Let $C$ be a nonsingular projective curve, and $D$ a divisor on $C$ of degree $d$.

(i) State the Riemann-Roch theorem for $D$, giving a brief explanation of each term. Deduce that if $d>2 g-2$ then $\ell(D)=1-g+d$.

(ii) Show that, for every $P \in C$,

$$\ell(D-P) \geqslant \ell(D)-1$$

Deduce that $\ell(D) \leqslant 1+d$. Show also that if $\ell(D)>1$, then $\ell(D-P)=\ell(D)-1$ for all but finitely many $P \in C$.

(iii) Deduce that for every $d \geqslant g-1$ there exists a divisor $D$ of degree $d$ with $\ell(D)=1-g+d$.