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For supersymmetric heterotic string compactifications on a Calabi-Yauthreefold $X$ endowed with a vector bundle $V$ the world-sheet superpotential$W$ is a sum of contributions from isolated rational curves $\C$ in $X$; theindividual contribution is given by an exponential in the K\"ahler class of thecurve times a prefactor given essentially by the Pfaffian which depends on themoduli of $V$ and the complex structure moduli of $X$. Solutions of $DW=0$ (oreven of $DW=W=0$) can arise either by nontrivial cancellations between theindividual terms in the summation over all contributing curves or because eachof these terms is zero already individually. Concerning the latter caseconditions on the moduli making a single Pfaffian vanish (for special modulivalues) have been investigated. However, even if corresponding moduli -fulfilling these constraints - for the individual contribution of one curve areknown it is not at all clear whether {\em one} choice of moduli exists whichfulfills the corresponding constraints {\em for all contributing curvessimultaneously}. Clearly this will in general happen only if the conditions onthe 'individual zeroes' had already a conceptual origin which allows them tofit together consistently. We show that this happens for a class of cases. Inthe special case of spectral cover bundles we show that a relevant solution sethas an interesting location in moduli space and is related to transitions whichchange the generation number.

On the Heterotic World-sheet Instanton Superpotential and its individual Contributions