  Box graphs succinctly and comprehensively characterize singular fibers of
elliptic fibrations in codimension two and three, as well as flop transitions
connecting these, in terms of representation theoretic data. We develop a
framework that provides a systematic map between a box graph and a crepant
algebraic resolution of the singular elliptic fibration, thus allowing an
explicit construction of the fibers from a singular Weierstrass or Tate model.
The key tool is what we call a fiber face diagram, which shows the relevant
information of a (partial) toric triangulation and allows the inclusion of more
general algebraic blowups. We shown that each such diagram defines a sequence
of weighted algebraic blowups, thus providing a realization of the fiber
defined by the box graph in terms of an explicit resolution. We show this
correspondence explicitly for the case of SU(5) by providing a map between box
graphs and fiber faces, and thereby a sequence of algebraic resolutions of the
Tate model, which realizes each of the box graphs.
