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The fan $\{0,{\mathbb{R}}_{\ge 0},{\mathbb{R}}_{\le 0}\}$ defines the complex projective line ${\mathbb{P}}^1$.

Before blowing up, the fiber over zero is the smooth curveand the blowup center $p$ lies on it. Its total transform is the union of the strict transform ${\tilde{F}}_0$ and the exceptional divisor $E$. Blowing up a smooth point of a smooth curve does not change the curve, so ${\tilde{F}}_0\cong {\mathbb{P}}^1$, while $E\cong {\mathbb{P}}^1$. They meet transversely in the single point recording the tangent direction of $F_0$ at $p$. Thereforeis two copies of the complex projective line meeting at one point.

Part (iii) givestwo complex projective lines meeting at one point. If $x\ne 0$, the fiber ${\mathbb{P}}^1\times \{x\}$ misses the blowup center. The blowup map is an isomorphism of algebraic varieties away from its center, sofor every nonzero closed point $x\in {\mathbb{A}}^1$.

Writing $\Phi =({\varphi }_1,{\varphi }_2{)}^T$, the scalar sector has a global $SU(2)$ flavour symmetry in addition to the gauged common $U(1)$ phase. A vacuum can be chosen as ${\Phi }_0=(0,v{)}^T$. A diagonal combination of the gauge phase and the $SU(2)$ rotation generated by ${\sigma }_3$ leaves it invariant.

The radial scalar has $m_h^2=2\lambda v^2$, the gauge boson has $m_A^2=2e^2{v}^2$, and two real scalars remain massless. The four real scalar directions consist of one radial mode and three angular modes; one angular mode is eaten, leaving the two Goldstone bosons expected from the two broken physical global generators. The vacuum set before quotienting is $S^3$, and the manifold of gauge-inequivalent vacua is the complex projective line