Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 114 / 3 / c

Properness implies $f(v)\ne 0$ for $v\ne 0$: otherwise the whole noncompact complex line $\mathbb{C}v$ would lie in $f^{-1}(0)$. Homogeneity therefore definesOn the fiber over $[v]$, defineThe identity $f(\lambda v)=\lambda f(v)$ makes this independent of the representative $v$, and it is a nonzero complex-linear map on every fiber. It is therefore the bundle isomorphism