  We investigate aspects of certain stringy invariants of singular elliptic
fibrations which arise in engineering Grand Unified Theories in F-theory. In
particular, we exploit the small resolutions of the total space of these
fibrations provided recently in the physics literature to compute `stringy
characteristic classes', and find that numerical invariants obtained by
integrating such characteristic classes are predetermined by the topology of
the base of the elliptic fibration. Moreover, we derive a simple (dimension
independent) formula for pushing forward powers of the exceptional divisor of a
blowup, which one may use to reduce any integral (in the sense of Chow
cohomology) on a small resolution of a singular elliptic fibration to an
integral on the base. We conclude with a speculatory note on the cohomology of
small resolutions of GUT vacua, where we conjecture that certain simple
formulas for their Hodge numbers may be given solely in terms of the first
Chern class and Hodge numbers of the base.}
