  We present models of heterotic compactification on Calabi-Yau threefolds and
compute the non-perturbative superpotential for vector bundle moduli. The key
feature of these models is that the threefolds, which are elliptically fibered
over del Pezzo surfaces, have homology classes with a unique holomorphic,
isolated genus-zero curve. Using the spectral cover construction, we present
vector bundles for which we can explicitly calculate the Pfaffians associated
with string instantons on these curves. These are shown to be non-zero, thus
leading to a non-vanishing superpotential in the 4D effective action. We
discuss, in detail, why such compactifications avoid the Beasley-Witten residue
theorem.
