Explain what is meant by a meromorphic differential on a compact connected Riemann surface $S$. Show that if $f$ is a meromorphic function on $S$ then $d f$ defines a meromorphic differential on $S$. Show also that if $\eta$ and $\omega$ are two meromorphic differentials on $S$ which are not identically zero then $\eta=h \omega$ for some meromorphic function $h$. Show that zeros and poles of a meromorphic differential are well-defined and explain, without proof, how to obtain the genus of $S$ by counting zeros and poles of $\omega$.

Let $V_{0} \subset \mathbb{C}^{2}$ be the affine curve with equation $u^{2}=v^{2}+1$ and let $V \subset \mathbb{P}^{2}$ be the corresponding projective curve. Show that $V$ is non-singular with two points at infinity, and that $d v$ extends to a meromorphic differential on $V$.

[You may assume without proof that that the map

$$(u, v)=\left(\frac{t^{2}+1}{t^{2}-1}, \frac{2 t}{t^{2}-1}\right), \quad t \in \mathbb{C} \backslash\{-1,1\},$$

is onto $V_{0} \backslash\{(1,0)\}$ and extends to a biholomorphic map from $\mathbb{P}^{1}$ onto $V$.]