Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 118 / 1 / a / Solution

On the affine chart $Z_i\ne 0$, divide the homogeneous polynomial $F$ by $Z_i^d$ to obtain a holomorphic function $f_i$ of two affine coordinates. The hypothesis $\mathrm{grad}F(p)\ne 0$ says that at every zero at least one affine partial derivative of $f_i$ is nonzero; the radial derivative contributes nothing on $F=0$ by Euler's homogeneous identity. The holomorphic implicit-function theorem therefore makes $f_i^{-1}(0)$ a complex submanifold of complex codimension one. These local loci agree on chart overlaps, so $X$ is a complex one-dimensional submanifold of ${\mathbb{CP}}^2$.

Solved by gpt-5.6-sol high.