axio-dilaton in F-theory compactification. This formulation allows the axio-dilaton to have $SL(2,\Z)$ monodromy. Genus-one fibrations do not necessarily admit a global section; there are situations in which they have a global section, and those in which they do not. F-theory compactifications on elliptic fibrations with a global section have been investigated in recent studies, for example, in [@MorrisonPark; @MPW; @BGK; @BMPWsection; @CKP; @BGK1306; @CGKP; @CKP1307; @CKPS; @AL; @EKY1410; @LSW; @CKPT; @CGKPS; @MP2; @BMW2017; @CL2017; @BMW1706; @EKY1712; @KimuraMizoguchi; @EK1802; @Kimura1802; @LRW2018; @EK1808; @MizTani2018; @CMPV1811; @TT2019]. Although Calabi–Yau genus-one fibration lacking a global section cannot be expressed in the Weierstrass form, when the Jacobian fibration of it exists, the Jacobian fibration yields an elliptic fibration with a global section. Calabi–Yau genus-one fibration $Y$ and the Jacobian fibration $J(Y)$ have the identical types of the singular fibers, and they have the same discriminant loci.

Genus-one fibers degenerate over the codimension 1 locus in the base space, and this locus is referred to as the discriminant locus. Such degenerate fibers are called the singular fibers. When genus-one fiber degenerates, it becomes either $\P^1$ with a single singularity, or a sum of smooth $\P^1$’s meeting in specific ways. The types of the singular fibers of 