The two standard <affine scheme>[affine] charts of the blowup restrict to an affine open cover of $X_0$. On the $x$-chart write $y=xu$. Since $xy=x^2u$, the fiber is
$$
U_x\cap X_0=\operatorname{Spec}k[x,u]/(x^2u).
$$
On the $y$-chart write $x=yv$, giving
$$
U_y\cap X_0=\operatorname{Spec}k[y,v]/(y^2v).
$$
On their overlap, $u$ and $v$ are invertible with $v=u^{-1}$ and $y=xu$. The squared factors $x^2$ and $y^2$ display the doubled exceptional component explicitly.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 152 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-152/3/c/solution