= Solution
Choose a nonconstant rational function $t$ on $C$. Functions in $L(D)$ are integral over a finite-dimensional bounded-pole space over $k(t)$; equivalently, evaluation of sufficiently many principal parts embeds $L(D)$ into a finite-dimensional vector space. Hence $L(D)$ is finite dimensional.
If $E=D+\operatorname{div}(h)$, multiplication by $h$ gives the isomorphism $L(E)\to L(D)$. A <canonical divisor> is the divisor of a nonzero rational differential. <Riemann-Roch theorem> says $\ell(D)-\ell(K-D)=\deg D+1-g$. Taking $D=0$ gives $1-g=1+g-\deg K$, hence $\deg K=2g-2$.
For a smooth plane curve of degree $d$, the adjunction formula gives $K_C\sim(d-3)H|_C$. A cubic has $d=3$, so $K_C\sim0$.
Back to article page
