Consider the rational curve (conic) given by image of the map


$$ u([z,w])=[z^2,-z^2,w^2,-w^2,zw] \in \mathbb{P}^4 $$


which lies in quintic 3-fold $X: x\_1^5+\cdots+x\_5^5- x\_1\cdots x\_5=0$.


By Grothendick theorem and the fact that $X$ is Calabi-Yau, we know that $u^\*N\_C^X= O(a) \oplus O(b)$, for some $a+b=-2$, where $N\_C^X$ is the normal bundle of $C=imgae(u)$ in $X$.


How should I calculate $a,b$ for this (or any other) explicitly given map?



Calculating the decomposition of a vector bundle over rational curve
Let $(z,w) \mapsto (f\_1(z,w),\dots,f\_5(z,w)$, $\deg f\_i = s$, be a map $P^1 \to P^4$ and $g(x\_1,\dots,x\_5)$, $\deg g = d$, be an equation of a hypersurface containing the image. Then the normal bundle is the middle cohomology of the following complex
$$
O(1)^2 \to O(s)^5 \to O(ds)
$$
where the first map is given by the matrix $(\partial f\_i/\partial z,\partial f\_i/\partial w)$, and the second map is given by the matrix $(\partial g/\partial x\_i)(f\_1,\dots,f\_5)$.
In the case of a conic in a quintic the complex takes form
$$
O(1)^2 \to O(2)^5 \to O(10),
$$
so the normal bundle should be $O(-1+a) \oplus O(-1-a)$. Then it is clear that $a = \dim H^0(N) = \dim H^1(N)$, you should use the above complex (with explicit differentials) to compute $H^0(N)$. Passing to the cohomology of the above complex you obtain
$$
k^4 \to k^{15} \to k^{11}.
$$
Then $H^1(N)$ is the cokernel of the second map. So, you just have to compute the rank of this map. My computation shows that in your specific case the map is surjective, hence $a = 0$. So, the final answer is $N = O(-1)^2$.


