In Vakil's "The Rising Sea", remark 14.2.3 says that the sheaf $O_X(D)$ for divisor $D$ comes along with a canonical rational section, namely the section corresponding to function $1$ defined in the complement of support of $D$. My question is what if the scheme under consideration has only finitely many points and hence possibly the complement of the support is empty, e.g., we can consider a DVR.
Here our schemes are taken to be integral and normal.
A divisor $D$ is defined to be a formal sum of codimension-one subvarieties. No codimension-one subvariety contains the generic point of an integral scheme, so it cannot be the case that the complement of the support of $D$ is empty.