the perturbative interpretations. We demonstrate that these theories can be seen as deformations of the stable degenerations as a result of the coincident 7-branes on the F-theory side. We also discuss applications to $SO(32)$ heterotic strings. We derive the Weierstrass equations of K3 elliptic fibrations appearing as the compactification spaces of the F-theory duals of some $SO(32)$ heterotic strings. We determine the gauge groups that arise on F-theory compactifications, including the global structures of the gauge groups.

We build elliptically fibered Calabi–Yau 3-folds in Section \[sec4\] by fibering examples of elliptic K3 surfaces, which belong to the F-theory side of the moduli of eight-dimensional unbroken $\mathfrak{e}_7\mathfrak{e}_7$ non-geometric heterotic strings, over $\P^1$. We analyze F-theory compactifications on the resulting elliptic Calabi–Yau 3-folds. First, we consider the higher-dimensional analog of the construction of genus-one fibered K3 surfaces without a global section[^6] to build genus-one fibered Calabi–Yau 3-folds without a section. This construction ensures that the resulting 3-folds in fact satisfy the Calabi–Yau condition. Similar constructions of genus-one fibered Calabi–Yau 4-folds without a section using double covers can be found in [@KCY4]. Taking the Jacobian fibration[^7] of the resulting genus-one fibered Calabi–Yau 3-folds yields elliptically fibered Calabi–Yau 3-folds with a global section. K3 