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Jan 24, 2015 at 15:49 vote accept Brenin
Jan 21, 2015 at 20:36 comment added user47305 Whoops, I see you want a Calabi-Yau threefold. Problem (2) can occur verbatim in that case, but (1) can't since it involves blowing stuff up. However, I think you can get basically the same issue on a CY3 using other constructions: there can be a finite set of homologous rational curves, all with normal bundle $\mathcal O(-1) + \mathcal O(-1)$.
Jan 21, 2015 at 20:06 history answered user47305 CC BY-SA 3.0