  Several dimensionful parameters needed for model building can be engineered
in a certain class of SU(5) F-theory GUTs by adding extra singlet fields which
are localized along pairwise intersections of D7-branes. The values of these
parameters, however, depend on dynamics external to the GUT which causes the
singlets to acquire suitable masses or expectation values. In this note, we
demonstrate that D3-instantons which wrap the same 4-cycle as one of the
intersecting D7's can provide precisely the needed dynamics to generate several
important scales, including the supersymmetry-breaking scale and the
right-handed neutrino mass. Furthermore, these instantons seem unable to
directly generate the \mu term suggesting that, at least in this class of
models, it should perhaps be tied to one of the other scales in the problem.
More specifically, we study the simple system consisting of a pair of D7-branes
wrapping del Pezzo surfaces which intersect along a curve $\Sigma$ of genus 0
or 1 and classify all instanton configurations which can potentially contribute
to the superpotential. This allows one to formulate topological conditions
which must be imposed on \Sigma for various model-building applications. Along
the way, we also observe that the construction of arXiv:0808.1286 which
engineers a linear superpotential in fact realizes an O'Raifeartaigh model at
the KK scale whose 1-loop Coleman-Weinberg potential generically leads to a
metastable, long-lived SUSY-breaking vacuum.
