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?



Calculating normal bundle
``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 $\mathcal O(1)\oplus\mathcal O(-3)$.


