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

The two standard affine charts of the blowup restrict to an affine open cover of $X_0$. On the $x$-chart write $y=xu$. Since $xy=x^{2}u$, the fiber isOn the $y$-chart write $x=yv$, givingOn 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.