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In this section we briefly discuss the geometry of an elliptically fibered Calabi-Yau three- foldX, although we note that the following holds for general complex dimension ofX.

By definition an elliptic fibration over a baseBis defined by a holomorphic projection

over a generic point pt in B given by elliptic curve with a zero point and a number of rational points. As mentioned before we can always describe an elliptic fibration over B by its Weierstrass model (3.2), 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 bundlesK_B−4respec- tively K_B−6, where K_B denotes the canonical bundle of the base. If it exists globally we can also construct the Tate form1 (3.1) where the coefficients a_i take values in K_B−i. The holomorphic zero section in the Tate and Weierstrass form is given byz=0.

However, when having global questions in mind, such as the global resolution of singu- larities of the elliptic fibrationX 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 dis- cussed at the end of section 3.1.1, 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 indP₂, cf. section 3.2.1.

When we fiber an elliptic curve E over a base B its zero point P becomes the zero section ˆs_Pand its rational pointsQ_mlift to rational sections ˆs_m≡sˆ_Qmof the elliptic fibration

π : X → B. All of these sections define injective maps ˆsP,sˆm : B,→X and the group

generated by the ˆsm is the Mordell-Weil group of the elliptic fibrationX. A holomorphic

section, that is in the literature typically denoted by σ, defines a holomorphic injection σ:B,→Xon all ofB. 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 [97, 20]. Thus, rational sections ˆs_mcan only be defined on the blow-upπB: ˆB→BofBalong the singular

loci of ˆsm withπB denoting the blow-down map. Consequently, a rational section defines

only a birational map ˆsm: ˆB→B,→X of the baseBintoX.

In sections 3.2.1 and 3.3, we study elliptic fibrations with general fiber given by an elliptic curveE with two rational pointsQ, andR and a zero sectionP. As we see there, in these cases even the zero section ˆsPis 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 3.3, the behavior of the three rational sections ˆs_Pand ˆs_Q, ˆs_R leads to a rich structure of charged matter.

The group of divisors, or its dual group H(1,1)(Xˆ), on the smooth elliptic Calabi-Yau manifold ˆX→X arising fromX by resolving all singularities, is generated by divisorsD_A that fall into four different classes of divisors:

• the zero section ˆsP of the fibration with homology class SP, which in the case of a

holomorphic sectionσ agrees with the class of the baseB,

• the rational sections ˆsm,m=1, . . . ,r, with divisor classesSmgenerating the Mordell-

Weil group of rational sections of the elliptic fibration ofX,

• the vertical divisors D_α =π∗(Db_α), α =1, . . . ,h(1,1)(B), of the fibration that are in-

herited from divisorsDb_α in the baseB,

• exceptional divisorsD_iI resolving singularities ofX from singularities in its elliptic fibration at irreducible components∆I =0 of the discriminant locus∆=0 inB.

We summarize this basis of divisors on ˆX as

D_A= (B,S_m,D_α,D_i), A=0,1, . . . ,h(1,1)(Xˆ). (3.7) We conclude with some key intersection properties of the divisorsDA. The exceptional

divisorsD_iI over one common discriminant locus intersect as

D_iI·D_iJ·D_α =−C_i(I)

IjIS(I)·B·Dα (3.8)

whereC_i(I)

IjI denotes the Cartan matrix of an ADE-groupG(I) in case of an ADE-singularity in the fibration. In this case we denote theD_iI as the Cartan divisors of the corresponding ADE group. Here the divisors S_(I) =π∗(Sb_(I)) are related to the components of the dis-

criminant and theSb_(I) are the loci in the baseB wrapped by 7-branes supporting a gauge groupG_(I)in F-theory. Upon intersecting the Cartan divisorsD_iI with a divisor ˜DinBthat intersectsS_(I) in a point, we obtain a rational curve that is localized overS_(I). The relation (3.8) teaches us that this curve is to be identified with minus the rootαiI of the Lie-algebra of the ADE-group under consideration, i.e.

C−α_iI :=DiI·D˜ for D˜·S(I)·B=1. (3.9) Next we turn to the divisorsS_P andS_mof the sections. By construction the divisors corre- sponding to a section will intersect the general fiberF∼=E as

SP·F=Sm·F=1. (3.10)

In addition we note that any section ofX also has to obey [97]

B2=−[c1(B)]·B, S2m·Dα =−[c1(B)]·Sm·Dα, (3.11) where the first relation holds in homology andc₁(B)is the first Chern class of the base.