TITLE: Calculating the decomposition of a vector bundle over rational curve
QUESTION [10 upvotes]: 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?

REPLY [4 votes]: Since you know the explicit equation of the conic, you can compute everything by using Macaulay2.
The following script should be clear:
i1 : k=ZZ/32003;

i2 : ringP1=k[x, y];

i3 : ringP4=k[z1, z2, z3, z4, z5];

i4 : I= ideal(z1^5+z2^5+z3^5+z4^5+z5^5-z1*z2*z3*z4*z5);

o4 : Ideal of ringP4

i5 : ringQuintic=ringP4/I;

i6 : conicMap=map(ringP1, ringQuintic, {x^2, -x^2, y^2, -y^2, x*y});

o6 : RingMap ringP1 <--- ringQuintic

i7 : conic=image conicMap;

i8 : IC=ideal conic;

o8 : Ideal of ringQuintic

i9 : ConormalModuleConic = IC/IC^2;

i10 : ConormalSheafConic= sheaf ConormalModuleConic;

i11 : NormalSheafConic= dual sheaf ConormalModuleConic;

i12 : HH^0(ConormalSheafConic)

       4
o12 = k

o12 : k-module, free

i13 : HH^1(ConormalSheafConic)

o13 = 0

o13 : k-module

i14 : HH^0(NormalSheafConic)

o14 = 0

o14 : k-module

i15 : HH^1(NormalSheafConic)

o15 = 0

o15 : k-module

The output reads
$h^0(X, N_{C|X}^{*})=4, \quad h^1(X, N_{C|X}^{*})=0, \quad h^0(X, N_{C|X})=0, \quad h^1(X, N_{C|X})=0$,
hence  $N_{C|X}=\mathcal{O}_{\mathbb{P}^1}(-1) \oplus \mathcal{O}_{\mathbb{P}^1}(-1)$, according to Sasha's and Sandor's answers.