the curve E in dP 2 and the analysis of the associated elliptic fibrationsX, its rational sections, its Tate 4 and Weierstrass form and codimension two singularities.
• We first derive the representation of an elliptic curve E over a field K with two rational points Q, R and a zero point P . To this end we consider a degree three line bundle M = O(P + Q + R) with holomorphic sections u, v and w and find a cubic relation among these holomorphic sections. This naturally leads to an embedding of the elliptic curve E as the non-generic cubic curve in P 2 , which is resolved into the Calabi-Yau onefold in the del Pezzo surface dP 2 . We determine the birational map of this elliptic curve in dP 2 to its Tate form and Weierstrass form with respect to the zero point P . This birational map allows us to determine the coordinates of the rational points Q and R in Weierstrass form. Then we construct an elliptically fibered Calabi-Yau threefold E →X → B over a base B with general fiber being the elliptic curve E in dP 2 . The point P becomes the zero sectionŝ P of the fibration and the rational points Q, R lift to rational sectionsŝ R ,ŝ Q .
• We find the matter content of an F-theory compactification onX by analyzing the codimension two singularities of the elliptic fibration, by deriving the Weierstrass representation of the fibration and the rational sections. The resolved fiber in dP 2 over all codimension two loci is a reducible I 2 curve, i.e. the elliptic curve E splits into two rational curves E = c 1 + c 2 intersecting in two points. This can be viewed as the Dynkin diagram of an extended SU(2) group arising in codimension two. The behavior of the rational sections is crucial for the analysis of matter. In the generic elliptic fibration with our dP 2 -fiber we identify six different codimension two loci. Along three loci one of the sectionsŝ P ,ŝ Q andŝ R are ill-behaved and we have to resolveX in the base B along the corresponding loci. In the resolved space the section wraps an entire fiber component. Physically, these loci are distinguished as the loci of matter of more exotic U(1)×U(1)-charges with charges (−1, 1), (−1, −2) and (0, 2). Further matter is located whenever one of our two rational sectionsŝ Q , s R intersect the zero sectionŝ P . These degenerations contribute matter of charges (1, 0) and (0, 1) depending on which section touches the zero section. Finally, a third type of singularity arises whenŝ Q andŝ R coincide and touch the zero section s R . This is still a codimension two phenomenon leading to matter of charge (1,1).
• Finally, we determine the multiplicities of matter fields. We demonstrate that this counting is comparably involved since the various codimension two loci corresponding to different matter fields intersect. Concretely, assuming matter of charges (−1, 1), (−1, −2) and (0, 2) is located at a collection of points loc 1 and matter with charges (1, 0), (0, 1) and (1, 1) is located at points loc 2 , we find that loc 2 is automatically obeyed at loc 1 , i.e. loc 1 ∩ loc 2 = 0. The appropriate multiplicity of the latter matter fields at loc 2 is then determined by subtracting the multiplicities of the former matter fields at loc 1 , taking into account the order of vanishing of loc 2 along loc 1 . This complication also arose in [43]. We show that the appropriate treatment involves the application of the resultant of loc 2 with loc 1 as its root.
We would also like to emphasize that we find all six-dimensional anomalies to be canceled for the matter charges and multiplicities found. This gives us some evidence for the completeness of our analysis of codimension two singularities of the fibrationX. Since the matter content is a codimension two phenomenon, we expect our results for the matter charges to hold in general, in particular for Calabi-Yau fourfolds.
We construct three global models as toric Calabi-Yau hypersurfaces with base P 2 . We present two different models with U(1)×U(1) gauge group, that differ in the complexity of the behavior of the rational sectionsŝ Q ,ŝ R . In the first model both sections never intersect the zero sections and only charge (1, 0), (0, 1), (1, 1) matter is realized. In the second model we allow for a non-trivial behavior of the sectionŝ R and confirm the existence of all six different matter multiplets. In both cases we confirm by anomaly cancellation the consistency and completeness of our analysis. We explicitly write down the 4d polytope realizing a model with SU(5)×U(1) 2 gauge group. We observe that matter in the 5-representation splits into five matter representation 5 (q 1 ,q 2 ) differing by their U(1) 2 -charges. The 10 matter curves does not split.
Finally we exploit the use of birational geometry to study extremal transitions between elliptic fibrations of two different fiber types [51] yielding two equivalent, but different descriptions of the same F-theory vacua with one U(1). We perform the extremal transition from the elliptic curve in P 2 (1, 2, 3), which is the standard Tate form, to the curve in Bl (1,0,0) P 2 (1, 2, 3), which is still in Tate form, but with Tate coefficient a 6 ≡ 0. Ultimately, simple toric techniques allow us to readily obtain the birational map from the latter model to the resolved quartic curve in Bl (0,1,0) P 2 (1, 1, 2), which immediately reproduces the map found in [43]. Extending this birational map to elliptic fibrations, this thereby establishes the birational equivalence of the U(1)-restricted Tate-model [39] and the fibration with elliptic curve in Bl (0,1,0) P 2 (1, 1, 2). We emphasize, however, that this equivalence only holds for elliptic fibrations of a certain class. This paper is organized as follows. In section 2 we introduce the basic concepts and constructions for elliptic fibrations with a Mordell-Weil group for F-theory compactifications with an Abelian gauge theory sector. In section 3 we derive the form of the general elliptic Calabi-Yau manifoldX with two rational sections. For this purpose we first derive the presentation of an elliptic curve E over a field K with two rational points as the hypersurface in dP 2 and then readily generalize to Calabi-Yau elliptic fibrations, that we classify for base B = P 2 . We elucidate the structure of the resolved geometry of the elliptic curve E in dP 2 . In section 4 we study in detail the general matter spectrum of the F-theory compactification on the resolved threefoldX. We show that both its singular Weierstrass model as well as the resolution toX with dP 2 -fiber are crucial to understand the structure of all codimension two singularities. We determine both U(1)×U(1)-charges and matter multiplicities. In section 5 we check that our spectrum is in general consistent with anomaly cancellation in 6d. We present three concrete toric realizations of elliptic threefoldsX over B = P 2 in section 6. Two examples have U(1)×U(1) gauge symmetry whereas we add an SU(5)-GUT sector in a third example. We conclude by an application of birational maps in section 7 to show the equivalence of elliptic fibrations with fibers of two different types, both of which with rank one MW-group. Section 8 contains our outlook. We have two appendices A and B with details on the Weierstrass model of the elliptic curve E and Nagell's algorithm of the cubic.
While we were preparing this manuscript the paper [52] appeared which has some overlap with our discussion. 6

Basics of Abelian Gauge Sectors in F-theory
In this section we review the notion of the Mordell-Weil group as the group of rational points on an elliptic curve in section 2.1. Then we discuss the geometry of Calabi-Yau manifolds formed as elliptic fibrations with general fiber given by such an elliptic curve in section 2.2. The rational points are lifted to rational sections of the elliptic fibration and contribute additional cycles to the homology group of these manifolds. The Abelian sector of an F-theory compactification on these elliptically fibered Calabi-Yau manifolds, as explained in section 2.3, is determined by the structure of these rational sections. The reader interested in the results of our analysis can skip this section and directly proceed with section 3.1.

Elliptic Curves with Multiple Rational Points
To set the stage for our later discussion, we begin with the most concrete definition of an elliptic curve E over a field K in terms of a Weierstrass model. The Tate form of the Weierstrass model reads with a i denoting numbers in a field K. In elliptic fibrations of E over a base B, K will be identified with the function field of B. In the chosen projectivization (2.1) is a one-dimensional Calabi-Yau hypersurface in the weighted projective space P 2 (1, 2, 3) with homogeneous coordinates [z : x : y]. The Tate form can be brought into the reduced Weierstrass formỹ 2 =x 3 + fxz 4 + gz 6 (2.2) by the variable transformatioñ with the following definitions Here the quantity ∆ defined in the last line of (2.4) is the discriminant of the Weierstrass equation (2.2). If ∆ = 0 then the elliptic curve defined by (2.2) is singular.
Both the Tate form (2.1) as well as the reduced Weierstrass form (2.2) have one distinguished point P at [z : x : y] = [0 : 1 : 1], referred to as the zero point. In general, a rational point on the elliptic curve E is defined as a point with coordinates in the designated field K. The set of rational points on E form an Abelian group under addition that is naturally defined in the Weierstrass form (2.2). The group of rational points, with P as the zero, is the Mordell-Weil group of the curve E. The Mordell-Weil group is finitely generated, i.e. it is the sum of a torsion subgroup and Z r , and r is the Mordell-Weil rank. We introduce a basis of the torsionless subgroup of the Mordell-Weil group as Q m , m = 1, . . . , r.
We note that the presence of a rational point can imply a factorization of the Weierstrass form. A case of particular interest later in this work is a point Q of the form [z :x :ỹ] = [1 : A : B] for given numbers A, B in K. In this case, the Weierstrass equation (2.2) factorizes as as is easily checked by plugging in the point Q. This implies that the coefficients f , g in (2.2) as well as the discriminant in (2.4) can be parametrized as This parametrization will prove useful in the discussion of matter with only charge 1 under the Abelian gauge field corresponding to the rational point Q [43], i.e. codimension two singularities only due to the presence of the point Q in the fiber E alone. We add that factorization properties of the Tate form (2.1) due to rational points of the form z = 0 have been discussed in [45].

