USp(4)-models

We study the geometry of elliptic fibrations satisfying the conditions of Step 2 of Tate's algorithm with a discriminant of valuation 4. We call such geometries USp(4)-models, as the dual graph of their special fiber is the twisted affine Dynkin diagram of type C$_2$. These geometries are used in string theory to model gauge theories with the non-simply-laced Lie group USp(4) on a smooth divisor S of the base. Starting with a singular Weierstrass model of a USp(4)-model, we present a crepant resolution of its singularities. We study the fiber structure of this smooth elliptic fibration and identify the fibral divisors up to isomorphism as schemes over S. These are P1-bundles over S or double covers of P1-bundles over S. We compute basic topological invariants such as the triple intersections of the fibral divisors and the Euler characteristic of the USp(4)-model. In the case of Calabi-Yau threefolds, we also compute the Hodge numbers. We study the compactfications of M/F theory on a USp(4)-model Calabi-Yau threefold.