Let $X$ be a smooth projective curve of genus 2, defined over the complex numbers. Show that there is a morphism $f: X \rightarrow \mathbf{P}^{1}$ which is a double cover, ramified at six points.

Explain briefly why $X$ cannot be embedded into $\mathbf{P}^{2}$.

For any positive integer $n$, show that there is a smooth affine plane curve which is a double cover of $\mathbf{A}^{1}$ ramified at $n$ points.

[State clearly any theorems that you use.]