Small resolution of a variety Suppose $Y$ is a hypersurface in $\mathbb{A}_{\mathbb{Q}}^4$ defined by 
\begin{equation}
X\,W-Y\,Z=0
\end{equation}
where $X,Y,Z$ and $W$ are the coordinates of $\mathbb{A}_{\mathbb{Q}}^4$. The singular locus of $Y$ is just a double point at origin and it admits a small resolution in a simple way: take a product $\mathbb{A}_{\mathbb{Q}}^4 \times \mathbb{P}^1_{\mathbb{Q}}$, then the variety $\hat{Y}$ is the variety defined by
\begin{equation}
X \,t_0+Y\,t_1=0 \\
Z \,t_0+W \,t_1=0
\end{equation} 
The natural projection $\mathbb{A}_{\mathbb{Q}}^4 \times \mathbb{P}^1_{\mathbb{Q}}\rightarrow \mathbb{A}_{\mathbb{Q}}^4$ defines a birational morphism
\begin{equation}
\pi: \hat{Y} \rightarrow Y
\end{equation}
The exceptional curve is just $\mathbb{P}^1_{\mathbb{Q}}$. If now $Z$ is the hypersurface in $\mathbb{A}_{\mathbb{Q}}^4 $ defined by
\begin{equation}
X^2+Y^2+Z^2+W^2=0
\end{equation}
What is the small resolution $\hat{Z}$ of $Z$? Is the exceptional curve now $\mathbb{P}^1_{\mathbb{Q}}$ or a conic (with no rational point)
If extend the field to $\mathbb{Q}(i)$, the hypersurfaces $Y$ and $Z$ are isomorphic, I guess this isomorphism extends to an isomorphism between $\hat{Y}$ and $\hat{Z}$ (over $\mathbb{Q}(i)$), what happens on the exceptional curve under this isomorphism?
 A: Indeed, the exceptional curve of $\hat{Z}$ is a conic with no rational points. This can be seen as follows. The blowup of the exceptional curve on a small resolution is isomorphic to the blowup $\tilde{Z}$ of $Z$ at the origin. Thus, to understand $\hat{Z}$ one needs to find a contraction of $\tilde{Z}$. Note that the exceptional divisor of $\tilde{Z}$ is isomorphic to the quadric $Q$
$$
x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0
$$
in $\mathbb{P}^3$. The contraction of $\tilde{Z}$ contracts $Q$ onto the exceptional curve of $\hat{Z}$, its fibers form a family of lines on $Q$. Thus the exceptional curve of $\hat{Z}$ is the connected component of the Hilbert scheme of lines on $Q$. It remains to note that this Hilbert scheme indeed has two components defined over $\mathbb{Q}$, one of them is defined by the equations
$$
x_{12} + x_{34} = x_{13} - x_{24} = x_{14} + x_{23} = x_{12}^2 + x_{13}^2 + x_{14}^2 = 0,
$$
and the other by the equations
$$
x_{12} - x_{34} = x_{13} + x_{24} = x_{14} - x_{23} = x_{12}^2 + x_{13}^2 + x_{14}^2 = 0
$$
inside the Plucker space $\mathbb{P}^5$. Both conics have no rational points.