Elliptic Calabi-Yau Manifolds with Rational Sections
In this section we briefly discuss the geometry of an elliptically fibered Calabi-Yau threefold X, although we note that the following holds for general complex dimension of X.
By definition an elliptic fibration over a base B is defined by a holomorphic projection π : X → B to the base B. We will be interested in fibrations with general fiber E = π −1 (pt) over a generic point pt in B given by elliptic curve with a zero point and 8 a number of rational points. As mentioned before we can always describe an elliptic fibration over B by its Weierstrass model (2.2), where the field K is replaced by the function field of B. Concretely, by the Calabi-Yau condition f , g are sections of the line bundles K −4 B respectively K −6 B , where K B denotes the canonical bundle of the base. If it exists globally we can also construct the Tate form 5 (2.1) where the coefficients a i take values in K −i B . The holomorphic zero section in the Tate and Weierstrass form is given by z = 0.
However, when having global questions in mind, such as the global resolution of singularities of the elliptic fibration X or the construction of rational sections, it is of advantage to consider elliptic fibrations X with the general elliptic fiber E given as the Calabi-Yau hypersurface in one of the 16 two-dimensional toric varieties. It is always possible, as discussed at the end of section 2.1, to obtain the Weierstrass form of these fibrations by a birational map. In this note we will focus on the elliptic fibration with general fiber in dP 2 , cf. section 3.1.
When we fiber an elliptic curve E over a base B its zero point P becomes the zero sectionŝ P and its rational points Q m lift to rational sectionsŝ m ≡ŝ Qm of the elliptic fibration π : X → B. All of these sections define injective mapsŝ P ,ŝ m : B → X and the group generated by theŝ m is the Mordell-Weil group of the elliptic fibration X. A holomorphic section, that is in the literature typically denoted by σ, defines a holomorphic injection σ : B → X on all of B. However, a rational section does in general not vary holomorphically over the base B. Indeed, over codimension two or higher the rational section can be ill-defined and wrap components of the reducible fiber over the singular loci [43,46]. Thus, rational sectionsŝ m can only be defined on the blow-up π B :B → B of B along the singular loci ofŝ m with π B denoting the blow-down map. Consequently, a rational section defines only a birational mapŝ m :B → B → X of the base B into X.
In sections 3.1 and 4, we study elliptic fibrations with general fiber given by an elliptic curve E with two rational points Q, and R and a zero section P . As we see there, in these cases even the zero sectionŝ P is not defined over codimension two and wraps fiber components of the elliptic fiber. A Calabi-Yau elliptic fibration without a holomorphic zero section still defines a valid F-theory background, although these most general fibrations have only recently drawn attention in the F-theory literature. As we discuss in section 4, the behavior of the three rational sectionsŝ P andŝ Q ,ŝ R leads to a rich structure of charged matter. The group of divisors, or its dual group H (1,1) (X), on the smooth elliptic Calabi-Yau manifoldX → X arising from X by resolving all singularities, is generated by divisors D A that fall into four different classes of divisors: • the zero sectionŝ P of the fibration with homology class S P , which in the case of a holomorphic section σ agrees with the class of the base B, • the rational sectionsŝ m , m = 1, . . . , r, with divisor classes S m generating the Mordell-Weil group of rational sections of the elliptic fibration of X, , of the fibration that are inherited from divisors D b α in the base B, • exceptional divisors D i I resolving singularities of X from singularities in its elliptic fibration at irreducible components ∆ I = 0 of the discriminant locus ∆ = 0 in B.
We summarize this basis of divisors onX as We conclude with some key intersection properties of the divisors D A . The exceptional divisors D i I over one common discriminant locus intersect as i I j I denotes the Cartan matrix of an ADE-group G (I) in case of an ADEsingularity in the fibration. In this case we denote the D i I as the Cartan divisors of the corresponding ADE group. Here the divisors S (I) = π * (S b (I) ) are related to the components of the discriminant and the S b (I) are the loci in the base B wrapped by 7-branes supporting a gauge group G (I) in F-theory. Upon intersecting the Cartan divisors D i I with a divisorD in B that intersects S (I) in a point, we obtain a rational curve that is localized over S (I) . The relation (2.8) teaches us that this curve is to be identified with minus the root α i I of the Lie-algebra of the ADE-group under consideration, i.e.
Next we turn to the divisors S P and S m of the sections. By construction the divisors corresponding to a section will intersect the general fiber F ∼ = E as In addition we note that any section of X also has to obey [43] where the first relation holds in homology and c 1 (B) is the first Chern class of the base.

Abelian Gauge Sectors in F-Theory
In an F-theory compactification on X gauge fields A a arise by expanding the M-theory three-form C 3 in the dual M-theory compactification 6 along appropriate (1, 1)-forms ω a , These are gauge fields along the Cartan generators of all G (I) and along U(1) groups. The U(1)-gauge fields correspond to rational sectionsŝ m and are constructed from the cohomology classes dual to the divisors S m of the rational sections by the Shioda map.
The Shioda map roughly speaking describes an orthogonalization procedure in the (co)homology group ofX. It is defined as the map from the Mordell-Weil group to H (1,1) (X) given by [43,55,44] where the curves C −α have been defined in (2.9) and C −1 (I) is the inverse of the Cartan matrix determined in (2.8). The divisorS P = S P + 1 2 π * c 1 (B) has been introduced for convenience, and is of relevance for the match of the F-and M-theory dual effective actions [56,57]. The matrix η αβ is the inverse of the intersection form of the divisors on the base, (2.14) The Shioda map (2.13) maps into the orthogonal complement 7 in H (1,1) (X) generated by the zero section S P , the vertical divisors D α and the Cartan divisors D i I . Thus it ensures that the gauge field associated to σ(ŝ m ) by the reduction (2.12) along the (1, 1)-form ω m defined as the Poincare dual of σ(ŝ m ) is a U(1)-gauge field.
Having defined the Shioda map (2.13) it is straightforward to calculate U(1)-charges of matter fields in F-theory. The matter fields arise from M2-branes wrapping rational curves c in the fiber of the elliptic fibration that are localized in codimension two in the base B. Under the assumption of a holomorpic zero section with S P = B, the charge of the M2-brane state under the U(1)-gauge field corresponding to the rational section σ m is then calculated as [43] This follows readily using the definition of the Shioda map (2.13) and the fact, that the isolated curves c neither intersect vertical divisors nor the holomorphic zero section.
We note that the intersection matrix between the σ(ŝ m ) projected into the homology of the base by π :X → B is Here, we introduced the definition of the Néron-Tate height pairing for a curve C and employed the normalized coroot matrix of the I-th 7-brane gauge group In this expression we denote the inner product on the Lie-algebra by ·, · and introduced the length squared λ I = 2 α 0 ,α 0 of the maximal root α 0 . The intersections (2.16) will be relevant for the discussion of anomaly cancellation in section 5.
We conclude by noting that if the zero sectionŝ P is only rational, the formula for the U(1)-charge has to be modified. The reason for this is that we have to relax the condition B · c = 0 and in general assume a non-trivial intersection where S P denotes the homology class of the rational zero section. Then (2.15) has to be replaced by In contrast, the formula (2.16) for the intersections of sections does not have to be modified. We note that (2.20) has also been used in [46].

Elliptic Fibrations with Two Rational Sections
In this section we determine an elliptic fibration with two rational sections and a zero section, i.e. an elliptic curve with Mordell-Weil group of rank two. This curve serves as the model for the general elliptic fiber in an elliptically fibered Calabi-Yau manifold. As discussed in section 2 the F-theory compactification on such a Calabi-Yau admits two U(1)-gauge groups.
We first find in section 3.1 that any elliptic curve E with three marked points (two rational and the zero point) has a representation as a non-generic cubic in P 2 . We argue further that this elliptic curve should be properly viewed as the generic Calabi-Yau onefold in the del Pezzo surface dP 2 , which is the blow up of P 2 at two generic points. However, we first focus on the singular model of E by neglecting the blow-ups in P 2 . This allows us to compute the Weierstrass model for E with respect to one of the three rational points on it and derive the location of the two other rational points in Weierstrass coordinates.
Then, in section 3.2, we discuss the resolution geometry of E in dP 2 that we obtain by performing the two blow-ups in P 2 . The understanding of the resolved geometry is relevant to resolve elliptic fibrations with general fiber being the curve E. We conclude by constructing elliptic fibrations with E and find that all elliptic fibrations with base B = P 2 are classified by two integers n 2 and n 12 . We will consequently denote the families of dP 2 -fibrations over P 2 we have found as dP 2 (n 2 , n 12 ).
Our discussion partly follows and extends the techniques of appendix B of [43].

