A toric morphism is induced by a lattice map carrying each source cone into a target cone. Compose a hypothetical fan map from the fan of $\mathbb P^2$ to the product fan with either coordinate projection to the fan of $\mathbb P^1$. If the three primitive source rays are $r_1,r_2,r_3$ with $r_1+r_2+r_3=0$, their scalar images must have each adjacent pair in one half-line. This is impossible for three numbers summing to zero unless all are zero. Both coordinate projections of the lattice map vanish, so the map itself is zero and the toric morphism is constant. This proves <no nonconstant toric morphism from the projective plane to the product of projective lines>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 152 / 2 / b / Solution

Every fan morphism from the fan of $\mathbb P^2$ to the fan of $\mathbb P^1$ is zero: the three source ray images sum to zero, while each adjacent pair must lie in one half-line. Composing a fan morphism to the product fan of $\mathbb P^1\times\mathbb P^1$ with both projections therefore makes it zero.


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