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

The product fan for ${\mathbb{P}}^1\times {\mathbb{A}}^1$ has rays generated byand maximal cones $\mathrm{Cone}(e_1,e_2)$ and $\mathrm{Cone}(e_2,-e_1)$. The given fan is obtained by the star subdivision of the first maximal cone throughBy the toric blowup at a torus-fixed point, this subdivision blows up the torus-fixed point corresponding to $\mathrm{Cone}(e_1,e_2)$. Hencefor that torus-fixed point $p$.

The fan of the affine line consists of the zero cone and ${\mathbb{R}}_{\ge 0}{e}^{\prime }$. Under $\phi (e_1)=0$ and $\phi (e_2)=e^{\prime }$, the four primitive source rays map asEvery cone of the source fan consequently maps into ${\mathbb{R}}_{\ge 0}{e}^{\prime }$. The defining criterion for a toric morphism is satisfied, so $\phi$ inducesUnder the blowup description from part (i), this is the blowup map followed by the projection ${\mathbb{P}}^1\times {\mathbb{A}}^1\to {\mathbb{A}}^1$.

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.

The inverse image of the dense torus ${\mathbb{A}}^1\setminus \{0\}={\mathbb{G}}_m$ is represented by the subfan formed from cones mapping to the zero cone of the target fan. These areThis is the fan of the projective line in the $e_1$-axis, together with the zero fan in the complementary $e_2$-direction. By the product fan construction,It also follows geometrically because the blowup center lies over zero, so the blowup is unchanged over ${\mathbb{A}}^1\setminus \{0\}$.

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 $x\in {\mathbb{A}}^1$.