  Local aspects of singular F-theory compactifications for SUSY GUT
model-building are fairly well understood in terms of Higgs bundles and their
spectral data. Several global issues remain, however, including a description
of G-fluxes, which are key to constructing chiral matter and stabilizing
moduli, and the global realization of U(1) symmetries that can forbid
phenomenologically unfavorable couplings. In this paper, we sharpen our earlier
proposal for describing G-fluxes through "spectral divisors" and introduce a
distinguished "Tate divisor", which can be used to describe both G-flux and
U(1)s when present. As an application, we give a general discussion of
M5-instanton contributions in the presence of G-flux and exemplify this in a
concrete example, where we comment on the ability of instanton induced
superpotential couplings to stabilize Kahler moduli.
