The Geometry of the SU(2)$\times$ G$_2$-model

We study elliptic fibrations that geometrically engineer an SU(2)$\times$ G$_2$ gauge theory realized by Weierstrass model for the collision III+$\text{I}_0^{*\text{ns}}$. We construct the four possible crepant resolutions of such a Weierstrass model and show that they form a chain of four minimal models connected by flops. We compute the generating function for the Euler characteristic of these crepant resolutions. In the case of a Calabi-Yau threefold, we consider the compactification of M-theory and F-theory on an SU(2)$\times$ G$_2$-model to a five and six-dimensional supergravity with eight supercharges. By matching each crepant resolution with each Coulomb chamber of the five-dimensional theory, we determine the number of multiplets and compute the prepotential in each Coulomb chamber. In particular, we discuss counting number of hypermultiplets in presence of singularities. We discuss in detail the cancellation of anomalies of the six-dimensional theory.