Constructing an Elliptic Curve with two Rational Points
Our discussion in this section is based on the following basic mathematical fact: Given an algebraic variety X with a very ample line bundle 8 M defined over it, we can find an embedding of X into projective space. If this is the case, we will have enough independent global sections a 0 , · · · , a n such that for every point x ∈ X there exists at least one section not vanishing at this point. Thus, there is an immersion The algebraic variety we are interested in is an elliptic curve E over a field K. We find the embedding of the curve E into projective space, or more generally a toric variety, explicitly as a hypersurface. For this purpose we consider the global sections of powers of M k . For sufficiently large k, in our case k = 3, not all sections are independent and the relation between them yield the desired hypersurface equation.
As a warm-up we use this logic to derive the Tate form (2.1) of an elliptic curve E with only the zero point P as follows. We introduce the degree one line bundle M = O(P ) over E and consider the homogeneous coordinates [z : x : y] on P 2 (1, 2, 3) as sections of the line bundles M , M 2 and M 3 , respectively. We recall that in the case of an elliptic curve E the Riemann-Roch theorem tells us that the number of independent global holomorphic sections of a line bundle M of degree d is h 0 (X, M ) = d. Then the bundle M 6 has six independent holomorphic sections, however, we can construct seven sections from z, x, y. Thus, there has to be a relation between these sections which yields precisely the Tate form (2.1) in P 2 (1, 2, 3).
The same strategy applies to finding the equation that describes an elliptic curve with rational points Q m . Concretely, we first find the sections [z : x : y] of O(kP ), k = 1, 2, 3, and then determine the relation between sections of higher degree generated from [z : x : y]. We perform this procedure in the following for an elliptic curve E with a zero point P and two rational points denoted Q and R. In the same spirit as before we first start with a general line bundle M of degree three that we then specialize to M = O(P + Q + R). The group of holomorphic section H 0 (M ) is generated by three sections denoted u, v, w. The space H 0 (2M ) has dimension six with sections u 2 , v 2 , w 2 , uv, vw and wu. The space H 0 (3M ) must have nine independent sections, however we know ten of them. Consequently, there has to be a linear relation of the form with coefficients s i in the field K. This equation can be interpreted as the cubic hypersurface in P 2 . Even more it is a section of its anti-canonical bundle O(3H), see figure  1, where H denotes the hyperplane class in P 2 . Thus the zero of this section defines the one-dimensional Calabi-Yau manifold, i.e. the torus 9 . Now we specialize to M = O(P + Q + R). Let us assume that the section u vanishes at the three points P , Q and R, then equation (3.2) simplifies to Performing appropriate shifts of the coordinates v and w we can always get rid of the coefficients s 4 and s 10 . However, these variable transformations involve square roots of 9 There is a one-to-one correspondence between elliptic curves and two-tori.
14 Figure 1: Fan of P 2 on the left and its dual polytope on the right.
the coefficients s i that are generically not defined over the field K. Thus, we specialize the constraint (3.2) by setting the coefficients s 4 and s 10 to zero. This specialization can also be viewed as changing the toric ambient variety such that the coefficients s 4 and s 10 are automatically absent by means of the toric construction. This is achieved by going from P 2 to dP 2 , which is the blow-up of P 2 at two generic points. In figure 2 we have depicted the polytope of dP  This latter perspective on the specialization of the constraint (3.2) as the Calabi-Yau hypersurface in dP 2 is of particular relevance for the construction of smooth elliptic fibrations in F-theory. However, the blown-up geometry dP 2 is of no importance for the determination of the Weierstrass model of the curve elliptic curve E with the rational points Q and R, with which we proceed in the remainder of this section. Thus, we discuss the resolved geometry of dP 2 with all divisor classes including the exceptional divisors E i separately in section 3.2 and work for the following in the patch e 1 = 1, e 2 = 1.
The restricted hypersurface following from (3.2) with s 4 = s 10 = 0 takes now the form Then, this hypersurface constraint specializes at u = 0 to vw(s 7 v + s 9 w) = 0 , We emphasize that the points P and Q coincide with precisely those points at which we blow up P 2 into dP 2 , see section 3.2 for a more detailed discussion.
We note that we could set even more coefficients s i to zero or to one by using the automorphisms of the ambient space, e.g. by shifting or rescaling variables and the equation, if we were interested in the properties of the elliptic curve alone. For example the shift symmetry of w by an appropriate factor of u is still unbroken in the presence of the rational points Q and R since these are left invariant. For example, we can perform the transformation w → w − s 5 2s 8 u to eliminate the term u 2 w in (3.4) 10 . However, we use later in section 6 the elliptic curve defined by (3.4) as the general elliptic fiber in an elliptically fibered Calabi-Yau manifold. Then the coefficients s i are lifted to sections over the base of the fibration. In this case the embedding of the fiber into the Calabi-Yau manifold will in general forbid eliminating more s i than those in (3.4). This can be seen explicitly in the toric example constructed in section 6. Therefore, we work with the most general form (3.4) of an elliptic curve with two rational points. 10 The coefficients will take the form with the other coefficients unchanged.
To obtain the Weierstrass equation with respect to P , we need to find the three sections of H 0 (kP ) = H 0 (kM − kQ − kR) for k = 1, 2, 3. These sections are precisely the projective coordinates z,x,ỹ of the Weierstrass model in P 2 (1, 2, 3). For the purpose of constructing these sections we determine sections of H 0 (kM ) on the ambient space dP 2 that simultaneously vanish k times both at Q and R 11 when restricted to E.
(3.10) 11 We note that this somewhat counter-intuitive fact follows from the fact that we implicitly divide by a function g vanishing at P + Q + R. Indeed, we recall the definition of the line bundle associated to a Cartier divisor D, O(D) := { f g |div( f g ) + D ≥ 0} for local functions f , g, cf. [58]. Applying this for O(P + Q + R) we obtain that g has three zeros precisely at P , Q, R, with f of degree three vanishing elsewhere. For O(P ) we obtain g has to vanish at P and f is of degree one. Then, sections of O(P ) are a subset of sections of O(P + Q + R) by demanding that f in the latter vanishes precisely at Q and R.
We have summarized the somewhat lengthy expressions for f , g along with the Tate coefficient a i and the discriminant in the appendix A.
Along similar lines outlined below, we obtain the coordinates for the generators of the Mordell-Weil group of E given by the rational points Q and R in Weierstrass form. The result reads x R = 1 12 (12s 2 7 s 2 8 + s 2 9 (s 2 6 + 8s 3 s 8 − 4s 2 s 9 ) + 4s 7 s 9 (−3s 6 s 8 + 2s 5 s 9 )) , y R = 1 2 (2s 3 7 s 3 8 + s 3 s 3 9 (−s 6 s 8 + s 5 s 9 ) + s 2 7 s 8 s 9 (−3s 6 s 8 + 2s 5 s 9 ) +s 7 s 2 9 (s 2 6 s 8 + 2s 3 s 2 8 − s 5 s 6 s 9 − s 2 s 8 s 9 + s 1 s 2 9 ) , z R = s 9 . (3.12) In the remainder of the section we derive the final results (3.8) and (3.10). The reader less interest in these details can safely skip to section 3.2. Beginning with H 0 (M ), we already know that the section u = 0, by assumption, vanishes at the three points P , Q and R. Thus we set z := u . (3.13) Next we need two find a section of H 0 (2M ) with double zeros at Q and R on E. There are two such sections, one of which is given by the section u 2 = z 2 . The other one is constructed from an appropriate linear combination of the other elements of H 0 (2M ). We make the ansatz x := av 2 + buv + cw 2 + dvw + euw . (3.14) The coefficients are fixed requiring that the section x vanishes at degree two on both points Q and R. This gives us four equations. The other coefficient can be rescaled away using the scaling of x.
A nice pictorial way to think about these conditions on the coefficients in x is to realize that x = 0 and p = 0 in (3.4) are two curves in the ambient space dP 2 . Then the first two of the four conditions on the coefficients enforce that x vanishes at the points P , Q, which are also automatically the intersections of p = 0 and x = 0. The other two conditions on the coefficients then make Q, R a double zero or equivalently let the curves intersect tangentially.
A systematic way of finding the coefficients is to solve for one of the variables u, v, w in one of the equations p = 0, x = 0 and then to use the other one to determine the unknown coefficients in (3.14) in order to obtain the right order of vanishing. First we note that both Q, and R can be described in the affine patch v = 1. To solve for one of the remaining variables u, w instead of dealing with radicals we can approximate the curve E by a Taylor expansion. We solve for w ≡ w(u) order by order in u by considering f (u, 1, w(u)) = 0 as an implicit function for w in terms of u. Expanding the curve p = 0 first around Q, then around R with coordinates (3.6) we obtain, omitting terms of third order and higher in u, Now we can pull back the section x to the curve E by simply plugging the solution for w(u) into the ansatz (3.14) for x. Since we require an order of vanishing of two we have to set the coefficients of u 0 and u 1 to zero. Employing the expansion (3.15) at Q yields  Next, we need three elements of H 0 (3M ) with triple zeros at Q and R on the curve E. We know ten sections of this bundle, only nine are independent and we already know two combinations that vanish appropriately, namely u 3 and ux. The other possible section has to be a linear combination of the remaining seven sections and we make the ansatz Proceeding in the same way as before, we use the solutions (3.15) to pull the section y to the curve E, but now require that it vanishes at order u 3 both at Q and R. This yields three conditions for each point, i.e. a total of six conditions, by demanding that the terms u 0 , u 1 and u 2 are absent. Setting c = 1 by rescaling y and cleaning some denominators, we obtain Now that we know the form of the sections x, y we can determine the Tate model (2.1) by plugging in the solutions (3.18), (3.20) and solving for the Tate coefficients a i . We note that in order to find a solution we have to reduce the Tate form, which is a degree six polynomial in u, v, w, in the ideal generated by the original cubic constraint (3.4). We present the results for the a i in (A.2) of appendix A from which we immediately calculate the coefficients b 2 , b 4 and b 6 in (3.10). From the Tate model, the Weierstrass form (2.2) can be obtained by the variable transformation (2.3) that takes the form of (3.8) as claimed. We readily obtain the functions f and g in the Weierstrass model and the discriminant and refer to appendix A for their explicit form.
Next, we focus on finding the rational points Q and R in Weierstrass form. Recalling that there is an implicit multiplication by a meromorphic function with poles of degree k in each case, we obtain the coordinates for Q (R) if we look for a section x of degree two that vanishes three times at Q (R). We have to include all elements of H 0 (2M ): x := av 2 + buv + cw 2 + dvw + euw +f u 2 , . (3.21) The new parameterf is fixed by the additional constraint from the vanishing of the coefficient of u 2 , while all the other coefficients take the same values as before. Noting that z ≡ u, we then obtain at Q which fixes the x-coordinate of Q. After scaling and going to the Weierstrass coordinates we obtain the generator as in (3.11), where theỹ-coordinate of Q is obtained by solving (3.9) forỹ givenx Q . Following the same procedure for the point R we confirm the result (3.12). 20

Resolved Elliptic Curve in dP 2 and its Elliptic Fibrations
In this section we discuss the key geometric properties of the resolution of the elliptic curve (3.4) in dP 2 . To this end we have to take into account the exceptional divisors E i and their sections e i , i = 1, 2 that we have set to one in the previous discussion. Then, smoothness of E is ensured by the toric construction since the curve (3.4) lifts to the generic Calabi-Yau onefold in dP 2 . The smoothness property even holds in elliptic fibrations X with general fiber given by dP 2 , if the most generic toric Calabi-Yau hypersurface is considered. Indeed, as we discuss explicitly in section 4 the divisors E 1 , E 2 resolve the singularities of the fibration at codimension two and higher. In particular, this allows us to determine the matter spectrum in F-theory from the splitting of the elliptic curve in dP 2 over codimension two from the presence of the new projective coordinates e i .
We construct dP 2 as follows. We blow up P 2 at the two points This is the blow-down map π : dP 2 → P 2 , where e 1 , e 2 are holomorphic sections vanishing at the exceptional divisors E 1 , E 2 . We see that the original codimension two loci u = w = 0, respectively, u = v = 0 in P 2 are replaced by e 1 = 0, respectively, e 2 = 0, i.e. by entire divisors E i with the geometry of P 1 . We summarize the toric realization of dP 2 encoded in its polytope ∆ dP 2 and its homogeneous coordinates as vertices divisor class Here we denote the vertices of ∆ dP 2 by ν i in the first row and write them explicitly as two-dimensional row-vectors in the second and third column as depicted in the polytope in figure 2. The only integral internal point is the origin. In the third and fourth column we summarize the homogeneous coordinates on dP 2 and their divisor classes, where H is the pullback of the hyperplane class on P 2 and the E i are the exceptional divisors. This in particular implies that the anti-canonical bundle K −1 dP 2 of dP 2 , which is computed as the negative of the sum of all toric divisors, has changed to K −1 We note the intersections in the star triangulation of ∆ dP 2 with star given by the origin as Here we made use of the exceptional set, the Stanley-Reissner ideal SR of all vertices not sharing a two dimensional cone, reading We observe that the polytope of P 2 is embedded by the first three points ν 1 , ν 2 and ν 3 . The addition of the vertices ν 4 , ν 5 not only introduces the divisors E i but also removes the corners (−1, 2) and (−1, −1) of the dual polytope of P 2 , compare figures 1 and 2. These corners correspond precisely to the monomials s 4 v 3 and s 10 w 3 in P 2 , that are no longer admissible sections of the anti-canonical bundle on dP 2 . As one can see by either looking at dual polytope 2 or by writing down all sections of the anti-canonical bundle O(3H − E 1 − E 2 ) of dP 2 by hand with the charge assignments in (3.24), the generic Calabi-Yau hypersurface p on dP 2 reads p = u (s 1 u 2 e 2 1 e 2 2 +s 2 u v e 1 e 2 2 +s 3 v 2 e 2 2 +s 5 u w e 2 1 e 2 +s 6 v w e 1 e 2 +s 8 w 2 e 2 1 )+s 7 v 2 w e 2 +s 9 v w 2 e 1 .

