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A balanced smooth rational curve in a calabi-Yau X is a smooth rational curve whose normal bundle is $O(-1)\oplus O(-1)$.

We usually like these curves because of their rigidity.

But, Is there any theorem that guaranty the existence of at least one such curve. For example for Quintic? Or for any other example?

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As far as I know, there is still no a priori argument known which would prove that a given projective simply-connected Calabi-Yau threefold X contains a rational curve (let alone a balanced one). – Balazs Jun 11 '10 at 14:50
up vote 4 down vote accepted

Perhaps you already know this: but we don't even know how to show that there are finitely many rational curves of a given degree $d$ on the general quintic threefold. This was originally conjectured by Clemens. However, for low degrees (up to $d = 11$ or something close to that), the conjecture is verified in the "strong form": any smooth rational curve of low degree (again, at most $11$) on the general quintic has normal bundle $O(-1) \oplus O(-1)$. As far as I know, that's the current state of affairs. It can often be difficult to produce rational curves with the "expected" normal bundle in any given situation!

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so based on what you said, we are aware of existence of such curves. where can I found the proof of the claim you mentioned above.(that low degree curves are all balanced ) – Mohammad F. Tehrani Jun 8 '10 at 20:43
yes, for low degrees we know existence. Here is a paper by E. Cotterill proving it for d = 11, you should be able to find references for lower degrees in the bibliography. – mdeland Jun 8 '10 at 21:17

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