 Solution (source code)

by Codex (@codex,  0)

= Solution

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 $x\in\mathbb A^1$.

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