(3.27)
Now we investigate the pullback under π of the points P , Q and R in (3.6) that we by abuse of notation denote by the same symbols. We readily infer from the map (3.23) that the section u ∈ O(P + Q + R) splits into three components as expected and we identify P as e 2 = 0, Q as e 1 = 0 and the image of R as D u := {u = 0}. The value of all coordinates in the notation [u : v : w : e 1 : e 2 ] is obtained by inserting these components into the hypersurface equation for the coordinates of the points P , Q and R on the resolved curve in dP 2 .
We emphasize the dependence of the points P , Q, R on the s i , in particular the situation when certain s i vanish. Note that it is manifest in the form (3.28), that the points P , Q and R are ill-defined if s i = s j = 0 simultaneously for the combinations (i, j) in (3.28) due to the Stanley-Reissner ideal (3.26). In particular, when considering elliptic fibrationsX with elliptic fiber given by (3.27) this translates to a special behavior of the corresponding sectionsŝ P ,ŝ Q ,ŝ R at codimension two loci s i = s j = 0. Note that the s i lift to sections on the base B of the fibrationX, see the next paragraph, and can generically vanish. As we discuss in detail in the next section 4 this behavior is the key to understand the matter spectrum of an F-theory compactification onX.
We would like to conclude by discussing the general construction of an elliptic fibration X over a base B. For concreteness we focus on fibrations over B = P 2 noting that the following can readily be generalized to different bases. First, we construct all fibrations of dP 2 over P 2 . The total space of this fibration of toric varieties is classified only by two integers, that we denote by n 2 and n 12 for reasons that will become clear below. We denote the total space as dP 2 → dP 2 (n 2 , n 12 ) → P 2 . (3.29) It is clear that the fibration dP 2 (n 2 , n 12 ) is only specified by two integers because we can associate each projective coordinates where H B is the hyperplane of the P 2 -base. However, by means of the three C * -actions on dP 2 we can always eliminate three k i and denote remaining two degrees as n 2 , n 12 . Then, a possible assignment is with all other coordinates taking values in the trivial bundle on P 2 . Next, we note that the anti-canonical bundle of dP 2 (n 2 , n 12 ) is calculated by adjunction as From this and the Calabi-Yau constraint (3.27) we can finally read off the degrees of the coefficients s i , that now lift to sections of P 2 as well: s 1 s 2 s 3 s 5 s 6 s 7 s 8 s 9 9 − n 2 − n 12 6 − n 2 3 + n 12 − n 2 6 − n 12 3 n 12 3 + n 2 − n 12 n 2 (3.32) Here the second line denotes the first Chern class or in other words the degree of the section s i with respect to the hyperplane class H B on the base.
Finally, (3.32) illuminates the meaning of the integers n 2 and n 12 . The sections s 7 and s 9 vanish at curves of degree n 2 respectively n 12 in P 2 . But as we see from (3.28) these are precisely the loci where the points Q and R, now lifted to rational sectionsŝ Q , ŝ R , respectively the points P , also lifted to a sectionŝ P , and R coincide. In other words, these integers calculate the following intersections, cf. (2.17), where we denoted the divisor classes of the sections by capital letters following the conventions of section 2.2.

Matter Spectrum: Codimension Two Singularities
In this section we determine the generic matter spectrum of a six-dimensional F-theory compactification on the elliptic threefoldX over B with the general elliptic fiber E in dP 2 found in section 3.1. We mostly consider the case B = P 2 , although our discussion is readily generalizable to other geometries.
Determining the matter spectrum requires an analysis of the codimension two singularities of this three-section elliptic fibration 12 . We see that the singularity structure that is generic to these fibrations is completely governed by the behavior of the three sections. This study of the behavior of the sections can be conveniently performed in the Weierstrass model (3.9) of the elliptic curve E. We determine the charges in three steps in subsections 4.1, 4.2 and 4.3. The multiplicities of matter fields are found in 4.4. This analysis requires a careful counting of the codimension two points since the loci in the base B supporting matter fields of different charges intersect. The right counting requires subtraction of multiplicities of intersecting loci according to their mutual order of vanishing, which is determined by the resultant of the polynomial systems determining the intersecting codimension two loci. Since the matter is a codimension two effect, the matter charges we find immediately carry over to four-dimensional F-theory compactifications on Calabi-Yau fourfolds. Our analysis extends the methods of [43] to elliptic fibrations with two-section models.
The following discussion is organized by the three qualitatively different codimension two singularities that can occur. The codimension two loci of the first two types are readily found in the Weierstrass model, whereas the third type has to be addressed in the birationally related dP 2 -model of section 3.2. The three types we distinguish correspond to the following behavior of the sections: • Away from the collision of a sectionŝ with the zero section z = 0 in the Weierstrass form, the existence of the section implies the factorization (2.5) of the Weierstrass form. This factorization makes manifest the presence of conifold singularities in X.
Their resolution leads to matter of charge 1 under the U(1) associated to σ(ŝ). We discuss the factorized Weierstrass form for the two sectionsŝ Q ,ŝ R in section 4.1.
• Two rational sectionsŝ,ŝ both away from the zero section z = 0 can collide and contribute charge (1, 1)-matter under the two associated U(1)× U(1)-symmetry. This behavior is discussed in section 4.2.
• A sectionŝ can be ill-defined at codimension two. This is easily seen in the dP 2model of section 3.2. The sectionŝ is ill-defined if its corresponding point (3.28) in the fiber passes through the Stanley-Reissner ideal (3.26) of dP 2 . This happens at codimension two loci s i = s j = 0 and we have to blow-up the base B at these loci. Our discussion in section 4.3 is organized according to which of these combinations vanish. Then the section wraps a P 1 fiber component, which is precisely the toric divisor in dP 2 corresponding to the ray that subdivides the 2d-cone corresponding to the two coordinates in the Stanley-Reissner ideal (3.26) .
All these singularities have in common that their resolved fiber is of I 2 -type, i.e. an SU(2)-fiber with the original singular curve and one isolated P 1 . The matter hypermultiplets in six dimensions arise from the isolated rational curve P 1 at the resolved codimension two singularities of the Calabi-Yau manifold. The intersection structure of the resolved fibers with the sectionsŝ P ,ŝ Q ,ŝ R determines the pattern of U(1)-charges of the matter. The multiplicities of matter fields is computed by counting the number of solutions to the codimension two constraints as demonstrated in section 4.4. We summarize our findings in table 4.1.

12
(−1, 1) n 12 (3 − n 2 + n 12 ) (0, 2) n 2 n 12 (−1, −2) n 2 (3 + n 2 − n 12 ) where the subscript indicates the degree of the polynomials on P 2 . The intersection of the sectionŝ with the zero section z = 0 in the Weierstrass model is the integer n, and we denote it by n 1 , n 2 for the sectionsŝ Q andŝ R respectively 14 . The intersection ofŝ Q , s R is denoted as n 12 , cf. We begin by spelling out the general strategy for the identification of these loci and specialize toŝ Q ,ŝ R below. We emphasize that the structure of the Weierstrass model of the fibration X with fiber E with two sections implies a particular factorization. This facilitates the search for codimension two singularities in the fiber in X that coincide where we set g 6 = g 6+2n g 2 n , g 9 = g 9+3n g 3 n .
Then, the Weierstrass model factors in the form with appropriate g 12 , cf. (2.6).
This is a singular Weierstrass model with singularities located at the points satisfying p 1 = dp 1 = 0, where d denotes the exterior derivative. The solutions to these equations are y = 0 , x = g 6 , g 9 = 0 ,ĝ 12 = g 12 + 2g 2 6 = 0 . We readily confirm that the discriminant (2.6) vanishes quadratically at g 9 =ĝ 12 = 0. The singularities are conifold singularities 15 that are resolved by the birational transformation (3.8). The original singularity at y = (x − g 6 ) = 0 is replaced by a rational curve P 1 and the whole fiber factors into two rational curves intersecting at two points, i.e. an I 2 -curve. The original singular fiber is identified as the rational curve intersected by the zero sectionŝ P . Then the other curve has to be the isolated exceptional P 1 . We can confirm this identification by using the birational transformation to the Weierstrass form (4.3) and by checking explicitly which curve is mapped to the singular point (4.4) at y = (x − g 6 ) = 0. Let us study this explicitly for the two sectionsŝ Q ,ŝ R of X.

Factorization at theŝ Q section
The MW-generator of this section has been worked out in equation (3.11). In this case, the section is located at z = 1 and therefore z = 0 globally. The Weierstrass equation thus factorizes with g 12 = 1 72 −s 4 6 + 4s 2 6 (2s 5 s 7 + 5s 3 s 8 + 2s 2 s 9 ) − 36s 6 (s 2 s 7 s 8 + s 3 s 5 s 9 + s 1 s 7 s 9 ) (4. We can use these two constraints to solve for any two coefficients s i , s j , as long as the solutions are rational. Plugging these solutions into the resolved elliptic fiber (3.4) reveals its I 2 nature. In fact, solving and replacing s 1 and s 2 yields a split of the fiber class as E = c 1 + c 2 for two rational curves c 1 , c 2 . The equations of the rational curves read The curve c 1 does not intersect s P norŝ R and is the isolated rational curve, see figure 3 for a depiction of this situation. This is also confirmed by noting that c 1 is blown-down to the singular point (4.4) by the map (3.8). It thus gives rise to a massless hypermultiplet. Since c 1 clearly intersects the rational sectionŝ Q = [u = 0 : v = 1 : w = 0] once, the hypermultiplet, according to (2.20) in the absence of Cartan divisors D i I , has U(1)×U(1)-charges (q 1 , q 2 ) = (1, 0) as claimed. We note that we could have performed the same analysis on the resolved geometry (3.27) with the same result.

