(i) Explain how a linear system on a curve $C$ may induce a morphism from $C$ to projective space. What condition on the linear system is necessary to yield a morphism $f: C \rightarrow \mathbb{P}^{n}$ such that the pull-back of a hyperplane section is an element of the linear system? What condition is necessary to imply the morphism is an embedding?

(ii) State the Riemann-Roch theorem for curves.

(iii) Show that any divisor of degree 5 on a curve $C$ of genus 2 induces an embedding.