For $N \geq 1$, let $V_{N}$ be the (irreducible) projective plane curve $V_{N}: X^{N}+Y^{N}+Z^{N}=0$ over an algebraically closed field of characteristic zero.

Show that $V_{N}$ is smooth (non-singular). For $m, n \geq 1$, let $\alpha_{m, n}: V_{m n} \rightarrow V_{m}$ be the morphism $\alpha_{m, n}(X: Y: Z)=\left(X^{n}: Y^{n}: Z^{n}\right)$. Determine the degree of $\alpha_{m, n}$, its points of ramification and the corresponding ramification indices.

Applying the Riemann-Hurwitz formula to $\alpha_{1, n}$, determine the genus of $V_{n}$.