Factorization at theŝ R section
The coordinates of this section in Weierstrass form are given in (3.12). We have to take into account that the z-coordinate of the section reads z = s 9 and can vanish. When it is non-zero, the Weierstrass equation factorizes as (4.3) along the loci g R The curve c 2 intersects the zero section,ŝ P and alsoŝ Q , but notŝ R . It is the original singular fiber as can also be checked using the birational map (3.8). The curve c 1 intersects the rational sectionŝ R at one point but does not intersectŝ P andŝ Q . It is blown-down to the singular point (4.4) in the birational map (3.8). Thus it is an isolated rational curve and contributes a hypermultiplet of charges (q 1 , q 2 ) = (0, 1) according to (2.20).

Doubly-Factorized Weierstrass Form: charge (1, 1)
Hypermultiplets charged under both sections can and should exist, both from a geometric point of view and for anomaly cancellation, see section 5. Their location in the base can be found by determining singular points of the elliptic fiber E due to a factorization (4.3) of the Weierstrass form, where in addition both sectionsŝ Q ,ŝ R coincide. As we will demonstrate next, this indeed is a codimension two phenomenon.
From the perspective of the dP 2 fiber, the collision of the two sections happens when s 7 = 0. In the Weierstrass form there are even more loci, where the two sections collide and at which the Weierstrass model becomes automatically singular. We first summarize the structure that we find at these loci, before going into specifics. In the appropriate patch, z = s 9 , we first require the coincidence conditionŝ Q =ŝ R , that reads If this is true, then also s 3 9 g Q 9 = g R 9+3n 2 because the y-coordinates of the sectionsŝ Q ,ŝ R have been determined in section 3.1 by plugging in the x-and z-coordinates into the Weierstrass form and solving for y. But since s 2 9 x Q := s 2 9 g Q 6 = g R 6+2n 2 =: x R , the equality s 3 9 g Q 9 = g R 9+3n 2 is automatic. For the loci under consideration, it requires only a single further constraint to make the Weierstrass model singular, which reads These two conditions (4.14) and (4.15) bring the Weierstrass equation (4.3), using the notation g 6 ≡ s 2 9 g Q 6 = g R 6+2n 2 , into the form for f 6 = −2g 6 , where the double zero at x = g 6 , y = 0 is manifest. We note that the polynomialĝ Q 12+4n 2 = s 4 9ĝ R 12 = 0 in (4.6) respectively (4.10) and we see that fiber is automatically singular if the sections coincide and g 9 = 0. Here we denote the third root in x by f 6 , however the fiber is smooth x = f 6 .

(4.17)
We see that s 7 = 0 is a solution as expected, but also the vanishing of the polynomial in parenthesis makes the sections collide. In the first case, s 7 = 0, the fiber becomes singular iff in addition either s 3 = 0 or s 9 = 0. However, these cases have to be treated differently, as discussed in section 4.3, because the rational sections are ill-defined at these loci. Thus, we assume s 7 = 0 in the following and explore the second possibility, the vanishing of the expression in the parenthesis in (4.17). By solving δg 6 = 0 we see explicitly that g 9 ≡ s 3 9 g Q 9 = g R 9+3n 2 now take the same form. Solving δg 6 = 0 e.g. for s 6 we determine g 9 = 1 2 s 7 s 2 9 (s 3 s 2 8 − s 2 s 8 s 9 + s 1 s 2 9 ) . Noting that s 9 = 0 by assumption, we drop now the coefficients of s 7 and s 9 in front of δg 6 and g 9 . Then the codimension two loci for matter read To check the order of vanishing of the discriminant, cf. appendix A and section 3.1, along the codimension two locus we consider a small neighborhood around (4.14) and (4.15) defined as δg 6 = , g 9 = , s 9 = 0 , s 7 = 0 . There is another way of looking for singularities of the elliptic fibration, that contribute charged matter in F-theory, which is the third type mentioned at the beginning of this section. Qualitatively, the strategy is as follows. For non-trivial sections s 9 and s 7 over B, the sectionŝ R = [0 : −s 9 : s 7 ] can 'move' on the fiber E when varying the point on the base B. This implies that there are loci in B where two or more sections intersect in the fiber E. This is achieved when s 7 = 0 or s 9 = 0 as can be seen from (3.6) in the P 2 -model for E, see figure 4. Plugging this ansatz into the discriminant ∆ admits a factorization.  We emphasize that at each of these loci one sectionŝ P ,ŝ Q ,ŝ R is ill-defined. This can be seen most easily on the resolved model of the elliptic curve E in dP 2 introduced in section 3. To understand what is happening when one of the sectionsŝ P ,ŝ Q ,ŝ R , collectively denoted asŝ for the following paragraph, is singular, we first look at the sections at a location in the base where the fiber is not singular. In this case, after performing the blowups in the ambient space to dP 2 , the exceptional P 1 's intersect the Calabi-Yau polynomial at exactly one point in the fiber with coordinates shown in (4.24). At any of the loci of table 4.2, when two coefficients, s i and s j vanish simultaneously, the two coordinates of the exceptional P 1 vanish simultaneously and the coordinates stop making sense. This is another way of saying that the section passes through the Stanley-Reissner ideal (4.25). To resolve this problem we blow up the base at this locus by introducing coordinates [ 1 : 2 ] ∈ P 1 and writing (4.26) The coordinates (4.24) of the sectionŝ will be now given by the point [s j : s i ] = [ 1 : 2 ] in P 1 . When the two coordinates vanish simultaneously, s i = s j = 0, the values of 1 and 2 in (4.26) become unrestricted and we obtain the entire P 1 in the fiber. Another way of seeing this is to notice that the sectionŝ takes different values at the singular locus s i = s j = 0 depending on the direction (parametrized by [ 1 : 2 ]) at which we approach the locus in the base. Coming in all possible directions makes the section take all possible values at the same locus: the section becomes the entire P 1 of slopes. Now that we understand that the sections can become entire P 1 's, we demonstrate the factorization of the equation (3.27) into two rational curves along all codimension two loci in table 4.2. We discuss these loci successively and refer to figure 5 to keep track of which of the three sections wraps a P 1 in the resolved fiber. We notice using (4.24), (4.25) and the toric intersections (3.25) on dP 2 the intersections of the rational sections with the two curves as S P · c 1 = 0 , S P · c 2 = 1 , S Q · c 1 = −1 , S Q · c 2 = 0 , S R · c 1 = 1 , S R · c 2 = 0 , (4.29) where we denoted the homology classes of the sections by capital letters S • as before.
Here we note that the last two intersections follow, besides from the toric intersections (3.25), also from the defining property of a section, 1 = S R · E = S R · (c 1 + c 2 ), which with S R · c 2 = 0 immediately yields S R · c 1 = 1.
We note that the intersections (4.29) imply that c 2 is the original singular curve since it intersects the zero sectionŝ P and c 1 is the exceptional P 1 contributing a hypermultiplet. Thus it follows from (2.20) that the U(1) Q ×U(1) R -charges are (q 1 , q 2 ) = (−1, 1).
Singularity ofŝ R & blow-up at s 7 = s 9 = 0: charge (0, 2) Proceeding in a similar way as before we see that the sectionŝ R is blown up in a P 1 , which we identify with the toric divisor D u := {u = 0} in dP 2 . Furthermore, the polynomial (3.27) factorizes at s 7 = s 9 = 0 into two rational curves given by The intersections of the sections are evaluated as before employing (4.24), (4.25) and the toric intersections (3.25) on dP 2 as (4.32) Here we have determined the intersections of S R by noting that due to the blow-up in the base at s 7 = s 9 = 0, cf. (4.24), it wraps the entire P 1 corresponding to the ray subdividing the cone formed e 1 , e 2 . As we see by recalling figure 2 this is nothing else but the toric divisor with D u section u. This then implies that S P · c 2 = 2 because the equation for c 2 has two solutions with u = 0. From the defining property 1 = S P · E = S P · (c 1 + c 2 ) = S P · c 1 + 2 we infer S P · c 1 = −1 as claimed in (4.32).
From the location of the zero section we infer that c 1 is the original singular fiber and c 2 the isolated rational curve contributing a hypermultiplet. Its charges are calculated employing (2.20) as (q 1 , q 2 ) = (0, 2).
where the intersection S P · c 2 = 2 can be obtained by setting e 2 = 0 in (4.34). Alternatively, it can also be calculated using 1 = S P · E = S P · (c 1 + c 2 ) = −1 + S P · c 2 .
Here we have used that the blow-up in the base along s 8 = s 9 = 0 has blown-up the zero section with coordinates (4.24) into E 2 , which is the toric divisor of the ray subdividing the cone spanned by u and v. The curve c 1 is part of the zero section c 1 , thus it is the original singular fiber, whereas c 2 maps to the singular point of the original fiber. Thus c 2 is the isolated curve and it contributes according to (2.20) a hypermultiplet of total charge (q 1 , q 2 ) = (−1, −2). We note that this locus is special in the sense that it supports the only matter multiplet where the second term in (2.20) contributes to the charge due to the ill-defined zero sectionŝ P .

Calculating Matter Multiplicities
Now that we have determined the full matter representations and their codimension two loci in the base B, we can count their multiplicities. The main complication that arises in these calculations is due to the intersection of different codimension two loci. In order to avoid double-counting we thus have to appropriately subtract the number of intersection points of a codimension two loci under consideration with other codimension two loci.
For this analysis, we first have to recall the counting of the multiplicity of a root in a system of two polynomials with two independent variables. Given two polynomials as the resultant of f, g with respect to x. The degree of a root (x 0 , y 0 ) of (4.36) is given by the multiplicity of y 0 as a zero of h(y) 17 .
Now that we know how to count roots of a polynomial system, the strategy is simple: take the degrees of the polynomials for the codimension two loci we are interested, multiply them and then subtract all intersecting loci we are not interested in with the appropriate multiplicity. As a warm up and a demonstration of this technique we rederive the multiplicity of the charge one loci of the elliptic fibration over P 2 with elliptic fiber Bl (0,1,0) P 1,1,2 considered in [43].

