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Insert the primitive rayinside the singular cone. The corresponding star subdivision replaces $⟨v_0,v_1⟩$ by $⟨v_0,v_3⟩$ and $⟨v_3,v_1⟩$. Both new determinants have absolute value one, as do the two unchanged cones, so the subdivided fan is smooth. The induced proper birational toric morphism $\tilde{X}\to X$ is therefore a toric resolution of singularities. Its exceptional invariant curve has self-intersection $-2$, and $\tilde{X}$ is the Hirzebruch surface ${\mathbb{F}}_2$.

Solved by gpt-5.6-sol high.

A star subdivision inserts a ray through a lattice point in a cone and subdivides every cone containing that point. It induces a proper birational toric morphism.