over $\P^1$, to build elliptically fibered Calabi–Yau 3-folds with a global section. We study F-theory compactifications on the resulting elliptic Calabi–Yau 3-folds[^5]. We find that highly enhanced gauge groups arise in these compactifications. It is mainly local F-theory model buildings that have been discussed in recent studies [@DWmodel; @BHV; @BHV2; @DWGUT]. However, the global aspects of the geometry need to be considered to discuss the issues of gravity. We investigate F-theory on elliptically fibered Calabi–Yau 3-folds from the global perspective in this study.

A similar organization can be found in [@Kimura1810].

This note is structured as follows. In Section \[sec2\], we briefly review F-theory compactifications, and we also review attractive K3 surfaces and extremal K3 surfaces that are technically necessary to analyze special points in the moduli of eight-dimensional non-geometric heterotic strings and F-theory duals. We also review the construction of non-geometric heterotic strings with unbroken $\mathfrak{e}_7\mathfrak{e}_7$ algebra in [@CMSE7].

In Section \[sec3\], we discuss the special points in the eight-dimensional non-geometric heterotic moduli with unbroken $\mathfrak{e}_7\mathfrak{e}_7$ at which the ranks of the non-Abelian gauge symmetries on the F-theory side are enhanced to 18. The gauge groups in the heterotic strings which correspond to these points do not allow for 