Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 118 / 4 / c

The projective linear group acts transitively on ${\mathbb{CP}}^2$. A projective automorphism carrying $p$ to another point $q$ lifts, by the universal construction of the blowup of a complex manifold at a point, to a biholomorphism between the two blowups. Thus the biholomorphism type is independent of the center.

If $D_1\sim D_2$, then $D_1-D_2$ is the divisor of a meromorphic function $f$. Pulling back givesso the total inverse-image divisors are linearly equivalent on $\tilde{X}$. Here ${\pi }^{-1}(D_i)$ must mean the total transform; strict transforms need not be linearly equivalent if their multiplicities at the blown-up point differ.

Solved by gpt-5.6-sol high.