) model
The loci of the charge one hypermultiplets were given in equations (5.64) of [43] with b n = 0 and f 3+n = 0 because those were identified as the charge two loci. We see that f 3+n = b n = 0 obviously solves the system.
Viewing now the equations (4.38) as polynomials in the variables f 3+n and b n , the resultant with respect to the variable f 3+n is given by from where it can be seen that f 3+n = b n = 0 is a root of degree 16 of the system (4.38). Now we can calculate the multiplicity of the charge one hypermultiplets. The degrees of (4.38) are (9 + 3n) and (12 + 4n) respectively and there are n(3 + n) roots we are not interested in and that have to be subtracted, each of which of degree 16. Thus we obtain (9 + 3n)(12 + 4n) − 16n(3 + n) = 4(9 − n)(3 + n) (4.40) charge one hypermultiplets.

Matter multiplicities of the dP 2 model
In this subsection we will finally apply the resultant technique to determine the multiplicity of all six codimension two matter loci found in sections 4.1, 4.2 and 4.3. We denote the multiplicities by the same variables x i , i = 1, . . . , 6, as in (5.5) to facilitate a straightforward comparison with results from anomaly cancellation.
The basis of our analysis are the degrees of the coefficients s i calculated in (3.32) for the base B = P 2 . Let us recall that the degree of s 9 is equal to the intersection number n 2 of the zero sectionŝ P and the sectionŝ R and the degree n 12 of s 7 equals the intersection number ofŝ Q andŝ R , cf. (3.33). With these degrees, the multiplicities of the hypermultiplets of section 4.3 are straightforward: they are simply the multiplication of the degrees of the two coefficient that vanish, see table 4.3. For the other hypermultiplets we have to work a little more.

Anomaly Cancellation: a Consistency Check
In this section we check the consistency of the matter spectrum determined in the previous section via constraints arising from anomaly cancellation. We demonstrate that anomaly cancellation allows us to classify all F-theory compactifications with base B = P 2 and the two section elliptic fiber determined in section 3.1. This confirms the geometric analysis of the previous section that has led to determination of the charges of matter fields. We also see that the computed multiplicities agree with the ones predicted by anomaly cancellation. This analysis ensure a consistent low-energy theory of F-theory in six dimensions with two Abelian gauge fields.
Anomaly cancellation in six-dimensional F-theory vacua, based on general results on 6d anomaly cancellation [59,60], has been discussed in [55,43], whose notations and conventions we follow here. See also the analysis in [44] of anomaly cancellation in four dimensional F-theory compactifications. We note that the following discussion holds more generally for any Abelian gauge theory.
We are interested in anomaly cancellation in an Abelian gauge theory with matter arising from F-theory compactifications. In this case, there are only the purely gravitational anomaly, the mixed Abelian-gravitational and the pure Abelian gauge anomalies. We assume that there are H hypermultiplets, T tensor multiplets and V vector multiplets. The gauge and mixed anomalies induced by the six-dimensional charged matter in hypermultiplets have to be canceled by a classical effect, the generalized Green-Schwarz mechanism [61]. These cancellation conditions are the anomaly equations. The mixed Abelian-gravitational, purely Abelian and purely gravitational anomaly equations read x qm,qn,q k ,q l q m q n q k q l , Here we have labeled the r U(1)-fields in our theory by A m , where r denotes the rank of the Mordell-Weil group of the Calabi-Yau threefold X. On the right hand side of (5.1), the sum runs over all hypermultiplets with charge vector q = (q 1 , . . . , q r ). The integers x qm,qn respectively x qm,qn,q k ,q l denote the number of hypermultiplets with charges (q m , q n ) under U(1) m ×U(1) n respectively charges (q m , q n , q k , q l ) under U(1) m ×U(1) n ×U(1) k ×U(1) l . The left hand side of (5.1) is to be evaluated in the cohomology of the base B of the fibration. K B denotes the canonical class of B, · is the intersection pairing on B and the expressions b mn are defined as the curves b mn = −π(σ(ŝ m )· σ(ŝ n )) = −π(S m · S n ) − [K B ] + π(S m · S P ) + π(S n · S P ) .

(5.2)
We recall that S m denotes the divisors class of a sectionŝ m and the homology class S P of the zero sectionŝ P . Here the first line has been evaluated employing (2.16) in the presence of only Abelian gauge fields. We note that the pure gravitational anomaly even without further specification of the spectrum puts an upper bound on a theory with T = 0, which is the case for B = P 2 , as We now analyze the anomaly constraints (5.1) for the F-theory compactification on the elliptic fibration over P 2 with two U(1) gauge fields and the matter content determined in section 4. We see that anomaly cancellation fixes the possible matter multiplicities in terms of two integers. We first evaluate the anomaly coefficients b mn in (5.1). In the case of P 2 as the base, there is only one element generating H 2 (B), the hyperplane class H B . The anti-canonical line bundle is K P 2 = −3H B . Then the coefficients are just numbers, namely the coefficients of H B , and evaluated to b kl = 2(n k + 3), k = l 3 + n k + n l − n kl , k = l, where we defined n kl = π(S k · S l ) and n k = π(S k · B). In addition, we have employed the second relation in (2.11) to replace π(S k · S k ) = (S k · S k · H B )H B = K = 3H B . Then, we evaluate the right hand side of (5.1) using the spectrum found in section 4 with arbitrary multiplicities x i assigned as Upon inserting this into the anomaly constraints (5.1) for two sections S 1 ≡ S Q , S 2 ≡ S R and the zero section S P we obtain a set of linear equations for the x i reading 36(3 + n 1 ) = 12(3 + n 1 ) 2 = x 1 + x 3 + x 4 + x 6 , 18(3 + n 1 − n 12 + n 2 ) = 6(3 + n 1 )(3 + n 1 − n 12 + n 2 ) = x 3 − x 4 + 2x 6 , 12(3 + n 2 ) 2 = x 2 + x 3 + x 4 + 16x 5 + 16x 6 , 6(3 + n 2 )(3 + n 1 − n 12 + n 2 ) = x 3 − x 4 + 8x 6 , 36(3 + n 2 ) = x 2 + x 3 + x 4 + 4x 5 + 4x 6 , 4(3 + n 1 )(3 + n 2 ) + 2(3 + n 1 − n 12 + n 2 ) 2 = x 3 + x 4 + 4x 6 .
We conclude by evaluating two special cases. First we consider n 2 = n 12 = 0 where we obtain In this case, charge two loci are completely absent. A less trivial example can be constructed by setting n 2 = n 12 = 1 in which case we obtain The global geometries of both examples are constructed in section 6. As we see there, we will find perfect agreement with results from anomaly cancellation by using the general geometric techniques of section 4.

Toric Elliptic Calabi-Yau Manifolds with Two Rational Sections
In this section we construct explicitly toric elliptically fibered Calabi-Yau threefolds realizing concrete elliptic fibrations with two rational sections. In all examples the base of the fibration is B = P 2 and the generic fiber is the elliptic curve in dP 2 . The first two examples both have U(1)×U(1)-gauge symmetry, but with rational sections of different complexity and consequently different matter sectors. In a third example we add an SU(5)-GUT sector and explicitly present the corresponding four-dimensional toric polytope.
In all examples considered below, we construct the toric ambient variety V as a toric fibration of dP 2 over a toric base B, The presence of a toric fibration can be detected by analyzing the toric polytope ∆ V defining V . The polytope ∆ dP 2 of the fiber has to be cut out by a two-dimensional plane E through the origin 0 as Then a toric morphism ϕ to the base B is defined by first finding the two normal vectors n i , i = 1, 2, to E and then by defining the 2 × 4-matrix This matrix acts as ϕ T · p on the integral points p of the polytope ∆ V yielding all the integral points of the polytope ∆ B of the base B. We denote this in the suggestive form

Example 1
The first and simplest example we consider has three non-intersecting sections. As we have seen in section 4 even in this simple case, matter with charges (1, 0), (0, 1) and (1, 1) can exist and we check explicitly that this is the case. The vertices of the polytope along with their divisor classes and charge vectors (i) read: variable vertices divisor class We readily check by the condition (6.2) that the polytope (6.5) describes indeed a toric fibration with the hyperplane E = {(0, 0, x, y)|x, y ∈ R}. In other words, all points of the form (0, 0, x, y) describe the polytope of dP 2 , which are the fourth to eigth points in (6.5). The toric morphism ϕ in (6.3) is then constructed as the projection onto the first two coordinates of the vertices and reveals the polytope of the base P 2 as the first three points in (6.5). The toric Calabi-Yau hypersurfaceX in the toric variety V specified by (6.5) has Euler number and Hodge numbers It is satisfying to note that there are neither non-toric divisors nor non-toric complex structure deformations inX as expected by construction.
This model is characterized by n 2 = n 12 = 0 as can be seen for example by calculating the toric intersections (3.33). Using the general formula (3.32) or by constructing explicitly the sections of the anti-canonical bundle of (6.5) we can readily determine the coefficients s i . They are the most generic polynomials in the projective coordinates [z 0 : z 1 : z 2 ] on P 2 of the following degrees or line bundles (LB): The three sections are given by the intersection of the Calabi-Yau hypersurface with the three toric divisors D u := {u = 0}, E 1 and E 2 . We summarize this schematically aŝ s P : {e 2 = 0} ∩ pŝ Q : {e 1 = 0} ∩ p ,ŝ R : {u = 0} ∩ p , (6.8) where p is the polynomial of the Calabi-Yau hypersurface. All three sections intersect the fiber once and are holomorphic everywhere on the base B = P 2 . To be consistent with the notation in the rest of this work, we chooseŝ P as the zero section. The other two sectionsŝ Q andŝ R generate the rank two Mordell-Weill group, responsible for the Abelian U (1) × U (1)-sector of the theory.
The matter content of this example can be determined explicitly in this toric example or by using the results of sections 4. We summarized our findings in equation (5.9) and emphasize that anomaly cancellation is automatic.

