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

For a torus character ${\chi }^m$, its principal divisor on a toric variety isHere $t=xy={\chi }^{(1,1)}$, so on the blown-up fanThe scheme-theoretic fiber $X_0=V(f^{\ast }t)$ 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.