I just realize that even though I know what normal bundes are, I dont know how to compute them. The main objective is to show that a ration curve C on a quintic threefold doesnt move. If C is a line, then its normal bundle in the ambient space is $\mathcal O^{\oplus 3}(1)$. If we know that C is rigid on the quintic threefold, then its normal bundle, being a subundle of $\mathcal O^{\oplus 3}(1)$, must be $\mathcal O^{\oplus 2}(-1)$. But how do we prove this? What about higher degree rational curves?
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$\begingroup$ There must be a misprint when you write $\mathcal{O}^{\oplus 3}(1)$: how can the normal bundle of a curve inside a threefold have rank 3 ? $\endgroup$– QfwfqJan 6, 2011 at 16:05
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$\begingroup$ There it is meant: the normal bundle $N_{C/X}$ of $C$ in the quintic threefold $X$ is a subbundle of the normal bundle $N_{C/\mathbb{P}^4}$ of $C$ in $\mathbb{P}^4$, and $N_{C/\mathbb{P}^4}=\mathcal{O}(1)^{\oplus3}$ $\endgroup$– domenico fiorenzaJan 6, 2011 at 16:21
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2 Answers
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``Rigid'' can mean either that $C$ does not move (i.e., defines an isolated, but maybe non-reduced, point on the Hilbert scheme of lines on the smooth quintic $Q$) or that it defines an isolated reduced point. Moreover, it is possible for a line $C$ to move on $Q$; e.g., if there is a hyperplane section of $Q$ that is a cone, then the generators of the cone move on $Q$. Even if $C$ can be contracted to an isolated singularity (so certainly does not move), then its normal bundle can be $\mathcal O\oplus\mathcal O(-2)$ or $\mathcal O(1)\oplus\mathcal O(-3)$.
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A detailed treatment can be found in
Sheldon Katz, On the finiteness of rational curves on quintic threefolds, Compositio Math 60, 151-162 (1986) available as archive.numdam.org/article/CM_1986__60_2_151_0.pdf
answered Jan 6, 2011 at 12:39