Example 2
In order to create a concrete geometry with more complicated rational sections that in particular realize all the matter content found in section 4 it is necessary to fiber the dP 2 -polytope slightly differently over the base P 2 . This requires a change of the line bundle on the base B under which the coordinates [u : v : w; e 1 : e 2 ] transform. A simple way to realize this torically is by changing the location of the base polytope ∆ B in the polytope of the toric fibration as: The remaining points take the same form as in (6.5) of the previous example. Thus, the only change of the charge vectors occurs for (0) , whose fourth entry −3 is to be replaced by−2. This means that the coordinate u now transforms in the bundle O P 2 (−2) over P 2 .
As before we check the existence of an elliptic fibration by the condition (6.2) with the hyperplane E = {(0, 0, x, y)|x, y ∈ R}. The toric morphism ϕ in (6.3) is constructed as before as the projection onto the first two coordinates of the vertices. The toric Calabi-Yau hypersurfaceX in the toric variety V specified by (6.9) has Euler number and Hodge numbers and again there are neither non-toric divisors nor non-toric complex structure deformations inX as expected.
This model is characterized by n 2 = n 12 = 1 as can be seen for example by calculating the toric intersections (3.33). By means of the general formula (3.32) or by constructing explicitly the sections of its anti-canonical bundle we determine the coefficients s i that are the most generic polynomials in the projective coordinates [z 0 : z 1 : z 2 ] on P 2 of the following degrees: coefficients section of LB The three rational sections are given by the intersection of the toric divisors with the Calabi-Yau hypersurface p = 0 aŝ Since n 2 = n 12 = 1 there are loci where each of the sectionsŝ P ,ŝ Q andŝ R are illdefined, cf. table 4.2. Thus, the full spectrum analyzed in section 4 is realized. The matter multiplicities follow readily from the classification in section 4.4 and were calculated explicitly in (5.10). Again, it is seen that anomalies are canceled.

Example with SU(5)
In this concluding example we construct a concrete resolved elliptically fibered Calabi-Yau threefold over P 2 with an SU (5)-GUT sector. The geometry constructed below is an extension of the geometry constructed in section 6.1 by a resolved SU(5)-singularity on the divisor z 2 = 0. The full gauge symmetry of the F-theory compactification is thus SU(5)×U(1)×U(1).
The variables, vertices of the polytope describing the toric variety V and the independent toric divisor classes read: We note that the Abelian sector has not changed compared to section 6.1. The sections are still located at the following intersections of the Calabi-Yau hypersurface p = 0, A constructive way to understand the form of the polytope in (6.13) is to start with the singular geometry and perform the four blow-ups necessary to resolved codimension one and two singularities of the fibration following [19,20,21]. Then all one has to do is to translate the divisor classes back into toric charge vectors and find the vertices realizing these charge vectors as linear relations. This analysis is performed in the following and leads to the polytope in (6.13).
As usual an SU(5)-singularity at codimension one in the base B is engineered by specializing the coefficients in the Calabi-Yau hypersurface equation p = 0, that is specializing the form presented in (3.27). In the fibration with elliptic fiber dP 2 this means that we have to consider non-generic sections s i in (3.27). Concretely, we engineer an SU(5)-singularity on the divisor z 2 = 0 in P 2 . One way to realize an SU(5)-singularity is to consider non-generic coefficients s i of the form With these definitions we can check explicitly, employing the formulae of section 2.1 for f , g and ∆, that the choice (6.15) realizes an SU(5)-singularity at z 2 = 0, i.e. f , g do not vanish whereas ∆ ∼ z 5 2 as required by the Kodaira classification for an SU(5)singularity [16]. In fact, by inserting (6.15) into the discriminant we observe the structure ∆ = −z 5 2 β 4 5 P + z 2 β 2 5 (8β 4 P + β 5 R) + z 2 2 (16β 2 3 β 3 4 + β 5 Q) + z 3 2 S + z 4 2 T + z 5 2 U + O(z 6 2 ) (6.16) with the identification β 5 = s 6 , P := P 1 P 2 P 3 P 4 P 5 = (s 2 s 5 − s 1 s 6 )s 7 (−s 3 s 6 + s 2 s 7 )s 8 (−s 7 s 8 + s 6 s 9 ) . (6.17) All the other coefficients can be determined similarly but are not shown because we are only interested in the structure at codimension two. We readily see that the form of the discriminant agrees with the expected one from the local model analysis in the literature, compare e.g. to [19] whose notation we follow. However, in a local model the discriminant (6.16) contains only terms up to order five in z 2 . As expected for global models there are higher order terms in z 2 in (6.16), in this case terms up to z 10 2 . Next we turn to the resolution of the singular elliptic fibration over z 2 = 0. A completely smooth elliptic fibration is obtained by four consecutive blow-ups in the ambient space, two of which being ordinary blow-ups and the other two being small resolutions on the Calabi-Yau hypersurface. The first blow-up has to be performed at the codimension three locus w = v = z 2 = 0 in the ambient space. Upon introducing a new section d 1 corresponding to one exceptional divisor, we formulate the blow-up relation as The Calabi-Yau manifold is still singular atw = u = d 1 = 0. We have to perform another blow-up and upon introducing a new section d 2 we obtaiñ These blow-ups resolve all codimension one singularities in the elliptically fibered Calabi-Yau manifold. However, there are still codimension two singularities of the elliptic fibration that have to be resolved. This has to be done via a small resolution in order to maintain a flat fibration. In the following we will consider one of the six possible small resolutions in [19,20]. We perform the small resolutions atw =d 1 = 0 and atw = d 2 = 0 by introducing two new divisor sections d 3 , d 4 . Then, the resolution map reads It can be checked explicitly that these final resolutions render the geometry smooth. Combining all the four resolution maps, we obtain the total variable transformations w →d 1d In order to shorten our notation we will in the following drop all the tildes of the variables on the resolution by abuse of notation. A consistent assignment of divisor classes on the resolved geometry is presented in (6.13), where we note that the coordinates displayed there are the coordinates on the resolution after dropping tildes.
Finally, we use the resolution map (6.21) to obtain the proper transform of the Calabi-Yau polynomial p in (3.27). The fully resolved geometryX reads, after dropping superscripts , as p =s 6 (e 1 e 2 )uvw + We conclude with some remarks on codimension two singularities and the expected matter multiplets in the resolved geometry It is well-known, see e.g. [19] for a review, that the SU(5)-singularity in the fiber enhances further at the codimension two loci z 2 = 0 with either β 5 = 0 or P = 0. In our case, these loci were identified in (6.17). At the loci β 5 := s 6 ! = 0 the fiber enhances to a D 5 -type singularity which are the loci of 10-matter representations in F-theory. As we see from (6.17) the β 5 is generic. Thus there is only one 10-matter representation in an model with an additional U(1) 2 -sector at β 5 = 0. In contrast, the loci P = 0, where the fiber enhances to an I 6 -singularity and where 5matter is located, split into five different curves described by P i = 0, i = 1, . . . , 5. Thus in an F-theory compactification with SU(5)×U(1) 2 gauge symmetry there will be five different 5-matter representations with differing U(1)-charges. It would be interesting to investigate these geometries further and check anomaly cancellation in six dimensions by working out matter multiplicities along the lines of section 4.4.
7 Equivalence of Bl (1,0,0) P 2 (1, 2, 3) and Bl (0,1,0) P 2 (1, 1, 2) Elliptic Fibrations In this section we illustrate the power of birational transformations to show the equivalence of two different representations of an elliptic curve with one rational point. One representation is the U(1)-restricted Tate model [39] in the blow-up Bl (1,0,0) P 2 (1, 2, 3) and the other one is the hypersurface in Bl (0,1,0) P 2 (1, 1, 2) studied in [43]. We show that elliptic fibrations constructed from these models agree via a birational map and are related to models with no rational section by extremal transitions. We strongly suspect that the logic presented in the following holds more generally and can also be applied to prove the birational equivalence with dP 1 -model studied in [46] and equivalences of models with multiple sections. Our following discussion is based on the extremal transitions studied in [51] 18 .
The toric variety Bl (1,0,0) P 2 (1, 2, 3) is represented by a polytope with vertices with only interior point the origin. We note that there are three points interior to the one-dimensional faces of the polytope. As before we presented the vertices as row-vectors in the first two columns, introduced the projective coordinates in the third column and displayed the toric divisor classes in the fourth column. H denotes the proper transform of the hyperplane class on P 2 (1, 2, 3) and E 1 is the exceptional divisor. The charge vectors (i) in the fifth and sixth columns are the generators of the Mori cone, that describe the scaling relations among the coordinates. We note that the charge vector of P 2 (1, 2, 3) is obtained as ( We also obtain the charge vector˜ (1) = (1) + (2) = (1, 0, 1, 2) T , as denoted in the last column of (7.1), with respect to which [z : 1 : y 1 : e 1 ] scale as the homogeneous coordinates [u : v : w] in P (2) (1, 1, 2). In addition, we can make this manifest by forming the following C * -covariant combination of the coordinates [z : x 1 : y 1 : e 1 ] as We readily check using (7.1) that these coordinates enjoy the C * -actions  (1, 1, 2). Thus, we see the birational equivalence of the two models. In the following we work out the birational maps in detail and recover the birational map, that was found in [43] algebraically, by a straightforward toric analysis and one toric blow-up.
Before proceeding we present the toric data of Bl (0,1,0) P (2) (1, 1, 2). Its polytope reads vertices divisor class where again the only interior point is the origin and there is one point interior to an one-dimensional face. The homogeneous coordinates in the third column scale under the relations displayed in the last two columns, that form a basis of the two-dimensional Moricone. We recover the scaling relation of Bl (0,1,0) P (2) (1, 1, 2) in (7.4) as (1) =˜ (1) + 2˜ (2) .
We perform our derivation of the birational equivalence of these two presentation of elliptic curves by first matching their respective singular Tate models and then by performing the toric resolution. We begin with recalling the U(1)-restricted Tate model. The Tate form (2.1) of the elliptic curve takes the form y 2 + a 1 xyz + a 3 yz 3 = x 3 + a 2 x 2 z 2 + a 4 xz 4 (7.7) where we work in the affine patch with the coordinate of the exceptional divisor e 1 = 1. The coefficient a 6 has been removed torically, a 6 ≡ 0, by adding the last point in (7.1) of the exceptional divisor to the polytope of P 2 (1, 2, 3), which removes the point in the dual polytope to (7.1) corresponding to the monomial a 6 z 6 . Thus, an elliptically fibered Calabi-Yau manifold over a base B with the global Tate model of the form (7.7) with generic coefficients a i ∈ O(−iK B ) is readily constructed as the Calabi-Yau hypersurface or complete intersection in the ambient space obtained by fibering Bl (1,0,0) P 2 (1, 2, 3) over the base B. In particular if B admits a toric description, i.e. is either a toric variety or a toric complete intersection, we can construct this ambient space torically by fibering the polytope (7.1) over the polytope describing B. We refer to section 6 for the required structure of the polytope to construct such a toric fibration Now we turn to the hypersurface in Bl (0,1,0) P (2) (1, 1, 2). It has been shown in [43] that its Tate form reads y 2 +ã 1 x yz +ã 3 yz 3 = x 3 +ã 2 x 2 z 2 +ã 4 x z 4 +ã 6 z 6 (7.8) with the Tate coefficientsã i given bỹ We note immediately that these coefficients are not generic, namely we have five different coefficients parametrized by only four independent variablesã 1 ,ã 2 ,ã 3 andã 4 . Indeed we can eliminate this redundancy by a simple variable transformation, When considering an elliptic fibration, these coefficients are identified with generic sections of appropriate line bundles on the base B of the fibration. We then identify a i ≡ a i in the U(1)-restricted Tate model (7.7). Thus we have shown by matching the Tate forms (7.7) and (7.8) that the elliptic curves in Bl (1,0,0) P 2 (1, 2, 3), i.e. the U(1)-restricted Tate model, and in Bl (0,1,0) P 2 (1, 1, 2) are birationally equivalent.
We emphasize that this equivalence does not hold for general elliptically fibered Calabi-Yau manifolds. We claim that the equivalence only holds if the Calabi-Yau constraint with general fiber in Bl (0,1,0) P 2 (1, 1, 2) takes globally the form where it is crucial that the coefficient of w 2 is one, which puts certain restrictions on the construction of the elliptic fibration. Fibrations of these type have been studied in [43] for the example of B = P 2 . In this case the coefficients were identified as b ≡ b n , c 3 = f 3+n , c 2 = 3f 6 , c 1 = 2f 9−n , c 0 = f 12−2n . (7.13) where the integers in the subscript denote the degree of the polynomial on P 2 . We see with these identifications that the Tate coefficients (7.11) are indeed general polynomials and the fibration becomes equivalent to a generic U(1)-restricted Tate model (7.7) as claimed.
First we note that the Tate model (7.7) is singular at codimension two in the base when a 3 = a 4 = 0, or equivalently in the parametrization (7.11) when All these singular loci in the base are contained in the intersection of the surface x = y = 0 with the elliptic curve (7.8). Thus all these singularities are resolved by a single toric blow-up along at x = y = 0. We perform the toric blow-up in (7.7) by setting x = x 1 e 1 = W T , y = y 1 e 1 = V W T , z = U T (7.15) where we have also used the coordinates (7.3) and have at the same time performed the coordinate transformation (7.10) to eliminateã 6 in (7.8). Inserting this into (7.7) we obtain where we have canceled a factor T 2 W . We readily observe that this constraint is mapped under the blow-down map (7.5) to the quartic in P 2 (1, 1, 2) as claimed. Then we perform the simple variable transformation in order to bring (7.16) into the form (7.12), where we multiplied by b and, by abuse of notation, denoted the new coordinates again by W and T . In these coordinates (7.15) reads where we again denote the shifted coordinates (7.17) by W and T .
Finally, we relate the coordinatesx,ỹ defined in (2.3) that bring the U(1)-restricted Tate model (7.7) into Weierstrass form to the coordinates of [U : V : W : T ] of Bl (0,1,0) P 2 (1, 1, 2). This amounts to simply writing out the transformation (2.3) in terms of x and y given by (7.19). First, we evaluate using the Tate form (7.7) the parameters Then we readily plug this into (2.3) with (7.19) to obtaiñ where, as before, W and T are to be understood as the shifted coordinates in (7.17). We note that we can use the C * -action on [z :x :ỹ] to cancel the prefactors 1 b k , k = 1, 2, 3. We readily recognize the birational transformation 19 (7.21) as the birational map worked out in [43] to map the Weierstrass model of the elliptic fibration under consideration to the quartic (7.12). In fact, by using the C * -action we can eliminate the prefactor of T in (7.21) in front of z. Then we identify the coordinates of P 2 (1, 1, 2) in the conventions of [43] as Thus, we have found the same birational map from a simple toric resolution (7.15) and identification of charge vectors. Although both maps (7.15) and (7.21) are blow-ups at x = y = 0 respectivelyx =ỹ = 0, the former map is considerably more natural.

