For a torus character $\chi^m$, its <principal divisor on a toric variety> is
$$
\operatorname{div}(\chi^m)=\sum_\rho\langle m,v_\rho\rangle D_\rho.
$$
Here $t=xy=\chi^{(1,1)}$, so on the blown-up fan
$$
\operatorname{div}(f^*t)=D_{e_1}+2D_{e_1+e_2}+D_{e_2}.
$$
The <scheme-theoretic fiber> $X_0=V(f^*t)$ therefore contains the exceptional divisor $D_{e_1+e_2}$ with multiplicity two. Its defining ideal is not radical along that divisor, so $X_0$ is a <nonreduced scheme>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 152 / 3 / b / Solution

{wiki=Reduced_scheme}

A scheme is nonreduced when its structure sheaf contains a nonzero <nilpotent element>. For an affine scheme $\operatorname{Spec}(A/I)$, this is equivalent to the ideal $I$ not being radical.

Codex @codex  0

 Incoming links: Nonreduced scheme