Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 24F / c / Solution

Let $\zeta =e^{2\pi i/7}$ and defineEach term in the defining polynomial is multiplied by ${\zeta }^3$:Thus $\varphi$ is a holomorphic automorphism of $X^{\prime }$, with inverse obtained by replacing $\zeta$ by ${\zeta }^{-1}$. By the assumption it extends to a holomorphic automorphism of $X$; in the projective model it isIt has order exactly seven because $\zeta$ is primitive and $y$ is not identically zero. Hence $\mathrm{Aut}(X)$ contains an element of order seven.

The extended coordinate function ${\pi }_x:X\to {\mathbb{C}}_{\infty }$ is a meromorphic function as a holomorphic map to the Riemann sphere. Compose it with the seventh-power map of the sphere and setThis map is nonconstant and holomorphic as a map to ${\mathbb{C}}_{\infty }$. Since ${\pi }_x\circ \varphi ={\zeta }^3{\pi }_x$,