  We study Euclidean D3-branes wrapping divisors $D$ in Calabi-Yau orientifold
compactifications of type IIB string theory. Witten's counting of fermion zero
modes in terms of the cohomology of the structure sheaf $\mathcal{O}_D$ applies
when $D$ is smooth, but we argue that effective divisors of Calabi-Yau
threefolds typically have singularities along rational curves. We generalize
the counting of fermion zero modes to such singular divisors, in terms of the
cohomology of the structure sheaf $\mathcal{O}_{\overline{D}}$ of the
normalization $\overline{D}$ of $D$. We establish this by detailing
compactifications in which the singularities can be unwound by passing through
flop transitions, giving a physical incarnation of the normalization process.
Analytically continuing the superpotential through the flops, we find that
singular divisors whose normalizations are rigid can contribute to the
superpotential: specifically,
$h^{\bullet}_{+}(\mathcal{O}_{\overline{D}})=(1,0,0)$ and
$h^{\bullet}_{-}(\mathcal{O}_{\overline{D}})=(0,0,0)$ give a sufficient
condition for a contribution. The examples that we present feature infinitely
many isomorphic geometric phases, with corresponding infinite-order monodromy
groups $\Gamma$. We use the action of $\Gamma$ on effective divisors to
determine the exact effective cones, which have infinitely many generators. The
resulting nonperturbative superpotentials are Jacobi theta functions, whose
modular symmetries suggest the existence of strong-weak coupling dualities
involving inversion of divisor volumes.
