# Normal bundle of exceptional locus of the conifold

Let us consider the conifold singularity $xy-zw=0$ in $\mathbb{C}^4$. By blowing up along the divisor defined by $x=z=0$, we have a small resolution of the conifold with $\mathbb{P}^1$ as the exceptional locus. Here is my question. How can one see that the normal bundle of this $\mathbb{P}^1$ is given by $\mathcal{O}(-1)^{\oplus2}$?

Let us next consider the generalized conifold singularity $xy-(z-w^k)(z+w^k)=0$ in $\mathbb{C}^4$. Again, by blowing up along the divisor defined by $x=z-w^k=0$, we have a small resolution of the generalized conifold with $\mathbb{P}^1$ as the exceptional locus. How can one see that the normal bundle of this $\mathbb{P}^1$ is given by $\mathcal{O}\oplus\mathcal{O}(-2)$?

It would be nice if one can show these normal bundles with explicit coordinates. Thank you in advance.

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First let us describe the resolution explicitly. Let $A = k[x,y,z,w]/(xy-(z-w^k)(z+w^k))$. Note that the ideal $I = (x,z-w^k)$ has a resolution of the form $$A\oplus A \to A\oplus A \to I \to 0,$$ where the map is given by the matrix $\left(\begin{smallmatrix} x & z+w^k \\ z-w^k & y \end{smallmatrix}\right)$. It follows that the blowup $X$ of the ideal is embedded into $Spec(A)\times P^1 \subset A^4\times P^1$ with coordinates $(x,y,z,w)$ on $A^4$ and homogeneous coordinated $(u,v)$ on $P^1$ and is given their by the system of equations $$xu + (z+w^k)v = (z-w^k)u + yv = xy - (z-w^k)(z+w^k) = 0.$$ These equations are sections of the bundle $O(1) \oplus O(1) \oplus O$. Therefore the normal bundle to $P^1$ is the kernel of the map $$O^4 \to O(1) \oplus O(1) \oplus O$$ given by the matrix given by partial derivatives of the equations with respect to $x,y,z,w$ at $x = y = z = w = 0$, that is by $$\begin{pmatrix} u & 0 & v & 0 \\ 0 & v & u & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ if $k > 1$, and by $$\begin{pmatrix} u & 0 & v & v \\ 0 & v & u & -v \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ if $k = 1$. It follows easily that the kernel is $O \oplus O(-2)$ in the first case (because of zeroes in the last column) and $O(-1) \oplus O(-1)$ in the second.