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The units in the structure sheaf of a scheme form a multiplicative sheaf of abelian groups. On a smooth algebraic curve, a nonzero rational function is a unit at a point precisely when its discrete valuation there is zero. Its inclusion in the sheaf of nonzero rational functions on an irreducible variety relates divisor on an algebraic curve to the Picard group through the long exact sequence in sheaf cohomology.

Let $E$ be a smooth plane cubic with an inflection point $O$ as identity. A point $P$ with $3(P-O)=0$ in Picard group has a line whose intersection divisor with $E$ is $3P$. Indeed, ${\mathcal{O}}_E(3P)\cong {\mathcal{O}}_E(3O)\cong {\mathcal{O}}_E(1)$. On this genus-one curve, the Riemann-Roch theorem gives $h^0({\mathcal{O}}_E(1))=3$: the dual term has negative degree. Restriction of ambient linear forms is injective and its source also has dimension three, so it is an isomorphism. The section vanishing at $3P$ therefore comes from a line. Conversely, such a line gives $3P\sim 3O$, so $P$ is a three-torsion point. This includes $O$; points of exact order three are the other such points.