  We study the geometry of gauge fluxes in four-dimensional F-theory vacua with
gauge group SU(3)xSU(2)xU(1)xU(1) and its implications for phenomenology. The
models are defined by a previously introduced class of elliptic fibrations
whose fibre is given as a cubic hypersurface in ${\rm Bl}_2{\mathbb P}^2$, with
the non-abelian gauge group factors SU(3)xSU(2) engineered torically via the
top construction. To describe gauge fluxes on these fibrations we provide a
classification of the primary vertical middle cohomology group in a fashion
valid for any choice of base space. Using the ideal theoretic technique of
primary decomposition we compute the cohomology classes of the matter surfaces
associated with states charged under the non-abelian gauge group. These
expressions allow us to interpret the cancellation of the pure and mixed
non-abelian anomalies geometrically as a result of the general form of the
matter surfaces, without reference to a specific type of gauge flux. Explicit
results for the chiral indices of all matter states are obtained in terms of
intersection numbers of the base and can be directly applied to any choice of
base consistent with the fibration. As a demonstration we scan for globally
consistent F-theory vacua on $\mathbb P^3$, ${\rm Bl}_1\mathbb P^3$ and ${\rm
Bl}_2 \mathbb{P}^3$, and find a globally consistent flux configuration with the
chiral Standard Model spectrum plus an extra triplet pair, which may be lifted
by a recombination process.
