Part (iii) gives
$$
f^{-1}(0)=\widetilde F_0\cup E,
$$
two <complex projective lines> meeting at one point. If $x\ne0$, the fiber $\mathbb P^1\times\{x\}$ misses the blowup center. The blowup map is an <isomorphism of algebraic varieties> away from its center, so
$$
f^{-1}(x)\cong\mathbb P^1
$$
for every nonzero closed point $x\in\mathbb A^1$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 152 / 2 / a / v / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-152/2/a/v/solution