Flops and Fibral Geometry of E$_7$-models

An E$_7$-Weierstrass model is conjectured to have eight distinct crepant resolutions whose flop diagram is a Dynkin diagram of type E$_8$. In previous work, we explicitly constructed four distinct resolutions, for which the flop diagram formed a D$_4$ sub-diagram. The goal of this paper is to explore those properties of a resolved E$_7$-model which are not invariant under flops. In particular, we examine the fiber degenerations, identify the fibral divisors up to isomorphism, and study violation of flatness appearing over certain codimension-three loci in the base, where a component of the fiber grows in dimension from a rational curve to a rational surface. For each crepant resolution, we compute the triple intersection polynomial and the linear form induced by the second Chern class, as well as the holomorphic and ordinary Euler characteristics, and the signature of each fibral divisor. We identify the isomorphism classes of the rational surfaces that break the flatness of the fibration. Moreover, we explicitly show that the D$_4$ flops correspond to the crepant resolutions of the orbifold given by $\mathbb{C}^3$ quotiented by the Klein four-group.