Outlook
In this work we have analyzed F-theory vacua with U(1)×U(1) gauge group from compactifications on elliptically fibered Calabi-Yau manifolds with rank two Mordell-Weil group. We have derived from first principles that one has to depart from the Tate and Weierstrass form to represent an elliptic curve E with two rational points and its resolved elliptic fibrations: the appropriate presentation is the toric Calabi-Yau one-fold in dP 2 .
We have found the birational maps to its Tate and Weierstrass model and the coordinates of its two rational points in Weierstrass form. In addition we have explicitly studied its resolved elliptic fibrationsX over a base B, that we have classified for B = P 2 . We have seen explicitly that one has to give up the paradigm of a holomorphic zero section in F-theory. This has also been confirmed by analyzing the codimension two singularities of the fibrationX, that determine the F-theory matter spectrum. The spectrum that we work out in detail and in generality is only consistent with six-dimensional anomaly cancellation if the zero section wraps fiber components over codimension two loci.
fiber types in F-theory. A possible starting point to study such different fibrations is to construct and analyze elliptic fibrations with general elliptic fiber realized as a Calabi-Yau onefold in all the different toric varieties corresponding to the 16 two-dimensional reflexive polytopes. This might also be relevant for a partial classification of Abelian sectors in F-theory. Furthermore, the systematic inclusion of non-Abelian sectors as well as the study of their matter spectrum and anomaly cancellation along the lines of sections 4 and 5 would be desirable. In particular, the general check of the gravitational anomaly, done here only for examples, can be used to prove the completeness of the study of codimension two singularities. Mathematically this translates into the Euler number of the resolutionX [62]. We also refer to the early study of [63] on the connection between the rank of the Mordell-Weil group and the Euler number for elliptic fibrations of different fiber types. Finally and of immediate importance for the full understanding of four-dimensional F-theory vacua is the systematic extension of our techniques to Calabi-Yau fourfolds, codimension three phenomena and the structure of G 4 -fluxes. We will return to the above issues in a future work [64].
in all of the above cases reduces to finding an appropriate map to a P 1 such that the Calabi-Yau equation takes the form of a double cover over this P 1 . Upon mapping one branch point in t to infinity one immediately recovers the Weierstrass model.
The application of this idea to the cubic curve is the foundation of Nagell's algorithm that we present in the following. Consider a cubic curve in P 2 with projective coordinates [u : v : w] of the form p = s 1 u 3 + s 2 u 2 v + s 3 uv 2 + s 4 v 3 + s 5 u 2 w + s 6 uvw + s 7 v 2 w + s 8 uw 2 + s 9 vw 2 + s 10 w 3 . (B.2) For the moment we assume that the coefficients s i are in a field K, e.g. K = C. Then, for generic s i the cubic (B.2) defines a smooth elliptic curve E. In applications to elliptic fibrations π : X → B of E over a base B, the s i are sections of a line bundle L i and are locally represented by polynomials in local coordinates on the base B.
The cubic (B.2) can be brought into Weierstrass form by application of Nagell's algorithm, where we follow the exposition of [65]. In the case at hand, the above mentioned map to an appropriate P 1 , which turns the cubic (B.2) into a double cover of the form (B.1), is given by mapping to the slopes t of lines through a given point Q on E. Concretely, we first assume that s 10 = 0 which can be achieved by a coordinate transformation 21 of u and v. Then the point P = [0, 0, 1] is on the curve (B.2). Next we assume that s 9 = 0 without loss of generality since we can always interchange u and v and both s 8 = s 9 = 0 implies that the corresponding elliptic curve E is singular at P .
Then we define p = F 3 (u, v) + F 2 (u, v)w + F 1 (u, v)w 2 , (B.3) i.e. F i (u, v) is the coefficient polynomial of the term w 3−i in (B.2). Every line in P 2 meets the curve (B.2) in three points. The tangent at P is the line meeting E twice at P . The point P becomes a double point of (B.2) precisely when F 1 = 0 along the line v = t 0 u, thus, the tangent at P is described by the equation In other words, the slope t 0 of the tangent at P is t 0 = − s 8 s 9 . The tangent meets p in another point Q, see figure 6 for an illustration of this situation. Its coordinates are Q = [−e 2 s 9 : e 2 s 8 : e 3 ] where we defined e i = F i (s 9 , −s 8 ) 22 . If e 2 = 0, then P = Q Figure 6: Cubic curve E (in blue) with the two points P and Q. The tangent at P intersects E at the point Q. This implies that the slope t 0 of the tangent is a root of the discriminant δ(t) in (B.8), since δ(t) = 0 for double points of E along lines through Q. is a triple point or flex of E and if e 3 = 0, then Q = [−s 9 , s 8 , 0] is at infinity. Both e 2 = e 3 = 0 implies that p = 0 along the line F 1 = 0 or p = F 1 (u, v)q 2 (u, v, w) for a quadratic polynomial in [u : v : w]. This means that the curve E is reducible with components P 1 and the quadric in P 2 , thus, not elliptic.
Next we perform a variable transformation that centers Q at u = v = 0. The variable change reads e 3 = 0 : u = u − s 9 e 2 e 3 w , v = v + s 8 e 2 e 3 w , (B.5) where we have to distinguish again the cases e 3 = 0 and e 3 = 0. We denote the constraint p in the new coordinates as where f i denote homogeneous polynomials of degree i in (u , v ). Now we consider the lines v = tu through Q. Their intersections with E in generically two other points are determined by the intersection with the constraint p reading u (φ 3 (t)u 2 + φ 2 (t)u w + φ 1 (t)w 2 ) = 0 . (B.7) Here we readily cancel the factor u = 0 which is the point Q itself. Furthermore, we have defined φ i (t) = f i (1, t). We determine the two roots of this quadratic equation as a function of the slope t by multiplying (B.7) with 4φ 3 (t). Then we complete the square yielding (2φ 3 (t)u + φ 2 (t)w) 2 = δ(t)w 2 , where in the case at hand the φ i are lengthy polynomials in t that we omit. The double roots are the zeros t i of the discriminant δ(t) of the quadratic equation in u , of which there are generically four as δ(t) is a fourth order polynomial in t. Geometrically, the lines with the slopes t i are those lines through Q, that are a tangent or in other words are a double point of E at a different point. By construction, we already know one such line that is the tangent at P given by (B.4) with slope t 0 = − s 8 s 9 , cf. figure 6. In other words we know the linear factor (t − t 0 ) of δ(t). Thus we can perform the variable transformation t = t 0 + 1 τ to move this root to infinity τ = ∞. Consequently we obtain a third order polynomial ρ(τ ) = τ 4 δ(t 0 + 1 τ ) = cτ 3 + dτ 2 + eτ + k , (B.9) for coefficients c, d, e and k, that in the case at hand read