Canonical divisor of the projective line
= Canonical divisor of the projective line
{title2=$K_{\mathbb P^1}$}
For the affine coordinate $t$ on $\mathbb P^1$, put $s=t^{-1}$ at infinity. Since $dt=-s^{-2}ds$,
$$
(dt)=-2\infty,
$$
so $K_{\mathbb P^1}\sim-2\infty$. More generally,
$$
\ell(d\infty)=\max(d+1,0),
$$
because $L(d\infty)$ consists of polynomials of degree at most $d$ when $d\geq0$ and only the zero function when $d<0$.