Let $V$ be a smooth projective curve, and let $D$ be an effective divisor on $V$. Explain how $D$ defines a morphism $\phi_{D}$ from $V$ to some projective space. State the necessary and sufficient conditions for $\phi_{D}$ to be finite. State the necessary and sufficient conditions for $\phi_{D}$ to be an isomorphism onto its image.

Let $V$ have genus 2 , and let $K$ be an effective canonical divisor. Show that the morphism $\phi_{K}$ is a morphism of degree 2 from $V$ to $\mathbb{P}^{1}$.

By considering the divisor $K+P_{1}+P_{2}$ for points $P_{i}$ with $P_{1}+P_{2} \nsim K$, show that there exists a birational morphism from $V$ to a singular plane quartic.

[You may assume the Riemann-Roch Theorem.]